Metamath Proof Explorer

Theorem psdmul

Description: Product rule for power series. An outline is available at https://github.com/icecream17/Stuff/blob/main/math/psdmul.pdf . (Contributed by SN, 25-Apr-2025)

Ref Expression
Hypotheses psdmul.s ⊢ S = I mPwSer R
psdmul.b ⊢ B = Base S
psdmul.p ⊢ + ˙ = + S
psdmul.m ⊢ · ˙ = ⋅ S
psdmul.i ⊢ φ → I ∈ V
psdmul.r ⊢ φ → R ∈ CRing
psdmul.x ⊢ φ → X ∈ I
psdmul.f ⊢ φ → F ∈ B
psdmul.g ⊢ φ → G ∈ B
Assertion psdmul ⊢ φ → I mPSDer R ⁡ X ⁡ F · ˙ G = I mPSDer R ⁡ X ⁡ F · ˙ G + ˙ F · ˙ I mPSDer R ⁡ X ⁡ G

Proof

Step Hyp Ref Expression
1 psdmul.s ⊢ S = I mPwSer R
2 psdmul.b ⊢ B = Base S
3 psdmul.p ⊢ + ˙ = + S
4 psdmul.m ⊢ · ˙ = ⋅ S
5 psdmul.i ⊢ φ → I ∈ V
6 psdmul.r ⊢ φ → R ∈ CRing
7 psdmul.x ⊢ φ → X ∈ I
8 psdmul.f ⊢ φ → F ∈ B
9 psdmul.g ⊢ φ → G ∈ B
10 eqid ⊢ Base R = Base R
11 eqid ⊢ + R = + R
12 6 crngringd ⊢ φ → R ∈ Ring
13 12 ringcmnd ⊢ φ → R ∈ CMnd
14 13 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → R ∈ CMnd
15 simpr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
16 eqid ⊢ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin = h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
17 16 psrbagsn ⊢ I ∈ V → y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
18 5 17 syl ⊢ φ → y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
19 18 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
20 16 psrbagaddcl ⊢ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d + f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
21 15 19 20 syl2anc ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d + f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
22 16 psrbaglefi ⊢ d + f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∈ Fin
23 21 22 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∈ Fin
24 eqid ⊢ ⋅ R = ⋅ R
25 6 crnggrpd ⊢ φ → R ∈ Grp
26 25 grpmndd ⊢ φ → R ∈ Mnd
27 26 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → R ∈ Mnd
28 16 psrbagf ⊢ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d : I ⟶ ℕ 0
29 28 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d : I ⟶ ℕ 0
30 7 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → X ∈ I
31 29 30 ffvelcdmd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ⁡ X ∈ ℕ 0
32 peano2nn0 ⊢ d ⁡ X ∈ ℕ 0 → d ⁡ X + 1 ∈ ℕ 0
33 31 32 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ⁡ X + 1 ∈ ℕ 0
34 33 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → d ⁡ X + 1 ∈ ℕ 0
35 eqid ⊢ ⋅ R = ⋅ R
36 12 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → R ∈ Ring
37 1 10 16 2 8 psrelbas ⊢ φ → F : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
38 37 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → F : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
39 elrabi ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
40 39 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
41 38 40 ffvelcdmd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → F ⁡ u ∈ Base R
42 1 10 16 2 9 psrelbas ⊢ φ → G : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
43 42 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → G : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
44 eqid ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 = k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0
45 16 44 psrbagconcl ⊢ d + f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0
46 21 45 sylan ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0
47 elrabi ⊢ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
48 46 47 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
49 43 48 ffvelcdmd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
50 10 35 36 41 49 ringcld ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
51 10 24 27 34 50 mulgnn0cld ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
52 disjdifr ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∩ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d = ∅
53 52 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∩ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d = ∅
54 1nn0 ⊢ 1 ∈ ℕ 0
55 0nn0 ⊢ 0 ∈ ℕ 0
56 54 55 ifcli ⊢ if i = X 1 0 ∈ ℕ 0
57 56 nn0ge0i ⊢ 0 ≤ if i = X 1 0
58 29 ffvelcdmda ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → d ⁡ i ∈ ℕ 0
59 58 nn0red ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → d ⁡ i ∈ ℝ
60 56 nn0rei ⊢ if i = X 1 0 ∈ ℝ
61 60 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → if i = X 1 0 ∈ ℝ
62 59 61 addge01d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → 0 ≤ if i = X 1 0 ↔ d ⁡ i ≤ d ⁡ i + if i = X 1 0
63 57 62 mpbii ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → d ⁡ i ≤ d ⁡ i + if i = X 1 0
64 63 ralrimiva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∀ i ∈ I d ⁡ i ≤ d ⁡ i + if i = X 1 0
65 29 ffnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d Fn I
66 54 55 ifcli ⊢ if y = X 1 0 ∈ ℕ 0
67 66 elexi ⊢ if y = X 1 0 ∈ V
68 eqid ⊢ y ∈ I ⟼ if y = X 1 0 = y ∈ I ⟼ if y = X 1 0
69 67 68 fnmpti ⊢ y ∈ I ⟼ if y = X 1 0 Fn I
70 69 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → y ∈ I ⟼ if y = X 1 0 Fn I
71 5 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → I ∈ V
72 inidm ⊢ I ∩ I = I
73 65 70 71 71 72 offn ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d + f y ∈ I ⟼ if y = X 1 0 Fn I
74 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → d ⁡ i = d ⁡ i
75 eqeq1 ⊢ y = i → y = X ↔ i = X
76 75 ifbid ⊢ y = i → if y = X 1 0 = if i = X 1 0
77 56 elexi ⊢ if i = X 1 0 ∈ V
78 76 68 77 fvmpt ⊢ i ∈ I → y ∈ I ⟼ if y = X 1 0 ⁡ i = if i = X 1 0
79 78 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → y ∈ I ⟼ if y = X 1 0 ⁡ i = if i = X 1 0
80 65 70 71 71 72 74 79 ofval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → d + f y ∈ I ⟼ if y = X 1 0 ⁡ i = d ⁡ i + if i = X 1 0
81 65 73 71 71 72 74 80 ofrfval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ≤ f d + f y ∈ I ⟼ if y = X 1 0 ↔ ∀ i ∈ I d ⁡ i ≤ d ⁡ i + if i = X 1 0
82 64 81 mpbird ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ≤ f d + f y ∈ I ⟼ if y = X 1 0
83 82 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ≤ f d + f y ∈ I ⟼ if y = X 1 0
84 5 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → I ∈ V
85 16 psrbagf ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k : I ⟶ ℕ 0
86 85 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k : I ⟶ ℕ 0
87 29 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d : I ⟶ ℕ 0
88 16 psrbagf ⊢ d + f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d + f y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0
89 21 88 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d + f y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0
90 89 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d + f y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0
91 nn0re ⊢ q ∈ ℕ 0 → q ∈ ℝ
92 nn0re ⊢ r ∈ ℕ 0 → r ∈ ℝ
93 nn0re ⊢ s ∈ ℕ 0 → s ∈ ℝ
94 letr ⊢ q ∈ ℝ ∧ r ∈ ℝ ∧ s ∈ ℝ → q ≤ r ∧ r ≤ s → q ≤ s
95 91 92 93 94 syl3an ⊢ q ∈ ℕ 0 ∧ r ∈ ℕ 0 ∧ s ∈ ℕ 0 → q ≤ r ∧ r ≤ s → q ≤ s
96 95 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ q ∈ ℕ 0 ∧ r ∈ ℕ 0 ∧ s ∈ ℕ 0 → q ≤ r ∧ r ≤ s → q ≤ s
97 84 86 87 90 96 caoftrn ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ≤ f d ∧ d ≤ f d + f y ∈ I ⟼ if y = X 1 0 → k ≤ f d + f y ∈ I ⟼ if y = X 1 0
98 83 97 mpan2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ≤ f d → k ≤ f d + f y ∈ I ⟼ if y = X 1 0
99 98 ss2rabdv ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⊆ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0
100 undifr ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⊆ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ↔ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∪ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d = k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0
101 99 100 sylib ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∪ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d = k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0
102 101 eqcomd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 = k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∪ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
103 10 11 14 23 51 53 102 gsummptfidmsplit ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
104 eqid ⊢ 0 R = 0 R
105 ovex ⊢ ℕ 0 I ∈ V
106 105 rabex ⊢ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∈ V
107 106 rabex ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∈ V
108 107 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∈ V
109 ovex ⊢ F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ V
110 eqid ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ⟼ F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ⟼ F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
111 109 110 fnmpti ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ⟼ F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u Fn k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0
112 111 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ⟼ F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u Fn k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0
113 fvexd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → 0 R ∈ V
114 112 23 113 fndmfifsupp ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → finSupp 0 R⁡ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ⟼ F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
115 10 104 24 108 50 114 14 33 gsummulg ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = d ⁡ X + 1 ⋅ R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
116 difrab ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d = k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d
117 116 eleq2i ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ↔ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d
118 breq1 ⊢ k = u → k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ↔ u ≤ f d + f y ∈ I ⟼ if y = X 1 0
119 breq1 ⊢ k = u → k ≤ f d ↔ u ≤ f d
120 119 notbid ⊢ k = u → ¬ k ≤ f d ↔ ¬ u ≤ f d
121 118 120 anbi12d ⊢ k = u → k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ↔ u ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ u ≤ f d
122 121 elrab ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ↔ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ u ≤ f d
123 16 psrbagf ⊢ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u : I ⟶ ℕ 0
124 123 ffnd ⊢ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u Fn I
125 124 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u Fn I
126 73 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d + f y ∈ I ⟼ if y = X 1 0 Fn I
127 5 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → I ∈ V
128 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → u ⁡ i = u ⁡ i
129 65 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d Fn I
130 66 a1i ⊢ y ∈ I → if y = X 1 0 ∈ ℕ 0
131 68 130 fmpti ⊢ y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0
132 131 a1i ⊢ φ → y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0
133 132 ffnd ⊢ φ → y ∈ I ⟼ if y = X 1 0 Fn I
134 133 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → y ∈ I ⟼ if y = X 1 0 Fn I
135 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → d ⁡ i = d ⁡ i
136 78 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → y ∈ I ⟼ if y = X 1 0 ⁡ i = if i = X 1 0
137 129 134 127 127 72 135 136 ofval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → d + f y ∈ I ⟼ if y = X 1 0 ⁡ i = d ⁡ i + if i = X 1 0
138 125 126 127 127 72 128 137 ofrfval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ≤ f d + f y ∈ I ⟼ if y = X 1 0 ↔ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0
139 125 129 127 127 72 128 135 ofrfval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ≤ f d ↔ ∀ i ∈ I u ⁡ i ≤ d ⁡ i
140 139 notbid ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ¬ u ≤ f d ↔ ¬ ∀ i ∈ I u ⁡ i ≤ d ⁡ i
141 rexnal ⊢ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i ↔ ¬ ∀ i ∈ I u ⁡ i ≤ d ⁡ i
142 140 141 bitr4di ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ¬ u ≤ f d ↔ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i
143 138 142 anbi12d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ u ≤ f d ↔ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i
144 31 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → d ⁡ X ∈ ℕ 0
145 123 adantl ⊢ φ ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u : I ⟶ ℕ 0
146 7 adantr ⊢ φ ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → X ∈ I
147 145 146 ffvelcdmd ⊢ φ ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ⁡ X ∈ ℕ 0
148 147 adantlr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ⁡ X ∈ ℕ 0
149 148 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → u ⁡ X ∈ ℕ 0
150 nn0nlt0 ⊢ d ⁡ X ∈ ℕ 0 → ¬ d ⁡ X < 0
151 144 150 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → ¬ d ⁡ X < 0
152 29 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d : I ⟶ ℕ 0
153 152 ffvelcdmda ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → d ⁡ i ∈ ℕ 0
154 153 nn0cnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → d ⁡ i ∈ ℂ
155 154 addridd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → d ⁡ i + 0 = d ⁡ i
156 155 breq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → u ⁡ i ≤ d ⁡ i + 0 ↔ u ⁡ i ≤ d ⁡ i
157 156 biimpd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → u ⁡ i ≤ d ⁡ i + 0 → u ⁡ i ≤ d ⁡ i
158 ifnefalse ⊢ i ≠ X → if i = X 1 0 = 0
159 158 oveq2d ⊢ i ≠ X → d ⁡ i + if i = X 1 0 = d ⁡ i + 0
160 159 breq2d ⊢ i ≠ X → u ⁡ i ≤ d ⁡ i + if i = X 1 0 ↔ u ⁡ i ≤ d ⁡ i + 0
161 160 imbi1d ⊢ i ≠ X → u ⁡ i ≤ d ⁡ i + if i = X 1 0 → u ⁡ i ≤ d ⁡ i ↔ u ⁡ i ≤ d ⁡ i + 0 → u ⁡ i ≤ d ⁡ i
162 157 161 syl5ibrcom ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → i ≠ X → u ⁡ i ≤ d ⁡ i + if i = X 1 0 → u ⁡ i ≤ d ⁡ i
163 162 imp ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I ∧ i ≠ X → u ⁡ i ≤ d ⁡ i + if i = X 1 0 → u ⁡ i ≤ d ⁡ i
164 163 impancom ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I ∧ u ⁡ i ≤ d ⁡ i + if i = X 1 0 → i ≠ X → u ⁡ i ≤ d ⁡ i
165 164 necon1bd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I ∧ u ⁡ i ≤ d ⁡ i + if i = X 1 0 → ¬ u ⁡ i ≤ d ⁡ i → i = X
166 165 ancrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I ∧ u ⁡ i ≤ d ⁡ i + if i = X 1 0 → ¬ u ⁡ i ≤ d ⁡ i → i = X ∧ ¬ u ⁡ i ≤ d ⁡ i
167 166 ex ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → u ⁡ i ≤ d ⁡ i + if i = X 1 0 → ¬ u ⁡ i ≤ d ⁡ i → i = X ∧ ¬ u ⁡ i ≤ d ⁡ i
168 167 ralimdva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 → ∀ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → i = X ∧ ¬ u ⁡ i ≤ d ⁡ i
169 168 anim1d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → ∀ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → i = X ∧ ¬ u ⁡ i ≤ d ⁡ i ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i
170 169 imp ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → ∀ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → i = X ∧ ¬ u ⁡ i ≤ d ⁡ i ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i
171 rexim ⊢ ∀ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → i = X ∧ ¬ u ⁡ i ≤ d ⁡ i → ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → ∃ i ∈ I i = X ∧ ¬ u ⁡ i ≤ d ⁡ i
172 171 imp ⊢ ∀ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → i = X ∧ ¬ u ⁡ i ≤ d ⁡ i ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → ∃ i ∈ I i = X ∧ ¬ u ⁡ i ≤ d ⁡ i
173 fveq2 ⊢ i = X → u ⁡ i = u ⁡ X
174 fveq2 ⊢ i = X → d ⁡ i = d ⁡ X
175 173 174 breq12d ⊢ i = X → u ⁡ i ≤ d ⁡ i ↔ u ⁡ X ≤ d ⁡ X
176 175 notbid ⊢ i = X → ¬ u ⁡ i ≤ d ⁡ i ↔ ¬ u ⁡ X ≤ d ⁡ X
177 176 ceqsrexbv ⊢ ∃ i ∈ I i = X ∧ ¬ u ⁡ i ≤ d ⁡ i ↔ X ∈ I ∧ ¬ u ⁡ X ≤ d ⁡ X
178 177 simprbi ⊢ ∃ i ∈ I i = X ∧ ¬ u ⁡ i ≤ d ⁡ i → ¬ u ⁡ X ≤ d ⁡ X
179 172 178 syl ⊢ ∀ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → i = X ∧ ¬ u ⁡ i ≤ d ⁡ i ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → ¬ u ⁡ X ≤ d ⁡ X
180 31 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ⁡ X ∈ ℕ 0
181 180 nn0red ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ⁡ X ∈ ℝ
182 148 nn0red ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ⁡ X ∈ ℝ
183 181 182 ltnled ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ⁡ X < u ⁡ X ↔ ¬ u ⁡ X ≤ d ⁡ X
184 183 biimpar ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ¬ u ⁡ X ≤ d ⁡ X → d ⁡ X < u ⁡ X
185 179 184 sylan2 ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → i = X ∧ ¬ u ⁡ i ≤ d ⁡ i ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → d ⁡ X < u ⁡ X
186 170 185 syldan ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → d ⁡ X < u ⁡ X
187 breq2 ⊢ u ⁡ X = 0 → d ⁡ X < u ⁡ X ↔ d ⁡ X < 0
188 186 187 syl5ibcom ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → u ⁡ X = 0 → d ⁡ X < 0
189 151 188 mtod ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → ¬ u ⁡ X = 0
190 189 neqned ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → u ⁡ X ≠ 0
191 elnnne0 ⊢ u ⁡ X ∈ ℕ ↔ u ⁡ X ∈ ℕ 0 ∧ u ⁡ X ≠ 0
192 149 190 191 sylanbrc ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → u ⁡ X ∈ ℕ
193 elfzo0 ⊢ d ⁡ X ∈ 0 ..^ u ⁡ X ↔ d ⁡ X ∈ ℕ 0 ∧ u ⁡ X ∈ ℕ ∧ d ⁡ X < u ⁡ X
194 144 192 186 193 syl3anbrc ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → d ⁡ X ∈ 0 ..^ u ⁡ X
195 fzostep1 ⊢ d ⁡ X ∈ 0 ..^ u ⁡ X → d ⁡ X + 1 ∈ 0 ..^ u ⁡ X ∨ d ⁡ X + 1 = u ⁡ X
196 194 195 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → d ⁡ X + 1 ∈ 0 ..^ u ⁡ X ∨ d ⁡ X + 1 = u ⁡ X
197 149 nn0red ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → u ⁡ X ∈ ℝ
198 33 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → d ⁡ X + 1 ∈ ℕ 0
199 198 nn0red ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → d ⁡ X + 1 ∈ ℝ
200 7 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → X ∈ I
201 iftrue ⊢ i = X → if i = X 1 0 = 1
202 174 201 oveq12d ⊢ i = X → d ⁡ i + if i = X 1 0 = d ⁡ X + 1
203 173 202 breq12d ⊢ i = X → u ⁡ i ≤ d ⁡ i + if i = X 1 0 ↔ u ⁡ X ≤ d ⁡ X + 1
204 203 rspcv ⊢ X ∈ I → ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 → u ⁡ X ≤ d ⁡ X + 1
205 200 204 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 → u ⁡ X ≤ d ⁡ X + 1
206 205 imp ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 → u ⁡ X ≤ d ⁡ X + 1
207 206 adantrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → u ⁡ X ≤ d ⁡ X + 1
208 197 199 207 lensymd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → ¬ d ⁡ X + 1 < u ⁡ X
209 208 intn3an3d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → ¬ d ⁡ X + 1 ∈ ℕ 0 ∧ u ⁡ X ∈ ℕ ∧ d ⁡ X + 1 < u ⁡ X
210 elfzo0 ⊢ d ⁡ X + 1 ∈ 0 ..^ u ⁡ X ↔ d ⁡ X + 1 ∈ ℕ 0 ∧ u ⁡ X ∈ ℕ ∧ d ⁡ X + 1 < u ⁡ X
211 209 210 sylnibr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → ¬ d ⁡ X + 1 ∈ 0 ..^ u ⁡ X
212 196 211 orcnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → d ⁡ X + 1 = u ⁡ X
213 143 212 sylbida ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ u ≤ f d → d ⁡ X + 1 = u ⁡ X
214 213 anasss ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ u ≤ f d → d ⁡ X + 1 = u ⁡ X
215 122 214 sylan2b ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d → d ⁡ X + 1 = u ⁡ X
216 117 215 sylan2b ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d ⁡ X + 1 = u ⁡ X
217 216 oveq1d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
218 217 mpteq2dva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
219 218 oveq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
220 16 psrbaglefi ⊢ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∈ Fin
221 220 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∈ Fin
222 26 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → R ∈ Mnd
223 33 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d ⁡ X + 1 ∈ ℕ 0
224 12 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → R ∈ Ring
225 elrabi ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
226 37 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → F : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
227 226 ffvelcdmda ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → F ⁡ u ∈ Base R
228 225 227 sylan2 ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → F ⁡ u ∈ Base R
229 42 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → G : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
230 29 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d : I ⟶ ℕ 0
231 230 ffvelcdmda ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → d ⁡ i ∈ ℕ 0
232 231 nn0cnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → d ⁡ i ∈ ℂ
233 225 123 syl ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u : I ⟶ ℕ 0
234 233 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u : I ⟶ ℕ 0
235 234 ffvelcdmda ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → u ⁡ i ∈ ℕ 0
236 235 nn0cnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → u ⁡ i ∈ ℂ
237 56 nn0cni ⊢ if i = X 1 0 ∈ ℂ
238 237 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → if i = X 1 0 ∈ ℂ
239 232 236 238 subadd23d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → d ⁡ i - u ⁡ i + if i = X 1 0 = d ⁡ i + if i = X 1 0 - u ⁡ i
240 232 238 236 addsubassd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → d ⁡ i + if i = X 1 0 - u ⁡ i = d ⁡ i + if i = X 1 0 - u ⁡ i
241 239 240 eqtr4d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → d ⁡ i - u ⁡ i + if i = X 1 0 = d ⁡ i + if i = X 1 0 - u ⁡ i
242 241 mpteq2dva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → i ∈ I ⟼ d ⁡ i - u ⁡ i + if i = X 1 0 = i ∈ I ⟼ d ⁡ i + if i = X 1 0 - u ⁡ i
243 eqid ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d = k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
244 16 243 psrbagconcl ⊢ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
245 elrabi ⊢ d − f u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
246 244 245 syl ⊢ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
247 246 adantll ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
248 16 psrbagf ⊢ d − f u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d − f u : I ⟶ ℕ 0
249 247 248 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f u : I ⟶ ℕ 0
250 249 ffnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f u Fn I
251 69 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → y ∈ I ⟼ if y = X 1 0 Fn I
252 5 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → I ∈ V
253 230 ffnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d Fn I
254 234 ffnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u Fn I
255 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → d ⁡ i = d ⁡ i
256 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → u ⁡ i = u ⁡ i
257 253 254 252 252 72 255 256 ofval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → d − f u ⁡ i = d ⁡ i − u ⁡ i
258 78 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → y ∈ I ⟼ if y = X 1 0 ⁡ i = if i = X 1 0
259 250 251 252 252 72 257 258 offval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f u + f y ∈ I ⟼ if y = X 1 0 = i ∈ I ⟼ d ⁡ i - u ⁡ i + if i = X 1 0
260 simplr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
261 18 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
262 260 261 20 syl2anc ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d + f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
263 262 88 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d + f y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0
264 263 ffnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d + f y ∈ I ⟼ if y = X 1 0 Fn I
265 253 251 252 252 72 255 258 ofval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → d + f y ∈ I ⟼ if y = X 1 0 ⁡ i = d ⁡ i + if i = X 1 0
266 264 254 252 252 72 265 256 offval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d + f y ∈ I ⟼ if y = X 1 0 − f u = i ∈ I ⟼ d ⁡ i + if i = X 1 0 - u ⁡ i
267 242 259 266 3eqtr4d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f u + f y ∈ I ⟼ if y = X 1 0 = d + f y ∈ I ⟼ if y = X 1 0 − f u
268 16 psrbagaddcl ⊢ d − f u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d − f u + f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
269 247 261 268 syl2anc ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f u + f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
270 267 269 eqeltrrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
271 229 270 ffvelcdmd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
272 10 35 224 228 271 ringcld ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
273 10 24 222 223 272 mulgnn0cld ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
274 disjdifr ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ∩ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 = ∅
275 274 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ∩ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 = ∅
276 simpl ⊢ k ≤ f d ∧ k ⁡ X = 0 → k ≤ f d
277 276 a1i ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ≤ f d ∧ k ⁡ X = 0 → k ≤ f d
278 277 ss2rabi ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⊆ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
279 278 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⊆ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
280 undifr ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⊆ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ↔ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ∪ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 = k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
281 279 280 sylib ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ∪ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 = k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
282 281 eqcomd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d = k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ∪ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0
283 10 11 14 221 273 275 282 gsummptfidmsplit ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
284 eldifi ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
285 7 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → X ∈ I
286 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ X ∈ I → d ⁡ X = d ⁡ X
287 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ X ∈ I → u ⁡ X = u ⁡ X
288 253 254 252 252 72 286 287 ofval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ X ∈ I → d − f u ⁡ X = d ⁡ X − u ⁡ X
289 285 288 mpdan ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f u ⁡ X = d ⁡ X − u ⁡ X
290 284 289 sylan2 ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d − f u ⁡ X = d ⁡ X − u ⁡ X
291 290 oveq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u ⁡ X + d − f u ⁡ X = u ⁡ X + d ⁡ X - u ⁡ X
292 234 285 ffvelcdmd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u ⁡ X ∈ ℕ 0
293 284 292 sylan2 ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u ⁡ X ∈ ℕ 0
294 293 nn0cnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u ⁡ X ∈ ℂ
295 31 nn0cnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ⁡ X ∈ ℂ
296 295 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d ⁡ X ∈ ℂ
297 294 296 pncan3d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u ⁡ X + d ⁡ X - u ⁡ X = d ⁡ X
298 291 297 eqtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u ⁡ X + d − f u ⁡ X = d ⁡ X
299 298 oveq1d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u ⁡ X + d − f u ⁡ X + 1 = d ⁡ X + 1
300 249 285 ffvelcdmd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f u ⁡ X ∈ ℕ 0
301 284 300 sylan2 ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d − f u ⁡ X ∈ ℕ 0
302 301 nn0cnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d − f u ⁡ X ∈ ℂ
303 1cnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → 1 ∈ ℂ
304 294 302 303 addassd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u ⁡ X + d − f u ⁡ X + 1 = u ⁡ X + d − f u ⁡ X + 1
305 299 304 eqtr3d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d ⁡ X + 1 = u ⁡ X + d − f u ⁡ X + 1
306 305 oveq1d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = u ⁡ X + d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
307 26 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → R ∈ Mnd
308 peano2nn0 ⊢ d − f u ⁡ X ∈ ℕ 0 → d − f u ⁡ X + 1 ∈ ℕ 0
309 300 308 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f u ⁡ X + 1 ∈ ℕ 0
310 284 309 sylan2 ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d − f u ⁡ X + 1 ∈ ℕ 0
311 284 272 sylan2 ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
312 10 24 11 mulgnn0dir ⊢ R ∈ Mnd ∧ u ⁡ X ∈ ℕ 0 ∧ d − f u ⁡ X + 1 ∈ ℕ 0 ∧ F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R → u ⁡ X + d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
313 307 293 310 311 312 syl13anc ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u ⁡ X + d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
314 306 313 eqtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
315 314 mpteq2dva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
316 315 oveq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
317 difssd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⊆ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
318 221 317 ssfid ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ∈ Fin
319 10 24 222 292 272 mulgnn0cld ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
320 284 319 sylan2 ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
321 10 24 222 309 272 mulgnn0cld ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
322 284 321 sylan2 ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
323 eqid ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
324 eqid ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
325 10 11 14 318 320 322 323 324 gsummptfidmadd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
326 316 325 eqtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
327 7 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → X ∈ I
328 65 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d Fn I
329 elrabi ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
330 329 124 syl ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u Fn I
331 330 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u Fn I
332 5 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → I ∈ V
333 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ∧ X ∈ I → d ⁡ X = d ⁡ X
334 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ∧ X ∈ I → u ⁡ X = u ⁡ X
335 328 331 332 332 72 333 334 ofval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ∧ X ∈ I → d − f u ⁡ X = d ⁡ X − u ⁡ X
336 327 335 mpdan ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d − f u ⁡ X = d ⁡ X − u ⁡ X
337 fveq1 ⊢ k = u → k ⁡ X = u ⁡ X
338 337 eqeq1d ⊢ k = u → k ⁡ X = 0 ↔ u ⁡ X = 0
339 119 338 anbi12d ⊢ k = u → k ≤ f d ∧ k ⁡ X = 0 ↔ u ≤ f d ∧ u ⁡ X = 0
340 339 elrab ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ↔ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ≤ f d ∧ u ⁡ X = 0
341 340 simprbi ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u ≤ f d ∧ u ⁡ X = 0
342 341 simprd ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u ⁡ X = 0
343 342 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u ⁡ X = 0
344 343 oveq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d ⁡ X − u ⁡ X = d ⁡ X − 0
345 31 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d ⁡ X ∈ ℕ 0
346 345 nn0cnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d ⁡ X ∈ ℂ
347 346 subid1d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d ⁡ X − 0 = d ⁡ X
348 336 344 347 3eqtrrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d ⁡ X = d − f u ⁡ X
349 348 oveq1d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d ⁡ X + 1 = d − f u ⁡ X + 1
350 349 oveq1d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
351 350 mpteq2dva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
352 351 oveq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
353 326 352 oveq12d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
354 25 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → R ∈ Grp
355 106 rabex ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∈ V
356 355 difexi ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ∈ V
357 356 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ∈ V
358 320 fmpttd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u : k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟶ Base R
359 ovex ⊢ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ V
360 359 323 fnmpti ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u Fn k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0
361 360 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u Fn k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0
362 361 318 113 fndmfifsupp ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → finSupp 0 R⁡ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
363 10 104 14 357 358 362 gsumcl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
364 322 fmpttd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u : k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟶ Base R
365 ovex ⊢ d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ V
366 365 324 fnmpti ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u Fn k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0
367 366 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u Fn k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0
368 367 318 113 fndmfifsupp ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → finSupp 0 R⁡ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
369 10 104 14 357 364 368 gsumcl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
370 106 rabex ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ∈ V
371 370 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ∈ V
372 278 sseli ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
373 372 321 sylan2 ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
374 373 fmpttd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u : k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟶ Base R
375 eqid ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
376 365 375 fnmpti ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u Fn k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0
377 376 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u Fn k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0
378 221 279 ssfid ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ∈ Fin
379 377 378 113 fndmfifsupp ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → finSupp 0 R⁡ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⟼ d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
380 10 104 14 371 374 379 gsumcl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
381 10 11 354 363 369 380 grpassd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
382 283 353 381 3eqtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
383 219 382 oveq12d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d d ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
384 103 115 383 3eqtr3d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ⁡ X + 1 ⋅ R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
385 8 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → F ∈ B
386 9 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → G ∈ B
387 1 2 35 4 16 385 386 21 psrmulval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → F · ˙ G ⁡ d + f y ∈ I ⟼ if y = X 1 0 = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
388 387 oveq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ⁡ X + 1 ⋅ R F · ˙ G ⁡ d + f y ∈ I ⟼ if y = X 1 0 = d ⁡ X + 1 ⋅ R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
389 107 difexi ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∈ V
390 389 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∈ V
391 eldifi ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0
392 39 123 syl ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → u : I ⟶ ℕ 0
393 392 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → u : I ⟶ ℕ 0
394 7 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → X ∈ I
395 393 394 ffvelcdmd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → u ⁡ X ∈ ℕ 0
396 10 24 27 395 50 mulgnn0cld ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
397 391 396 sylan2 ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
398 397 fmpttd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u : k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟶ Base R
399 eqid ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
400 359 399 fnmpti ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u Fn k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
401 400 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u Fn k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
402 difssd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⊆ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0
403 23 402 ssfid ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∈ Fin
404 401 403 113 fndmfifsupp ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → finSupp 0 R⁡ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
405 10 104 14 390 398 404 gsumcl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
406 10 11 354 369 380 grpcld ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
407 10 11 354 405 363 406 grpassd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
408 384 388 407 3eqtr4d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ⁡ X + 1 ⋅ R F · ˙ G ⁡ d + f y ∈ I ⟼ if y = X 1 0 = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
409 408 mpteq2dva ⊢ φ → d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟼ d ⁡ X + 1 ⋅ R F · ˙ G ⁡ d + f y ∈ I ⟼ if y = X 1 0 = d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟼ ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
410 1 2 4 12 8 9 psrmulcl ⊢ φ → F · ˙ G ∈ B
411 1 2 16 5 6 7 410 psdval ⊢ φ → I mPSDer R ⁡ X ⁡ F · ˙ G = d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟼ d ⁡ X + 1 ⋅ R F · ˙ G ⁡ d + f y ∈ I ⟼ if y = X 1 0
412 25 grpmgmd ⊢ φ → R ∈ Mgm
413 1 2 5 412 7 8 psdcl ⊢ φ → I mPSDer R ⁡ X ⁡ F ∈ B
414 1 2 4 12 413 9 psrmulcl ⊢ φ → I mPSDer R ⁡ X ⁡ F · ˙ G ∈ B
415 1 2 5 412 7 9 psdcl ⊢ φ → I mPSDer R ⁡ X ⁡ G ∈ B
416 1 2 4 12 8 415 psrmulcl ⊢ φ → F · ˙ I mPSDer R ⁡ X ⁡ G ∈ B
417 1 2 11 3 414 416 psradd ⊢ φ → I mPSDer R ⁡ X ⁡ F · ˙ G + ˙ F · ˙ I mPSDer R ⁡ X ⁡ G = I mPSDer R ⁡ X ⁡ F · ˙ G + R f F · ˙ I mPSDer R ⁡ X ⁡ G
418 1 10 16 2 414 psrelbas ⊢ φ → I mPSDer R ⁡ X ⁡ F · ˙ G : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
419 418 ffnd ⊢ φ → I mPSDer R ⁡ X ⁡ F · ˙ G Fn h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
420 1 10 16 2 416 psrelbas ⊢ φ → F · ˙ I mPSDer R ⁡ X ⁡ G : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
421 420 ffnd ⊢ φ → F · ˙ I mPSDer R ⁡ X ⁡ G Fn h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
422 106 a1i ⊢ φ → h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∈ V
423 inidm ⊢ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∩ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin = h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
424 413 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → I mPSDer R ⁡ X ⁡ F ∈ B
425 1 2 35 4 16 424 386 15 psrmulval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → I mPSDer R ⁡ X ⁡ F · ˙ G ⁡ d = ∑ R b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b
426 355 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∈ V
427 12 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → R ∈ Ring
428 elrabi ⊢ b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → b ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
429 1 10 16 2 413 psrelbas ⊢ φ → I mPSDer R ⁡ X ⁡ F : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
430 429 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → I mPSDer R ⁡ X ⁡ F : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
431 430 ffvelcdmda ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ b ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → I mPSDer R ⁡ X ⁡ F ⁡ b ∈ Base R
432 428 431 sylan2 ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → I mPSDer R ⁡ X ⁡ F ⁡ b ∈ Base R
433 42 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → G : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
434 16 243 psrbagconcl ⊢ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
435 434 adantll ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
436 elrabi ⊢ d − f b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f b ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
437 435 436 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f b ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
438 433 437 ffvelcdmd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → G ⁡ d − f b ∈ Base R
439 10 35 427 432 438 ringcld ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b ∈ Base R
440 439 fmpttd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b : k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟶ Base R
441 ovex ⊢ I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b ∈ V
442 eqid ⊢ b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b = b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b
443 441 442 fnmpti ⊢ b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b Fn k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
444 443 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b Fn k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
445 444 221 113 fndmfifsupp ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → finSupp 0 R⁡ b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b
446 eqid ⊢ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⟼ u − f y ∈ I ⟼ if y = X 1 0 = u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⟼ u − f y ∈ I ⟼ if y = X 1 0
447 df-of ⊢ ∘ f ⁡ + = m ∈ V , n ∈ V ⟼ o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o
448 vex ⊢ u ∈ V
449 448 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ V
450 ssv ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⊆ V
451 450 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⊆ V
452 ssv ⊢ y ∈ I ⟼ if y = X 1 0 ⊆ V
453 452 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → y ∈ I ⟼ if y = X 1 0 ⊆ V
454 447 449 451 453 elimampo ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ↔ ∃ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∃ n ∈ y ∈ I ⟼ if y = X 1 0 u = o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o
455 454 biimpa ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → ∃ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∃ n ∈ y ∈ I ⟼ if y = X 1 0 u = o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o
456 elrabi ⊢ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
457 16 psrbagf ⊢ m ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → m : I ⟶ ℕ 0
458 457 ffund ⊢ m ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → Fun ⁡ m
459 456 458 syl ⊢ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → Fun ⁡ m
460 459 funfnd ⊢ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m Fn dom ⁡ m
461 460 ad2antrl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → m Fn dom ⁡ m
462 velsn ⊢ n ∈ y ∈ I ⟼ if y = X 1 0 ↔ n = y ∈ I ⟼ if y = X 1 0
463 funmpt ⊢ Fun ⁡ y ∈ I ⟼ if y = X 1 0
464 funeq ⊢ n = y ∈ I ⟼ if y = X 1 0 → Fun ⁡ n ↔ Fun ⁡ y ∈ I ⟼ if y = X 1 0
465 463 464 mpbiri ⊢ n = y ∈ I ⟼ if y = X 1 0 → Fun ⁡ n
466 465 funfnd ⊢ n = y ∈ I ⟼ if y = X 1 0 → n Fn dom ⁡ n
467 462 466 sylbi ⊢ n ∈ y ∈ I ⟼ if y = X 1 0 → n Fn dom ⁡ n
468 467 ad2antll ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → n Fn dom ⁡ n
469 vex ⊢ m ∈ V
470 469 dmex ⊢ dom ⁡ m ∈ V
471 470 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → dom ⁡ m ∈ V
472 vex ⊢ n ∈ V
473 472 dmex ⊢ dom ⁡ n ∈ V
474 473 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → dom ⁡ n ∈ V
475 eqid ⊢ dom ⁡ m ∩ dom ⁡ n = dom ⁡ m ∩ dom ⁡ n
476 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 ∧ o ∈ dom ⁡ m → m ⁡ o = m ⁡ o
477 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 ∧ o ∈ dom ⁡ n → n ⁡ o = n ⁡ o
478 461 468 471 474 475 476 477 offval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → m + f n = o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o
479 478 eqeq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → u = m + f n ↔ u = o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o
480 elsni ⊢ n ∈ y ∈ I ⟼ if y = X 1 0 → n = y ∈ I ⟼ if y = X 1 0
481 480 oveq2d ⊢ n ∈ y ∈ I ⟼ if y = X 1 0 → m + f n = m + f y ∈ I ⟼ if y = X 1 0
482 481 eqeq2d ⊢ n ∈ y ∈ I ⟼ if y = X 1 0 → u = m + f n ↔ u = m + f y ∈ I ⟼ if y = X 1 0
483 482 ad2antll ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → u = m + f n ↔ u = m + f y ∈ I ⟼ if y = X 1 0
484 5 ad3antrrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → I ∈ V
485 456 457 syl ⊢ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m : I ⟶ ℕ 0
486 485 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m : I ⟶ ℕ 0
487 131 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0
488 nn0cn ⊢ q ∈ ℕ 0 → q ∈ ℂ
489 nn0cn ⊢ r ∈ ℕ 0 → r ∈ ℂ
490 nn0cn ⊢ s ∈ ℕ 0 → s ∈ ℂ
491 addsubass ⊢ q ∈ ℂ ∧ r ∈ ℂ ∧ s ∈ ℂ → q + r - s = q + r - s
492 488 489 490 491 syl3an ⊢ q ∈ ℕ 0 ∧ r ∈ ℕ 0 ∧ s ∈ ℕ 0 → q + r - s = q + r - s
493 492 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ q ∈ ℕ 0 ∧ r ∈ ℕ 0 ∧ s ∈ ℕ 0 → q + r - s = q + r - s
494 484 486 487 487 493 caofass ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0 = m + f y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0
495 simpr ⊢ φ ∧ i ∈ I → i ∈ I
496 56 a1i ⊢ φ ∧ i ∈ I → if i = X 1 0 ∈ ℕ 0
497 68 76 495 496 fvmptd3 ⊢ φ ∧ i ∈ I → y ∈ I ⟼ if y = X 1 0 ⁡ i = if i = X 1 0
498 133 133 5 5 72 497 497 offval ⊢ φ → y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0 = i ∈ I ⟼ if i = X 1 0 − if i = X 1 0
499 498 oveq2d ⊢ φ → m + f y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0 = m + f i ∈ I ⟼ if i = X 1 0 − if i = X 1 0
500 499 ad3antrrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0 = m + f i ∈ I ⟼ if i = X 1 0 − if i = X 1 0
501 237 subidi ⊢ if i = X 1 0 − if i = X 1 0 = 0
502 501 mpteq2i ⊢ i ∈ I ⟼ if i = X 1 0 − if i = X 1 0 = i ∈ I ⟼ 0
503 fconstmpt ⊢ I × 0 = i ∈ I ⟼ 0
504 502 503 eqtr4i ⊢ i ∈ I ⟼ if i = X 1 0 − if i = X 1 0 = I × 0
505 504 oveq2i ⊢ m + f i ∈ I ⟼ if i = X 1 0 − if i = X 1 0 = m + f I × 0
506 0zd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → 0 ∈ ℤ
507 488 addridd ⊢ q ∈ ℕ 0 → q + 0 = q
508 507 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ q ∈ ℕ 0 → q + 0 = q
509 484 486 506 508 caofid0r ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f I × 0 = m
510 505 509 eqtrid ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f i ∈ I ⟼ if i = X 1 0 − if i = X 1 0 = m
511 494 500 510 3eqtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0 = m
512 simpr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
513 511 512 eqeltrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
514 oveq1 ⊢ u = m + f y ∈ I ⟼ if y = X 1 0 → u − f y ∈ I ⟼ if y = X 1 0 = m + f y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0
515 514 eleq1d ⊢ u = m + f y ∈ I ⟼ if y = X 1 0 → u − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ↔ m + f y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
516 513 515 syl5ibrcom ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u = m + f y ∈ I ⟼ if y = X 1 0 → u − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
517 516 adantrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → u = m + f y ∈ I ⟼ if y = X 1 0 → u − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
518 483 517 sylbid ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → u = m + f n → u − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
519 479 518 sylbird ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → u = o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o → u − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
520 519 rexlimdvva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → ∃ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∃ n ∈ y ∈ I ⟼ if y = X 1 0 u = o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o → u − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
521 455 520 mpd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
522 simpr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
523 5 mptexd ⊢ φ → y ∈ I ⟼ if y = X 1 0 ∈ V
524 elsng ⊢ y ∈ I ⟼ if y = X 1 0 ∈ V → y ∈ I ⟼ if y = X 1 0 ∈ y ∈ I ⟼ if y = X 1 0 ↔ y ∈ I ⟼ if y = X 1 0 = y ∈ I ⟼ if y = X 1 0
525 523 524 syl ⊢ φ → y ∈ I ⟼ if y = X 1 0 ∈ y ∈ I ⟼ if y = X 1 0 ↔ y ∈ I ⟼ if y = X 1 0 = y ∈ I ⟼ if y = X 1 0
526 68 525 mpbiri ⊢ φ → y ∈ I ⟼ if y = X 1 0 ∈ y ∈ I ⟼ if y = X 1 0
527 526 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → y ∈ I ⟼ if y = X 1 0 ∈ y ∈ I ⟼ if y = X 1 0
528 447 mpofun ⊢ Fun ⁡ ∘ f ⁡ +
529 528 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → Fun ⁡ ∘ f ⁡ +
530 xpss ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⊆ V × V
531 470 inex1 ⊢ dom ⁡ m ∩ dom ⁡ n ∈ V
532 531 mptex ⊢ o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o ∈ V
533 532 rgen2w ⊢ ∀ m ∈ V ∀ n ∈ V o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o ∈ V
534 447 dmmpoga ⊢ ∀ m ∈ V ∀ n ∈ V o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o ∈ V → dom ⁡ ∘ f ⁡ + = V × V
535 533 534 mp1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → dom ⁡ ∘ f ⁡ + = V × V
536 530 535 sseqtrrid ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⊆ dom ⁡ ∘ f ⁡ +
537 522 527 529 536 elovimad ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → v + f y ∈ I ⟼ if y = X 1 0 ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0
538 5 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → I ∈ V
539 elrabi ⊢ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → v ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
540 16 psrbagf ⊢ v ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → v : I ⟶ ℕ 0
541 539 540 syl ⊢ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → v : I ⟶ ℕ 0
542 541 ad2antll ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → v : I ⟶ ℕ 0
543 131 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0
544 492 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ q ∈ ℕ 0 ∧ r ∈ ℕ 0 ∧ s ∈ ℕ 0 → q + r - s = q + r - s
545 538 542 543 543 544 caofass ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → v + f y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0 = v + f y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0
546 133 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → y ∈ I ⟼ if y = X 1 0 Fn I
547 78 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → y ∈ I ⟼ if y = X 1 0 ⁡ i = if i = X 1 0
548 546 546 538 538 72 547 547 offval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0 = i ∈ I ⟼ if i = X 1 0 − if i = X 1 0
549 548 oveq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → v + f y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0 = v + f i ∈ I ⟼ if i = X 1 0 − if i = X 1 0
550 504 oveq2i ⊢ v + f i ∈ I ⟼ if i = X 1 0 − if i = X 1 0 = v + f I × 0
551 0zd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → 0 ∈ ℤ
552 nn0cn ⊢ p ∈ ℕ 0 → p ∈ ℂ
553 552 addridd ⊢ p ∈ ℕ 0 → p + 0 = p
554 553 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ p ∈ ℕ 0 → p + 0 = p
555 538 542 551 554 caofid0r ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → v + f I × 0 = v
556 550 555 eqtrid ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → v + f i ∈ I ⟼ if i = X 1 0 − if i = X 1 0 = v
557 545 549 556 3eqtrrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → v = v + f y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0
558 oveq1 ⊢ u = v + f y ∈ I ⟼ if y = X 1 0 → u − f y ∈ I ⟼ if y = X 1 0 = v + f y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0
559 558 eqeq2d ⊢ u = v + f y ∈ I ⟼ if y = X 1 0 → v = u − f y ∈ I ⟼ if y = X 1 0 ↔ v = v + f y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0
560 557 559 syl5ibrcom ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u = v + f y ∈ I ⟼ if y = X 1 0 → v = u − f y ∈ I ⟼ if y = X 1 0
561 18 ad3antrrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
562 16 psrbagaddcl ⊢ m ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → m + f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
563 456 561 562 syl2an2 ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
564 16 psrbagf ⊢ m + f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → m + f y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0
565 563 564 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0
566 565 adantrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → m + f y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0
567 feq1 ⊢ u = m + f y ∈ I ⟼ if y = X 1 0 → u : I ⟶ ℕ 0 ↔ m + f y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0
568 566 567 syl5ibrcom ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → u = m + f y ∈ I ⟼ if y = X 1 0 → u : I ⟶ ℕ 0
569 483 568 sylbid ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → u = m + f n → u : I ⟶ ℕ 0
570 479 569 sylbird ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → u = o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o → u : I ⟶ ℕ 0
571 570 rexlimdvva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → ∃ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∃ n ∈ y ∈ I ⟼ if y = X 1 0 u = o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o → u : I ⟶ ℕ 0
572 455 571 mpd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u : I ⟶ ℕ 0
573 572 adantrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u : I ⟶ ℕ 0
574 573 ffvelcdmda ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → u ⁡ i ∈ ℕ 0
575 574 nn0cnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → u ⁡ i ∈ ℂ
576 237 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → if i = X 1 0 ∈ ℂ
577 575 576 npcand ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → u ⁡ i - if i = X 1 0 + if i = X 1 0 = u ⁡ i
578 577 mpteq2dva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → i ∈ I ⟼ u ⁡ i - if i = X 1 0 + if i = X 1 0 = i ∈ I ⟼ u ⁡ i
579 573 ffnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u Fn I
580 579 546 538 538 72 offn ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u − f y ∈ I ⟼ if y = X 1 0 Fn I
581 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → u ⁡ i = u ⁡ i
582 579 546 538 538 72 581 547 ofval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ i ∈ I → u − f y ∈ I ⟼ if y = X 1 0 ⁡ i = u ⁡ i − if i = X 1 0
583 580 546 538 538 72 582 547 offval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u − f y ∈ I ⟼ if y = X 1 0 + f y ∈ I ⟼ if y = X 1 0 = i ∈ I ⟼ u ⁡ i - if i = X 1 0 + if i = X 1 0
584 573 feqmptd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u = i ∈ I ⟼ u ⁡ i
585 578 583 584 3eqtr4rd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u = u − f y ∈ I ⟼ if y = X 1 0 + f y ∈ I ⟼ if y = X 1 0
586 oveq1 ⊢ v = u − f y ∈ I ⟼ if y = X 1 0 → v + f y ∈ I ⟼ if y = X 1 0 = u − f y ∈ I ⟼ if y = X 1 0 + f y ∈ I ⟼ if y = X 1 0
587 586 eqeq2d ⊢ v = u − f y ∈ I ⟼ if y = X 1 0 → u = v + f y ∈ I ⟼ if y = X 1 0 ↔ u = u − f y ∈ I ⟼ if y = X 1 0 + f y ∈ I ⟼ if y = X 1 0
588 585 587 syl5ibrcom ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → v = u − f y ∈ I ⟼ if y = X 1 0 → u = v + f y ∈ I ⟼ if y = X 1 0
589 560 588 impbid ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u = v + f y ∈ I ⟼ if y = X 1 0 ↔ v = u − f y ∈ I ⟼ if y = X 1 0
590 446 521 537 589 f1o2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⟼ u − f y ∈ I ⟼ if y = X 1 0 : ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⟶ 1-1 onto k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
591 10 104 14 426 440 445 590 gsumf1o ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b = ∑ R b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b ∘ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⟼ u − f y ∈ I ⟼ if y = X 1 0
592 553 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ p ∈ ℕ 0 → p + 0 = p
593 484 486 506 592 caofid0r ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f I × 0 = m
594 505 593 eqtrid ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f i ∈ I ⟼ if i = X 1 0 − if i = X 1 0 = m
595 494 500 594 3eqtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0 = m
596 595 512 eqeltrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
597 596 515 syl5ibrcom ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u = m + f y ∈ I ⟼ if y = X 1 0 → u − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
598 597 adantrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → u = m + f y ∈ I ⟼ if y = X 1 0 → u − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
599 483 598 sylbid ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → u = m + f n → u − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
600 479 599 sylbird ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → u = o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o → u − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
601 600 rexlimdvva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → ∃ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∃ n ∈ y ∈ I ⟼ if y = X 1 0 u = o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o → u − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
602 455 601 mpd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
603 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⟼ u − f y ∈ I ⟼ if y = X 1 0 = u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⟼ u − f y ∈ I ⟼ if y = X 1 0
604 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b = b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b
605 fveq2 ⊢ b = u − f y ∈ I ⟼ if y = X 1 0 → I mPSDer R ⁡ X ⁡ F ⁡ b = I mPSDer R ⁡ X ⁡ F ⁡ u − f y ∈ I ⟼ if y = X 1 0
606 oveq2 ⊢ b = u − f y ∈ I ⟼ if y = X 1 0 → d − f b = d − f u − f y ∈ I ⟼ if y = X 1 0
607 606 fveq2d ⊢ b = u − f y ∈ I ⟼ if y = X 1 0 → G ⁡ d − f b = G ⁡ d − f u − f y ∈ I ⟼ if y = X 1 0
608 605 607 oveq12d ⊢ b = u − f y ∈ I ⟼ if y = X 1 0 → I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b = I mPSDer R ⁡ X ⁡ F ⁡ u − f y ∈ I ⟼ if y = X 1 0 ⋅ R G ⁡ d − f u − f y ∈ I ⟼ if y = X 1 0
609 602 603 604 608 fmptco ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b ∘ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⟼ u − f y ∈ I ⟼ if y = X 1 0 = u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⟼ I mPSDer R ⁡ X ⁡ F ⁡ u − f y ∈ I ⟼ if y = X 1 0 ⋅ R G ⁡ d − f u − f y ∈ I ⟼ if y = X 1 0
610 5 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → I ∈ V
611 6 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → R ∈ CRing
612 7 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → X ∈ I
613 8 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → F ∈ B
614 elrabi ⊢ u − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u − f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
615 602 614 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u − f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
616 1 2 16 610 611 612 613 615 psdcoef ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → I mPSDer R ⁡ X ⁡ F ⁡ u − f y ∈ I ⟼ if y = X 1 0 = u − f y ∈ I ⟼ if y = X 1 0 ⁡ X + 1 ⋅ R F ⁡ u − f y ∈ I ⟼ if y = X 1 0 + f y ∈ I ⟼ if y = X 1 0
617 572 ffnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u Fn I
618 131 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0
619 618 ffnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → y ∈ I ⟼ if y = X 1 0 Fn I
620 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ X ∈ I → u ⁡ X = u ⁡ X
621 iftrue ⊢ y = X → if y = X 1 0 = 1
622 1ex ⊢ 1 ∈ V
623 621 68 622 fvmpt ⊢ X ∈ I → y ∈ I ⟼ if y = X 1 0 ⁡ X = 1
624 623 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ X ∈ I → y ∈ I ⟼ if y = X 1 0 ⁡ X = 1
625 617 619 610 610 72 620 624 ofval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ X ∈ I → u − f y ∈ I ⟼ if y = X 1 0 ⁡ X = u ⁡ X − 1
626 612 625 mpdan ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u − f y ∈ I ⟼ if y = X 1 0 ⁡ X = u ⁡ X − 1
627 626 oveq1d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u − f y ∈ I ⟼ if y = X 1 0 ⁡ X + 1 = u ⁡ X - 1 + 1
628 nn0sscn ⊢ ℕ 0 ⊆ ℂ
629 628 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → ℕ 0 ⊆ ℂ
630 572 629 fssd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u : I ⟶ ℂ
631 630 612 ffvelcdmd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u ⁡ X ∈ ℂ
632 1cnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → 1 ∈ ℂ
633 631 632 npcand ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u ⁡ X - 1 + 1 = u ⁡ X
634 627 633 eqtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u − f y ∈ I ⟼ if y = X 1 0 ⁡ X + 1 = u ⁡ X
635 617 619 610 610 72 offn ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u − f y ∈ I ⟼ if y = X 1 0 Fn I
636 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ i ∈ I → u ⁡ i = u ⁡ i
637 78 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ i ∈ I → y ∈ I ⟼ if y = X 1 0 ⁡ i = if i = X 1 0
638 617 619 610 610 72 636 637 ofval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ i ∈ I → u − f y ∈ I ⟼ if y = X 1 0 ⁡ i = u ⁡ i − if i = X 1 0
639 572 ffvelcdmda ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ i ∈ I → u ⁡ i ∈ ℕ 0
640 639 nn0cnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ i ∈ I → u ⁡ i ∈ ℂ
641 237 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ i ∈ I → if i = X 1 0 ∈ ℂ
642 640 641 npcand ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ i ∈ I → u ⁡ i - if i = X 1 0 + if i = X 1 0 = u ⁡ i
643 610 635 619 617 638 637 642 offveq ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u − f y ∈ I ⟼ if y = X 1 0 + f y ∈ I ⟼ if y = X 1 0 = u
644 643 fveq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → F ⁡ u − f y ∈ I ⟼ if y = X 1 0 + f y ∈ I ⟼ if y = X 1 0 = F ⁡ u
645 634 644 oveq12d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u − f y ∈ I ⟼ if y = X 1 0 ⁡ X + 1 ⋅ R F ⁡ u − f y ∈ I ⟼ if y = X 1 0 + f y ∈ I ⟼ if y = X 1 0 = u ⁡ X ⋅ R F ⁡ u
646 616 645 eqtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → I mPSDer R ⁡ X ⁡ F ⁡ u − f y ∈ I ⟼ if y = X 1 0 = u ⁡ X ⋅ R F ⁡ u
647 28 ad2antlr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → d : I ⟶ ℕ 0
648 647 ffvelcdmda ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ i ∈ I → d ⁡ i ∈ ℕ 0
649 648 nn0cnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ i ∈ I → d ⁡ i ∈ ℂ
650 649 640 641 subsub3d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ i ∈ I → d ⁡ i − u ⁡ i − if i = X 1 0 = d ⁡ i + if i = X 1 0 - u ⁡ i
651 650 mpteq2dva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → i ∈ I ⟼ d ⁡ i − u ⁡ i − if i = X 1 0 = i ∈ I ⟼ d ⁡ i + if i = X 1 0 - u ⁡ i
652 65 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → d Fn I
653 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ i ∈ I → d ⁡ i = d ⁡ i
654 652 635 610 610 72 653 638 offval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → d − f u − f y ∈ I ⟼ if y = X 1 0 = i ∈ I ⟼ d ⁡ i − u ⁡ i − if i = X 1 0
655 652 619 610 610 72 offn ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → d + f y ∈ I ⟼ if y = X 1 0 Fn I
656 652 619 610 610 72 653 637 ofval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ i ∈ I → d + f y ∈ I ⟼ if y = X 1 0 ⁡ i = d ⁡ i + if i = X 1 0
657 655 617 610 610 72 656 636 offval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → d + f y ∈ I ⟼ if y = X 1 0 − f u = i ∈ I ⟼ d ⁡ i + if i = X 1 0 - u ⁡ i
658 651 654 657 3eqtr4d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → d − f u − f y ∈ I ⟼ if y = X 1 0 = d + f y ∈ I ⟼ if y = X 1 0 − f u
659 658 fveq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → G ⁡ d − f u − f y ∈ I ⟼ if y = X 1 0 = G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
660 646 659 oveq12d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → I mPSDer R ⁡ X ⁡ F ⁡ u − f y ∈ I ⟼ if y = X 1 0 ⋅ R G ⁡ d − f u − f y ∈ I ⟼ if y = X 1 0 = u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
661 12 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → R ∈ Ring
662 572 612 ffvelcdmd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u ⁡ X ∈ ℕ 0
663 662 nn0zd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u ⁡ X ∈ ℤ
664 37 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → F : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
665 simpllr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
666 18 ad3antrrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
667 simprl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
668 eqid ⊢ l ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | l ≤ f d + f y ∈ I ⟼ if y = X 1 0 = l ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | l ≤ f d + f y ∈ I ⟼ if y = X 1 0
669 16 243 668 psrbagleadd1 ⊢ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 ∈ l ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | l ≤ f d + f y ∈ I ⟼ if y = X 1 0
670 665 666 667 669 syl3anc ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → m + f y ∈ I ⟼ if y = X 1 0 ∈ l ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | l ≤ f d + f y ∈ I ⟼ if y = X 1 0
671 eleq1 ⊢ u = m + f y ∈ I ⟼ if y = X 1 0 → u ∈ l ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | l ≤ f d + f y ∈ I ⟼ if y = X 1 0 ↔ m + f y ∈ I ⟼ if y = X 1 0 ∈ l ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | l ≤ f d + f y ∈ I ⟼ if y = X 1 0
672 670 671 syl5ibrcom ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → u = m + f y ∈ I ⟼ if y = X 1 0 → u ∈ l ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | l ≤ f d + f y ∈ I ⟼ if y = X 1 0
673 483 672 sylbid ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → u = m + f n → u ∈ l ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | l ≤ f d + f y ∈ I ⟼ if y = X 1 0
674 479 673 sylbird ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → u = o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o → u ∈ l ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | l ≤ f d + f y ∈ I ⟼ if y = X 1 0
675 674 rexlimdvva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → ∃ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∃ n ∈ y ∈ I ⟼ if y = X 1 0 u = o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o → u ∈ l ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | l ≤ f d + f y ∈ I ⟼ if y = X 1 0
676 455 675 mpd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u ∈ l ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | l ≤ f d + f y ∈ I ⟼ if y = X 1 0
677 elrabi ⊢ u ∈ l ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | l ≤ f d + f y ∈ I ⟼ if y = X 1 0 → u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
678 676 677 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
679 664 678 ffvelcdmd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → F ⁡ u ∈ Base R
680 42 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → G : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
681 21 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → d + f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
682 16 668 psrbagconcl ⊢ d + f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ l ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | l ≤ f d + f y ∈ I ⟼ if y = X 1 0 → d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ l ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | l ≤ f d + f y ∈ I ⟼ if y = X 1 0
683 681 676 682 syl2anc ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ l ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | l ≤ f d + f y ∈ I ⟼ if y = X 1 0
684 elrabi ⊢ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ l ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | l ≤ f d + f y ∈ I ⟼ if y = X 1 0 → d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
685 683 684 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
686 680 685 ffvelcdmd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
687 10 24 35 mulgass2 ⊢ R ∈ Ring ∧ u ⁡ X ∈ ℤ ∧ F ⁡ u ∈ Base R ∧ G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R → u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
688 661 663 679 686 687 syl13anc ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
689 660 688 eqtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → I mPSDer R ⁡ X ⁡ F ⁡ u − f y ∈ I ⟼ if y = X 1 0 ⋅ R G ⁡ d − f u − f y ∈ I ⟼ if y = X 1 0 = u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
690 689 mpteq2dva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⟼ I mPSDer R ⁡ X ⁡ F ⁡ u − f y ∈ I ⟼ if y = X 1 0 ⋅ R G ⁡ d − f u − f y ∈ I ⟼ if y = X 1 0 = u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
691 609 690 eqtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b ∘ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⟼ u − f y ∈ I ⟼ if y = X 1 0 = u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
692 691 oveq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b ∘ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⟼ u − f y ∈ I ⟼ if y = X 1 0 = ∑ R u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
693 snex ⊢ y ∈ I ⟼ if y = X 1 0 ∈ V
694 355 693 xpex ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∈ V
695 694 funimaex ⊢ Fun ⁡ ∘ f ⁡ + → ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∈ V
696 528 695 mp1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∈ V
697 26 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → R ∈ Mnd
698 10 35 661 679 686 ringcld ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
699 10 24 697 662 698 mulgnn0cld ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 → u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
700 eqid ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
701 359 700 fnmpti ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u Fn k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0
702 701 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u Fn k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0
703 702 23 113 fndmfifsupp ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → finSupp 0 R⁡ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
704 460 ad2antlr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → m Fn dom ⁡ m
705 467 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → n Fn dom ⁡ n
706 470 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → dom ⁡ m ∈ V
707 473 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → dom ⁡ n ∈ V
708 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 ∧ o ∈ dom ⁡ m → m ⁡ o = m ⁡ o
709 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 ∧ o ∈ dom ⁡ n → n ⁡ o = n ⁡ o
710 704 705 706 707 475 708 709 offval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → m + f n = o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o
711 710 eqeq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ n ∈ y ∈ I ⟼ if y = X 1 0 → u = m + f n ↔ u = o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o
712 711 rexbidva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → ∃ n ∈ y ∈ I ⟼ if y = X 1 0 u = m + f n ↔ ∃ n ∈ y ∈ I ⟼ if y = X 1 0 u = o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o
713 18 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
714 oveq2 ⊢ n = y ∈ I ⟼ if y = X 1 0 → m + f n = m + f y ∈ I ⟼ if y = X 1 0
715 714 eqeq2d ⊢ n = y ∈ I ⟼ if y = X 1 0 → u = m + f n ↔ u = m + f y ∈ I ⟼ if y = X 1 0
716 715 rexsng ⊢ y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∃ n ∈ y ∈ I ⟼ if y = X 1 0 u = m + f n ↔ u = m + f y ∈ I ⟼ if y = X 1 0
717 713 716 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → ∃ n ∈ y ∈ I ⟼ if y = X 1 0 u = m + f n ↔ u = m + f y ∈ I ⟼ if y = X 1 0
718 712 717 bitr3d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → ∃ n ∈ y ∈ I ⟼ if y = X 1 0 u = o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o ↔ u = m + f y ∈ I ⟼ if y = X 1 0
719 718 rexbidva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∃ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∃ n ∈ y ∈ I ⟼ if y = X 1 0 u = o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o ↔ ∃ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d u = m + f y ∈ I ⟼ if y = X 1 0
720 breq1 ⊢ k = m + f y ∈ I ⟼ if y = X 1 0 → k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ↔ m + f y ∈ I ⟼ if y = X 1 0 ≤ f d + f y ∈ I ⟼ if y = X 1 0
721 breq1 ⊢ k = m + f y ∈ I ⟼ if y = X 1 0 → k ≤ f d ↔ m + f y ∈ I ⟼ if y = X 1 0 ≤ f d
722 fveq1 ⊢ k = m + f y ∈ I ⟼ if y = X 1 0 → k ⁡ X = m + f y ∈ I ⟼ if y = X 1 0 ⁡ X
723 722 eqeq1d ⊢ k = m + f y ∈ I ⟼ if y = X 1 0 → k ⁡ X = 0 ↔ m + f y ∈ I ⟼ if y = X 1 0 ⁡ X = 0
724 721 723 anbi12d ⊢ k = m + f y ∈ I ⟼ if y = X 1 0 → k ≤ f d ∧ k ⁡ X = 0 ↔ m + f y ∈ I ⟼ if y = X 1 0 ≤ f d ∧ m + f y ∈ I ⟼ if y = X 1 0 ⁡ X = 0
725 724 notbid ⊢ k = m + f y ∈ I ⟼ if y = X 1 0 → ¬ k ≤ f d ∧ k ⁡ X = 0 ↔ ¬ m + f y ∈ I ⟼ if y = X 1 0 ≤ f d ∧ m + f y ∈ I ⟼ if y = X 1 0 ⁡ X = 0
726 720 725 anbi12d ⊢ k = m + f y ∈ I ⟼ if y = X 1 0 → k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ↔ m + f y ∈ I ⟼ if y = X 1 0 ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ m + f y ∈ I ⟼ if y = X 1 0 ≤ f d ∧ m + f y ∈ I ⟼ if y = X 1 0 ⁡ X = 0
727 456 713 562 syl2an2 ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
728 simplr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
729 simpr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
730 16 243 44 psrbagleadd1 ⊢ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0
731 728 713 729 730 syl3anc ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0
732 720 elrab ⊢ m + f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ↔ m + f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m + f y ∈ I ⟼ if y = X 1 0 ≤ f d + f y ∈ I ⟼ if y = X 1 0
733 732 simprbi ⊢ m + f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 → m + f y ∈ I ⟼ if y = X 1 0 ≤ f d + f y ∈ I ⟼ if y = X 1 0
734 731 733 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 ≤ f d + f y ∈ I ⟼ if y = X 1 0
735 7 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → X ∈ I
736 485 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m : I ⟶ ℕ 0
737 736 ffnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m Fn I
738 133 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → y ∈ I ⟼ if y = X 1 0 Fn I
739 5 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → I ∈ V
740 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ X ∈ I → m ⁡ X = m ⁡ X
741 623 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ X ∈ I → y ∈ I ⟼ if y = X 1 0 ⁡ X = 1
742 737 738 739 739 72 740 741 ofval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ X ∈ I → m + f y ∈ I ⟼ if y = X 1 0 ⁡ X = m ⁡ X + 1
743 735 742 mpdan ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 ⁡ X = m ⁡ X + 1
744 736 735 ffvelcdmd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m ⁡ X ∈ ℕ 0
745 nn0p1nn ⊢ m ⁡ X ∈ ℕ 0 → m ⁡ X + 1 ∈ ℕ
746 744 745 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m ⁡ X + 1 ∈ ℕ
747 743 746 eqeltrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 ⁡ X ∈ ℕ
748 747 nnne0d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 ⁡ X ≠ 0
749 748 neneqd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → ¬ m + f y ∈ I ⟼ if y = X 1 0 ⁡ X = 0
750 749 intnand ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → ¬ m + f y ∈ I ⟼ if y = X 1 0 ≤ f d ∧ m + f y ∈ I ⟼ if y = X 1 0 ⁡ X = 0
751 734 750 jca ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ m + f y ∈ I ⟼ if y = X 1 0 ≤ f d ∧ m + f y ∈ I ⟼ if y = X 1 0 ⁡ X = 0
752 726 727 751 elrabd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m + f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0
753 eleq1 ⊢ u = m + f y ∈ I ⟼ if y = X 1 0 → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ↔ m + f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0
754 752 753 syl5ibrcom ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u = m + f y ∈ I ⟼ if y = X 1 0 → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0
755 breq1 ⊢ k = u − f y ∈ I ⟼ if y = X 1 0 → k ≤ f d ↔ u − f y ∈ I ⟼ if y = X 1 0 ≤ f d
756 elrabi ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
757 756 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
758 131 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0
759 756 123 syl ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u : I ⟶ ℕ 0
760 759 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u : I ⟶ ℕ 0
761 7 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → X ∈ I
762 760 761 ffvelcdmd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u ⁡ X ∈ ℕ 0
763 339 notbid ⊢ k = u → ¬ k ≤ f d ∧ k ⁡ X = 0 ↔ ¬ u ≤ f d ∧ u ⁡ X = 0
764 118 763 anbi12d ⊢ k = u → k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ↔ u ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ u ≤ f d ∧ u ⁡ X = 0
765 764 elrab ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ↔ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ u ≤ f d ∧ u ⁡ X = 0
766 765 simprbi ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ u ≤ f d ∧ u ⁡ X = 0
767 766 simpld ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u ≤ f d + f y ∈ I ⟼ if y = X 1 0
768 767 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u ≤ f d + f y ∈ I ⟼ if y = X 1 0
769 768 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 → u ≤ f d + f y ∈ I ⟼ if y = X 1 0
770 756 124 syl ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u Fn I
771 770 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u Fn I
772 771 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 → u Fn I
773 21 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → d + f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
774 88 ffnd ⊢ d + f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d + f y ∈ I ⟼ if y = X 1 0 Fn I
775 773 774 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → d + f y ∈ I ⟼ if y = X 1 0 Fn I
776 775 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 → d + f y ∈ I ⟼ if y = X 1 0 Fn I
777 5 ad3antrrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 → I ∈ V
778 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ∈ I → u ⁡ i = u ⁡ i
779 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ∈ I → d + f y ∈ I ⟼ if y = X 1 0 ⁡ i = d + f y ∈ I ⟼ if y = X 1 0 ⁡ i
780 772 776 777 777 72 778 779 ofrfval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 → u ≤ f d + f y ∈ I ⟼ if y = X 1 0 ↔ ∀ i ∈ I u ⁡ i ≤ d + f y ∈ I ⟼ if y = X 1 0 ⁡ i
781 769 780 mpbid ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 → ∀ i ∈ I u ⁡ i ≤ d + f y ∈ I ⟼ if y = X 1 0 ⁡ i
782 781 r19.21bi ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ∈ I → u ⁡ i ≤ d + f y ∈ I ⟼ if y = X 1 0 ⁡ i
783 782 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ∈ I ∧ i ≠ X → u ⁡ i ≤ d + f y ∈ I ⟼ if y = X 1 0 ⁡ i
784 65 ad3antrrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ≠ X → d Fn I
785 69 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ≠ X → y ∈ I ⟼ if y = X 1 0 Fn I
786 5 ad4antr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ≠ X → I ∈ V
787 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ≠ X ∧ i ∈ I → d ⁡ i = d ⁡ i
788 78 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ≠ X ∧ i ∈ I → y ∈ I ⟼ if y = X 1 0 ⁡ i = if i = X 1 0
789 784 785 786 786 72 787 788 ofval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ≠ X ∧ i ∈ I → d + f y ∈ I ⟼ if y = X 1 0 ⁡ i = d ⁡ i + if i = X 1 0
790 789 an32s ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ∈ I ∧ i ≠ X → d + f y ∈ I ⟼ if y = X 1 0 ⁡ i = d ⁡ i + if i = X 1 0
791 158 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ∈ I ∧ i ≠ X → if i = X 1 0 = 0
792 791 oveq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ∈ I ∧ i ≠ X → d ⁡ i + if i = X 1 0 = d ⁡ i + 0
793 29 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 → d : I ⟶ ℕ 0
794 793 ffvelcdmda ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ∈ I → d ⁡ i ∈ ℕ 0
795 794 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ∈ I ∧ i ≠ X → d ⁡ i ∈ ℕ 0
796 795 nn0cnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ∈ I ∧ i ≠ X → d ⁡ i ∈ ℂ
797 796 addridd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ∈ I ∧ i ≠ X → d ⁡ i + 0 = d ⁡ i
798 790 792 797 3eqtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ∈ I ∧ i ≠ X → d + f y ∈ I ⟼ if y = X 1 0 ⁡ i = d ⁡ i
799 783 798 breqtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ∈ I ∧ i ≠ X → u ⁡ i ≤ d ⁡ i
800 simpr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 → u ⁡ X = 0
801 29 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → d : I ⟶ ℕ 0
802 801 761 ffvelcdmd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → d ⁡ X ∈ ℕ 0
803 802 nn0ge0d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → 0 ≤ d ⁡ X
804 803 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 → 0 ≤ d ⁡ X
805 800 804 eqbrtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 → u ⁡ X ≤ d ⁡ X
806 805 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ∈ I → u ⁡ X ≤ d ⁡ X
807 175 799 806 pm2.61ne ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ∈ I → u ⁡ i ≤ d ⁡ i
808 807 ralrimiva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 → ∀ i ∈ I u ⁡ i ≤ d ⁡ i
809 65 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → d Fn I
810 809 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 → d Fn I
811 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 ∧ i ∈ I → d ⁡ i = d ⁡ i
812 772 810 777 777 72 778 811 ofrfval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 → u ≤ f d ↔ ∀ i ∈ I u ⁡ i ≤ d ⁡ i
813 808 812 mpbird ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ u ⁡ X = 0 → u ≤ f d
814 813 ex ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u ⁡ X = 0 → u ≤ f d
815 766 simprd ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → ¬ u ≤ f d ∧ u ⁡ X = 0
816 815 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → ¬ u ≤ f d ∧ u ⁡ X = 0
817 imnan ⊢ u ≤ f d → ¬ u ⁡ X = 0 ↔ ¬ u ≤ f d ∧ u ⁡ X = 0
818 816 817 sylibr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u ≤ f d → ¬ u ⁡ X = 0
819 818 con2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u ⁡ X = 0 → ¬ u ≤ f d
820 814 819 pm2.65d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → ¬ u ⁡ X = 0
821 820 neqned ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u ⁡ X ≠ 0
822 762 821 191 sylanbrc ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u ⁡ X ∈ ℕ
823 822 nnge1d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → 1 ≤ u ⁡ X
824 823 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → 1 ≤ u ⁡ X
825 173 breq2d ⊢ i = X → 1 ≤ u ⁡ i ↔ 1 ≤ u ⁡ X
826 824 825 syl5ibrcom ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → i = X → 1 ≤ u ⁡ i
827 826 imp ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I ∧ i = X → 1 ≤ u ⁡ i
828 760 ffvelcdmda ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → u ⁡ i ∈ ℕ 0
829 828 nn0ge0d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → 0 ≤ u ⁡ i
830 829 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I ∧ ¬ i = X → 0 ≤ u ⁡ i
831 827 830 ifpimpda ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → if- i = X 1 ≤ u ⁡ i 0 ≤ u ⁡ i
832 brif1 ⊢ if i = X 1 0 ≤ u ⁡ i ↔ if- i = X 1 ≤ u ⁡ i 0 ≤ u ⁡ i
833 831 832 sylibr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → if i = X 1 0 ≤ u ⁡ i
834 833 ralrimiva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → ∀ i ∈ I if i = X 1 0 ≤ u ⁡ i
835 69 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → y ∈ I ⟼ if y = X 1 0 Fn I
836 5 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → I ∈ V
837 78 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → y ∈ I ⟼ if y = X 1 0 ⁡ i = if i = X 1 0
838 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → u ⁡ i = u ⁡ i
839 835 771 836 836 72 837 838 ofrfval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → y ∈ I ⟼ if y = X 1 0 ≤ f u ↔ ∀ i ∈ I if i = X 1 0 ≤ u ⁡ i
840 834 839 mpbird ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → y ∈ I ⟼ if y = X 1 0 ≤ f u
841 16 psrbagcon ⊢ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0 ∧ y ∈ I ⟼ if y = X 1 0 ≤ f u → u − f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u − f y ∈ I ⟼ if y = X 1 0 ≤ f u
842 757 758 840 841 syl3anc ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u − f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u − f y ∈ I ⟼ if y = X 1 0 ≤ f u
843 842 simpld ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u − f y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
844 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → d ⁡ i = d ⁡ i
845 809 835 836 836 72 844 837 ofval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → d + f y ∈ I ⟼ if y = X 1 0 ⁡ i = d ⁡ i + if i = X 1 0
846 771 775 836 836 72 838 845 ofrfval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u ≤ f d + f y ∈ I ⟼ if y = X 1 0 ↔ ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0
847 768 846 mpbid ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0
848 847 r19.21bi ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → u ⁡ i ≤ d ⁡ i + if i = X 1 0
849 828 nn0red ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → u ⁡ i ∈ ℝ
850 60 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → if i = X 1 0 ∈ ℝ
851 801 ffvelcdmda ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → d ⁡ i ∈ ℕ 0
852 851 nn0red ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → d ⁡ i ∈ ℝ
853 849 850 852 lesubaddd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → u ⁡ i − if i = X 1 0 ≤ d ⁡ i ↔ u ⁡ i ≤ d ⁡ i + if i = X 1 0
854 848 853 mpbird ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → u ⁡ i − if i = X 1 0 ≤ d ⁡ i
855 854 ralrimiva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → ∀ i ∈ I u ⁡ i − if i = X 1 0 ≤ d ⁡ i
856 771 835 836 836 72 offn ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u − f y ∈ I ⟼ if y = X 1 0 Fn I
857 771 835 836 836 72 838 837 ofval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → u − f y ∈ I ⟼ if y = X 1 0 ⁡ i = u ⁡ i − if i = X 1 0
858 856 809 836 836 72 857 844 ofrfval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u − f y ∈ I ⟼ if y = X 1 0 ≤ f d ↔ ∀ i ∈ I u ⁡ i − if i = X 1 0 ≤ d ⁡ i
859 855 858 mpbird ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u − f y ∈ I ⟼ if y = X 1 0 ≤ f d
860 755 843 859 elrabd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u − f y ∈ I ⟼ if y = X 1 0 ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
861 828 nn0cnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → u ⁡ i ∈ ℂ
862 237 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → if i = X 1 0 ∈ ℂ
863 861 862 npcand ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 ∧ i ∈ I → u ⁡ i - if i = X 1 0 + if i = X 1 0 = u ⁡ i
864 863 mpteq2dva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → i ∈ I ⟼ u ⁡ i - if i = X 1 0 + if i = X 1 0 = i ∈ I ⟼ u ⁡ i
865 856 835 836 836 72 857 837 offval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u − f y ∈ I ⟼ if y = X 1 0 + f y ∈ I ⟼ if y = X 1 0 = i ∈ I ⟼ u ⁡ i - if i = X 1 0 + if i = X 1 0
866 760 feqmptd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u = i ∈ I ⟼ u ⁡ i
867 864 865 866 3eqtr4rd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → u = u − f y ∈ I ⟼ if y = X 1 0 + f y ∈ I ⟼ if y = X 1 0
868 oveq1 ⊢ m = u − f y ∈ I ⟼ if y = X 1 0 → m + f y ∈ I ⟼ if y = X 1 0 = u − f y ∈ I ⟼ if y = X 1 0 + f y ∈ I ⟼ if y = X 1 0
869 868 eqeq2d ⊢ m = u − f y ∈ I ⟼ if y = X 1 0 → u = m + f y ∈ I ⟼ if y = X 1 0 ↔ u = u − f y ∈ I ⟼ if y = X 1 0 + f y ∈ I ⟼ if y = X 1 0
870 754 860 867 869 rspceb2dv ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∃ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d u = m + f y ∈ I ⟼ if y = X 1 0 ↔ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0
871 454 719 870 3bitrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ↔ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0
872 871 eqrdv ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 = k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0
873 difrab ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 = k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0
874 872 873 eqtr4di ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 = k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0
875 difssd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⊆ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0
876 874 875 eqsstrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⊆ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0
877 703 876 113 fmptssfisupp ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → finSupp 0 R⁡ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⟼ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
878 difss ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⊆ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d
879 disjdif ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∩ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d = ∅
880 ssdisj ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ⊆ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∩ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d = ∅ → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ∩ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d = ∅
881 878 879 880 mp2an ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ∩ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d = ∅
882 881 ineqcomi ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∩ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 = ∅
883 882 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∩ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 = ∅
884 279 99 psdmullem ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∪ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 = k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0
885 874 884 eqtr4d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 = k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∪ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0
886 10 104 11 14 696 699 877 883 885 gsumsplit2 ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
887 692 886 eqtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b ∘ u ∈ ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⟼ u − f y ∈ I ⟼ if y = X 1 0 = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
888 425 591 887 3eqtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → I mPSDer R ⁡ X ⁡ F · ˙ G ⁡ d = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
889 415 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → I mPSDer R ⁡ X ⁡ G ∈ B
890 1 2 35 4 16 385 889 15 psrmulval ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → F · ˙ I mPSDer R ⁡ X ⁡ G ⁡ d = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d F ⁡ u ⋅ R I mPSDer R ⁡ X ⁡ G ⁡ d − f u
891 6 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → R ∈ CRing
892 9 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → G ∈ B
893 1 2 16 252 891 285 892 247 psdcoef ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → I mPSDer R ⁡ X ⁡ G ⁡ d − f u = d − f u ⁡ X + 1 ⋅ R G ⁡ d − f u + f y ∈ I ⟼ if y = X 1 0
894 267 fveq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → G ⁡ d − f u + f y ∈ I ⟼ if y = X 1 0 = G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
895 894 oveq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f u ⁡ X + 1 ⋅ R G ⁡ d − f u + f y ∈ I ⟼ if y = X 1 0 = d − f u ⁡ X + 1 ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
896 893 895 eqtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → I mPSDer R ⁡ X ⁡ G ⁡ d − f u = d − f u ⁡ X + 1 ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
897 896 oveq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → F ⁡ u ⋅ R I mPSDer R ⁡ X ⁡ G ⁡ d − f u = F ⁡ u ⋅ R d − f u ⁡ X + 1 ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
898 309 nn0zd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → d − f u ⁡ X + 1 ∈ ℤ
899 10 24 35 mulgass3 ⊢ R ∈ Ring ∧ d − f u ⁡ X + 1 ∈ ℤ ∧ F ⁡ u ∈ Base R ∧ G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R → F ⁡ u ⋅ R d − f u ⁡ X + 1 ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
900 224 898 228 271 899 syl13anc ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → F ⁡ u ⋅ R d − f u ⁡ X + 1 ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
901 897 900 eqtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → F ⁡ u ⋅ R I mPSDer R ⁡ X ⁡ G ⁡ d − f u = d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
902 901 mpteq2dva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ F ⁡ u ⋅ R I mPSDer R ⁡ X ⁡ G ⁡ d − f u = u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⟼ d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
903 902 oveq2d ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d F ⁡ u ⋅ R I mPSDer R ⁡ X ⁡ G ⁡ d − f u = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
904 10 11 14 221 321 275 282 gsummptfidmsplit ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
905 890 903 904 3eqtrd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → F · ˙ I mPSDer R ⁡ X ⁡ G ⁡ d = ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
906 419 421 422 422 423 888 905 offval ⊢ φ → I mPSDer R ⁡ X ⁡ F · ˙ G + R f F · ˙ I mPSDer R ⁡ X ⁡ G = d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟼ ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
907 417 906 eqtrd ⊢ φ → I mPSDer R ⁡ X ⁡ F · ˙ G + ˙ F · ˙ I mPSDer R ⁡ X ⁡ G = d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟼ ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∖ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u + R ∑ R u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u
908 409 411 907 3eqtr4d ⊢ φ → I mPSDer R ⁡ X ⁡ F · ˙ G = I mPSDer R ⁡ X ⁡ F · ˙ G + ˙ F · ˙ I mPSDer R ⁡ X ⁡ G