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.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.r ⊢ φ → R ∈ CRing
6 psdmul.x ⊢ φ → X ∈ I
7 psdmul.f ⊢ φ → F ∈ B
8 psdmul.g ⊢ φ → G ∈ B
9 eqid ⊢ Base R = Base R
10 eqid ⊢ + R = + R
11 5 crngringd ⊢ φ → R ∈ Ring
12 11 ringcmnd ⊢ φ → R ∈ CMnd
13 12 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → R ∈ CMnd
14 simpr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
15 reldmpsr ⊢ Rel ⁡ dom ⁡ mPwSer
16 1 2 15 strov2rcl ⊢ F ∈ B → I ∈ V
17 7 16 syl ⊢ φ → I ∈ V
18 eqid ⊢ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin = h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
19 18 psrbagsn ⊢ I ∈ V → y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
20 17 19 syl ⊢ φ → y ∈ I ⟼ if y = X 1 0 ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
21 20 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
22 18 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
23 14 21 22 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
24 18 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
25 23 24 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
26 eqid ⊢ ⋅ R = ⋅ R
27 5 crnggrpd ⊢ φ → R ∈ Grp
28 27 grpmndd ⊢ φ → R ∈ Mnd
29 28 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
30 18 psrbagf ⊢ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d : I ⟶ ℕ 0
31 30 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d : I ⟶ ℕ 0
32 6 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → X ∈ I
33 31 32 ffvelcdmd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ⁡ X ∈ ℕ 0
34 peano2nn0 ⊢ d ⁡ X ∈ ℕ 0 → d ⁡ X + 1 ∈ ℕ 0
35 33 34 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ⁡ X + 1 ∈ ℕ 0
36 35 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
37 eqid ⊢ ⋅ R = ⋅ R
38 11 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
39 1 9 18 2 7 psrelbas ⊢ φ → F : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
40 39 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
41 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
42 41 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
43 40 42 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
44 1 9 18 2 8 psrelbas ⊢ φ → G : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
45 44 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
46 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
47 18 46 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
48 23 47 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
49 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
50 48 49 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
51 45 50 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
52 9 37 38 43 51 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
53 9 26 29 36 52 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
54 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 = ∅
55 54 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 = ∅
56 1nn0 ⊢ 1 ∈ ℕ 0
57 0nn0 ⊢ 0 ∈ ℕ 0
58 56 57 ifcli ⊢ if i = X 1 0 ∈ ℕ 0
59 58 nn0ge0i ⊢ 0 ≤ if i = X 1 0
60 31 ffvelcdmda ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → d ⁡ i ∈ ℕ 0
61 60 nn0red ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → d ⁡ i ∈ ℝ
62 58 nn0rei ⊢ if i = X 1 0 ∈ ℝ
63 62 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → if i = X 1 0 ∈ ℝ
64 61 63 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
65 59 64 mpbii ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → d ⁡ i ≤ d ⁡ i + if i = X 1 0
66 65 ralrimiva ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → ∀ i ∈ I d ⁡ i ≤ d ⁡ i + if i = X 1 0
67 31 ffnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d Fn I
68 56 57 ifcli ⊢ if y = X 1 0 ∈ ℕ 0
69 68 elexi ⊢ if y = X 1 0 ∈ V
70 eqid ⊢ y ∈ I ⟼ if y = X 1 0 = y ∈ I ⟼ if y = X 1 0
71 69 70 fnmpti ⊢ y ∈ I ⟼ if y = X 1 0 Fn I
72 71 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → y ∈ I ⟼ if y = X 1 0 Fn I
73 17 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → I ∈ V
74 inidm ⊢ I ∩ I = I
75 67 72 73 73 74 offn ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d + f y ∈ I ⟼ if y = X 1 0 Fn I
76 eqidd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ i ∈ I → d ⁡ i = d ⁡ i
77 eqeq1 ⊢ y = i → y = X ↔ i = X
78 77 ifbid ⊢ y = i → if y = X 1 0 = if i = X 1 0
79 58 elexi ⊢ if i = X 1 0 ∈ V
80 78 70 79 fvmpt ⊢ i ∈ I → y ∈ I ⟼ if y = X 1 0 ⁡ i = if i = X 1 0
81 80 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
82 67 72 73 73 74 76 81 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
83 67 75 73 73 74 76 82 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
84 66 83 mpbird ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ≤ f d + f y ∈ I ⟼ if y = X 1 0
85 84 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
86 17 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → I ∈ V
87 18 psrbagf ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k : I ⟶ ℕ 0
88 87 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k : I ⟶ ℕ 0
89 31 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d : I ⟶ ℕ 0
90 18 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
91 23 90 syl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d + f y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0
92 91 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
93 nn0re ⊢ q ∈ ℕ 0 → q ∈ ℝ
94 nn0re ⊢ r ∈ ℕ 0 → r ∈ ℝ
95 nn0re ⊢ s ∈ ℕ 0 → s ∈ ℝ
96 letr ⊢ q ∈ ℝ ∧ r ∈ ℝ ∧ s ∈ ℝ → q ≤ r ∧ r ≤ s → q ≤ s
97 93 94 95 96 syl3an ⊢ q ∈ ℕ 0 ∧ r ∈ ℕ 0 ∧ s ∈ ℕ 0 → q ≤ r ∧ r ≤ s → q ≤ s
98 97 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
99 86 88 89 92 98 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
100 85 99 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
101 100 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
102 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
103 101 102 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
104 103 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
105 9 10 13 25 53 55 104 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
106 eqid ⊢ 0 R = 0 R
107 ovex ⊢ ℕ 0 I ∈ V
108 107 rabex ⊢ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∈ V
109 108 rabex ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d + f y ∈ I ⟼ if y = X 1 0 ∈ V
110 109 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
111 ovex ⊢ F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ V
112 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
113 111 112 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
114 113 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
115 fvexd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → 0 R ∈ V
116 114 25 115 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
117 9 106 26 110 52 116 13 35 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
118 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
119 118 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
120 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
121 breq1 ⊢ k = u → k ≤ f d ↔ u ≤ f d
122 121 notbid ⊢ k = u → ¬ k ≤ f d ↔ ¬ u ≤ f d
123 120 122 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
124 123 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
125 18 psrbagf ⊢ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u : I ⟶ ℕ 0
126 125 ffnd ⊢ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u Fn I
127 126 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u Fn I
128 75 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
129 17 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → I ∈ V
130 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
131 67 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d Fn I
132 68 a1i ⊢ y ∈ I → if y = X 1 0 ∈ ℕ 0
133 70 132 fmpti ⊢ y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0
134 133 a1i ⊢ φ → y ∈ I ⟼ if y = X 1 0 : I ⟶ ℕ 0
135 134 ffnd ⊢ φ → y ∈ I ⟼ if y = X 1 0 Fn I
136 135 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
137 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
138 80 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
139 131 136 129 129 74 137 138 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
140 127 128 129 129 74 130 139 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
141 127 131 129 129 74 130 137 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
142 141 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
143 rexnal ⊢ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i ↔ ¬ ∀ i ∈ I u ⁡ i ≤ d ⁡ i
144 142 143 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
145 140 144 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
146 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 ∈ ℕ 0
147 125 adantl ⊢ φ ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u : I ⟶ ℕ 0
148 6 adantr ⊢ φ ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → X ∈ I
149 147 148 ffvelcdmd ⊢ φ ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ⁡ X ∈ ℕ 0
150 149 adantlr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ⁡ X ∈ ℕ 0
151 150 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
152 nn0nlt0 ⊢ d ⁡ X ∈ ℕ 0 → ¬ d ⁡ X < 0
153 146 152 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
154 31 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d : I ⟶ ℕ 0
155 154 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
156 155 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
157 156 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
158 157 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
159 158 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
160 ifnefalse ⊢ i ≠ X → if i = X 1 0 = 0
161 160 oveq2d ⊢ i ≠ X → d ⁡ i + if i = X 1 0 = d ⁡ i + 0
162 161 breq2d ⊢ i ≠ X → u ⁡ i ≤ d ⁡ i + if i = X 1 0 ↔ u ⁡ i ≤ d ⁡ i + 0
163 162 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
164 159 163 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
165 164 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
166 165 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
167 166 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
168 167 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
169 168 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
170 169 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
171 170 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
172 171 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
173 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
174 173 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
175 fveq2 ⊢ i = X → u ⁡ i = u ⁡ X
176 fveq2 ⊢ i = X → d ⁡ i = d ⁡ X
177 175 176 breq12d ⊢ i = X → u ⁡ i ≤ d ⁡ i ↔ u ⁡ X ≤ d ⁡ X
178 177 notbid ⊢ i = X → ¬ u ⁡ i ≤ d ⁡ i ↔ ¬ u ⁡ X ≤ d ⁡ X
179 178 ceqsrexbv ⊢ ∃ i ∈ I i = X ∧ ¬ u ⁡ i ≤ d ⁡ i ↔ X ∈ I ∧ ¬ u ⁡ X ≤ d ⁡ X
180 179 simprbi ⊢ ∃ i ∈ I i = X ∧ ¬ u ⁡ i ≤ d ⁡ i → ¬ u ⁡ X ≤ d ⁡ X
181 174 180 syl ⊢ ∀ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → i = X ∧ ¬ u ⁡ i ≤ d ⁡ i ∧ ∃ i ∈ I ¬ u ⁡ i ≤ d ⁡ i → ¬ u ⁡ X ≤ d ⁡ X
182 33 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ⁡ X ∈ ℕ 0
183 182 nn0red ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ⁡ X ∈ ℝ
184 150 nn0red ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ⁡ X ∈ ℝ
185 183 184 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
186 185 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
187 181 186 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
188 172 187 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
189 breq2 ⊢ u ⁡ X = 0 → d ⁡ X < u ⁡ X ↔ d ⁡ X < 0
190 188 189 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
191 153 190 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
192 191 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
193 elnnne0 ⊢ u ⁡ X ∈ ℕ ↔ u ⁡ X ∈ ℕ 0 ∧ u ⁡ X ≠ 0
194 151 192 193 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
195 elfzo0 ⊢ d ⁡ X ∈ 0 ..^ u ⁡ X ↔ d ⁡ X ∈ ℕ 0 ∧ u ⁡ X ∈ ℕ ∧ d ⁡ X < u ⁡ X
196 146 194 188 195 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
197 fzostep1 ⊢ d ⁡ X ∈ 0 ..^ u ⁡ X → d ⁡ X + 1 ∈ 0 ..^ u ⁡ X ∨ d ⁡ X + 1 = u ⁡ X
198 196 197 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
199 151 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
200 35 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
201 200 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
202 6 ad2antrr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → X ∈ I
203 iftrue ⊢ i = X → if i = X 1 0 = 1
204 176 203 oveq12d ⊢ i = X → d ⁡ i + if i = X 1 0 = d ⁡ X + 1
205 175 204 breq12d ⊢ i = X → u ⁡ i ≤ d ⁡ i + if i = X 1 0 ↔ u ⁡ X ≤ d ⁡ X + 1
206 205 rspcv ⊢ X ∈ I → ∀ i ∈ I u ⁡ i ≤ d ⁡ i + if i = X 1 0 → u ⁡ X ≤ d ⁡ X + 1
207 202 206 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
208 207 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
209 208 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
210 199 201 209 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
211 210 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
212 elfzo0 ⊢ d ⁡ X + 1 ∈ 0 ..^ u ⁡ X ↔ d ⁡ X + 1 ∈ ℕ 0 ∧ u ⁡ X ∈ ℕ ∧ d ⁡ X + 1 < u ⁡ X
213 211 212 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
214 198 213 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
215 145 214 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
216 215 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
217 124 216 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
218 119 217 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
219 218 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
220 219 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
221 220 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
222 18 psrbaglefi ⊢ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∈ Fin
223 222 adantl ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∈ Fin
224 28 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
225 35 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
226 11 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
227 elrabi ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
228 39 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → F : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
229 228 ffvelcdmda ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∧ u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → F ⁡ u ∈ Base R
230 227 229 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
231 44 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
232 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 → d : I ⟶ ℕ 0
233 232 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
234 233 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
235 227 125 syl ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → u : I ⟶ ℕ 0
236 235 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
237 236 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
238 237 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
239 58 nn0cni ⊢ if i = X 1 0 ∈ ℂ
240 239 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
241 234 238 240 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
242 234 240 238 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
243 241 242 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
244 243 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
245 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
246 18 245 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
247 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
248 246 247 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
249 248 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
250 18 psrbagf ⊢ d − f u ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d − f u : I ⟶ ℕ 0
251 249 250 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
252 251 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
253 71 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
254 17 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
255 232 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
256 236 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
257 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
258 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
259 255 256 254 254 74 257 258 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 u ⁡ i = d ⁡ i − u ⁡ i
260 80 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
261 252 253 254 254 74 259 260 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
262 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
263 20 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
264 262 263 22 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
265 264 90 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
266 265 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
267 255 253 254 254 74 257 260 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
268 266 256 254 254 74 267 258 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 y ∈ I ⟼ if y = X 1 0 − f u = i ∈ I ⟼ d ⁡ i + if i = X 1 0 - u ⁡ i
269 244 261 268 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
270 18 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
271 249 263 270 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
272 269 271 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
273 231 272 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
274 9 37 226 230 273 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
275 9 26 224 225 274 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
276 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 = ∅
277 276 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 = ∅
278 simpl ⊢ k ≤ f d ∧ k ⁡ X = 0 → k ≤ f d
279 278 a1i ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ≤ f d ∧ k ⁡ X = 0 → k ≤ f d
280 279 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
281 280 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
282 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
283 281 282 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
284 283 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
285 9 10 13 223 275 277 284 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
286 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
287 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 → X ∈ I
288 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
289 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
290 255 256 254 254 74 288 289 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
291 287 290 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
292 286 291 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
293 292 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
294 236 287 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
295 286 294 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
296 295 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
297 33 nn0cnd ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → d ⁡ X ∈ ℂ
298 297 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
299 296 298 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
300 293 299 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
301 300 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
302 251 287 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
303 286 302 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
304 303 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
305 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
306 296 304 305 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
307 301 306 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
308 307 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
309 28 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
310 peano2nn0 ⊢ d − f u ⁡ X ∈ ℕ 0 → d − f u ⁡ X + 1 ∈ ℕ 0
311 302 310 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
312 286 311 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
313 286 274 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
314 9 26 10 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
315 309 295 312 313 314 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
316 308 315 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
317 316 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
318 317 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
319 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
320 223 319 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
321 9 26 224 294 274 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
322 286 321 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 → u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ Base R
323 9 26 224 311 274 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
324 286 323 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
325 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
326 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
327 9 10 13 320 322 324 325 326 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
328 318 327 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
329 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 ∧ k ⁡ X = 0 → X ∈ I
330 67 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
331 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
332 331 126 syl ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u Fn I
333 332 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
334 17 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
335 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
336 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
337 330 333 334 334 74 335 336 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
338 329 337 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
339 fveq1 ⊢ k = u → k ⁡ X = u ⁡ X
340 339 eqeq1d ⊢ k = u → k ⁡ X = 0 ↔ u ⁡ X = 0
341 121 340 anbi12d ⊢ k = u → k ≤ f d ∧ k ⁡ X = 0 ↔ u ≤ f d ∧ u ⁡ X = 0
342 341 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
343 342 simprbi ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u ≤ f d ∧ u ⁡ X = 0
344 343 simprd ⊢ u ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 → u ⁡ X = 0
345 344 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
346 345 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
347 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 ∧ k ⁡ X = 0 → d ⁡ X ∈ ℕ 0
348 347 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
349 348 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
350 338 346 349 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
351 350 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
352 351 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
353 352 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
354 353 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
355 328 354 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
356 27 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → R ∈ Grp
357 108 rabex ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∈ V
358 357 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
359 358 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
360 322 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 ⟼ 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
361 ovex ⊢ u ⁡ X ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ V
362 361 325 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
363 362 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
364 363 320 115 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
365 9 106 13 359 360 364 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
366 324 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
367 ovex ⊢ d − f u ⁡ X + 1 ⋅ R F ⁡ u ⋅ R G ⁡ d + f y ∈ I ⟼ if y = X 1 0 − f u ∈ V
368 367 326 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
369 368 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
370 369 320 115 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
371 9 106 13 359 366 370 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
372 108 rabex ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∧ k ⁡ X = 0 ∈ V
373 372 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
374 280 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
375 374 323 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
376 375 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
377 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
378 367 377 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
379 378 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
380 223 281 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
381 379 380 115 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
382 9 106 13 373 376 381 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
383 9 10 356 365 371 382 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
384 285 355 383 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
385 221 384 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
386 105 117 385 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
387 7 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → F ∈ B
388 8 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → G ∈ B
389 1 2 37 4 18 387 388 23 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
390 389 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
391 109 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
392 391 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
393 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
394 41 125 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
395 394 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
396 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 + f y ∈ I ⟼ if y = X 1 0 → X ∈ I
397 395 396 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
398 9 26 29 397 52 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
399 393 398 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
400 399 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
401 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
402 361 401 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
403 402 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
404 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
405 25 404 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
406 403 405 115 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
407 9 106 13 392 400 406 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
408 9 10 356 371 382 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
409 9 10 356 407 365 408 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
410 386 390 409 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
411 410 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
412 1 2 4 11 7 8 psrmulcl ⊢ φ → F · ˙ G ∈ B
413 1 2 18 6 412 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
414 27 grpmgmd ⊢ φ → R ∈ Mgm
415 1 2 414 6 7 psdcl ⊢ φ → I mPSDer R ⁡ X ⁡ F ∈ B
416 1 2 4 11 415 8 psrmulcl ⊢ φ → I mPSDer R ⁡ X ⁡ F · ˙ G ∈ B
417 1 2 414 6 8 psdcl ⊢ φ → I mPSDer R ⁡ X ⁡ G ∈ B
418 1 2 4 11 7 417 psrmulcl ⊢ φ → F · ˙ I mPSDer R ⁡ X ⁡ G ∈ B
419 1 2 10 3 416 418 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
420 1 9 18 2 416 psrelbas ⊢ φ → I mPSDer R ⁡ X ⁡ F · ˙ G : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
421 420 ffnd ⊢ φ → I mPSDer R ⁡ X ⁡ F · ˙ G Fn h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
422 1 9 18 2 418 psrelbas ⊢ φ → F · ˙ I mPSDer R ⁡ X ⁡ G : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
423 422 ffnd ⊢ φ → F · ˙ I mPSDer R ⁡ X ⁡ G Fn h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
424 108 a1i ⊢ φ → h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ∈ V
425 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
426 415 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → I mPSDer R ⁡ X ⁡ F ∈ B
427 1 2 37 4 18 426 388 14 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
428 357 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ∈ V
429 11 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
430 elrabi ⊢ b ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → b ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
431 1 9 18 2 415 psrelbas ⊢ φ → I mPSDer R ⁡ X ⁡ F : h ∈ ℕ 0 I | h -1 ℕ ∈ Fin ⟶ Base R
432 431 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
433 432 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
434 430 433 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
435 44 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
436 18 245 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
437 436 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
438 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
439 437 438 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
440 435 439 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
441 9 37 429 434 440 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
442 441 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
443 ovex ⊢ I mPSDer R ⁡ X ⁡ F ⁡ b ⋅ R G ⁡ d − f b ∈ V
444 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
445 443 444 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
446 445 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
447 446 223 115 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
448 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
449 df-of ⊢ ∘ f ⁡ + = m ∈ V , n ∈ V ⟼ o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o
450 vex ⊢ u ∈ V
451 450 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → u ∈ V
452 ssv ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⊆ V
453 452 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d ⊆ V
454 ssv ⊢ y ∈ I ⟼ if y = X 1 0 ⊆ V
455 454 a1i ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → y ∈ I ⟼ if y = X 1 0 ⊆ V
456 449 451 453 455 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
457 456 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
458 elrabi ⊢ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
459 18 psrbagf ⊢ m ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → m : I ⟶ ℕ 0
460 459 ffund ⊢ m ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → Fun ⁡ m
461 458 460 syl ⊢ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → Fun ⁡ m
462 461 funfnd ⊢ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m Fn dom ⁡ m
463 462 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
464 velsn ⊢ n ∈ y ∈ I ⟼ if y = X 1 0 ↔ n = y ∈ I ⟼ if y = X 1 0
465 funmpt ⊢ Fun ⁡ y ∈ I ⟼ if y = X 1 0
466 funeq ⊢ n = y ∈ I ⟼ if y = X 1 0 → Fun ⁡ n ↔ Fun ⁡ y ∈ I ⟼ if y = X 1 0
467 465 466 mpbiri ⊢ n = y ∈ I ⟼ if y = X 1 0 → Fun ⁡ n
468 467 funfnd ⊢ n = y ∈ I ⟼ if y = X 1 0 → n Fn dom ⁡ n
469 464 468 sylbi ⊢ n ∈ y ∈ I ⟼ if y = X 1 0 → n Fn dom ⁡ n
470 469 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
471 vex ⊢ m ∈ V
472 471 dmex ⊢ dom ⁡ m ∈ V
473 472 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
474 vex ⊢ n ∈ V
475 474 dmex ⊢ dom ⁡ n ∈ V
476 475 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
477 eqid ⊢ dom ⁡ m ∩ dom ⁡ n = dom ⁡ m ∩ dom ⁡ n
478 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
479 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
480 463 470 473 476 477 478 479 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
481 480 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
482 elsni ⊢ n ∈ y ∈ I ⟼ if y = X 1 0 → n = y ∈ I ⟼ if y = X 1 0
483 482 oveq2d ⊢ n ∈ y ∈ I ⟼ if y = X 1 0 → m + f n = m + f y ∈ I ⟼ if y = X 1 0
484 483 eqeq2d ⊢ n ∈ y ∈ I ⟼ if y = X 1 0 → u = m + f n ↔ u = m + f y ∈ I ⟼ if y = X 1 0
485 484 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
486 17 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
487 458 459 syl ⊢ m ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → m : I ⟶ ℕ 0
488 487 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
489 133 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
490 nn0cn ⊢ q ∈ ℕ 0 → q ∈ ℂ
491 nn0cn ⊢ r ∈ ℕ 0 → r ∈ ℂ
492 nn0cn ⊢ s ∈ ℕ 0 → s ∈ ℂ
493 addsubass ⊢ q ∈ ℂ ∧ r ∈ ℂ ∧ s ∈ ℂ → q + r - s = q + r - s
494 490 491 492 493 syl3an ⊢ q ∈ ℕ 0 ∧ r ∈ ℕ 0 ∧ s ∈ ℕ 0 → q + r - s = q + r - s
495 494 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
496 486 488 489 489 495 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
497 simpr ⊢ φ ∧ i ∈ I → i ∈ I
498 58 a1i ⊢ φ ∧ i ∈ I → if i = X 1 0 ∈ ℕ 0
499 70 78 497 498 fvmptd3 ⊢ φ ∧ i ∈ I → y ∈ I ⟼ if y = X 1 0 ⁡ i = if i = X 1 0
500 135 135 17 17 74 499 499 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
501 500 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
502 501 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
503 239 subidi ⊢ if i = X 1 0 − if i = X 1 0 = 0
504 503 mpteq2i ⊢ i ∈ I ⟼ if i = X 1 0 − if i = X 1 0 = i ∈ I ⟼ 0
505 fconstmpt ⊢ I × 0 = i ∈ I ⟼ 0
506 504 505 eqtr4i ⊢ i ∈ I ⟼ if i = X 1 0 − if i = X 1 0 = I × 0
507 506 oveq2i ⊢ m + f i ∈ I ⟼ if i = X 1 0 − if i = X 1 0 = m + f I × 0
508 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
509 490 addridd ⊢ q ∈ ℕ 0 → q + 0 = q
510 509 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
511 486 488 508 510 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
512 507 511 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
513 496 502 512 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
514 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
515 513 514 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
516 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
517 516 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
518 515 517 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
519 518 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
520 485 519 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
521 481 520 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
522 521 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
523 457 522 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
524 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
525 17 mptexd ⊢ φ → y ∈ I ⟼ if y = X 1 0 ∈ V
526 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
527 525 526 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
528 70 527 mpbiri ⊢ φ → y ∈ I ⟼ if y = X 1 0 ∈ y ∈ I ⟼ if y = X 1 0
529 528 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
530 449 mpofun ⊢ Fun ⁡ ∘ f ⁡ +
531 530 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 ⁡ +
532 xpss ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ⊆ V × V
533 472 inex1 ⊢ dom ⁡ m ∩ dom ⁡ n ∈ V
534 533 mptex ⊢ o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o ∈ V
535 534 rgen2w ⊢ ∀ m ∈ V ∀ n ∈ V o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o ∈ V
536 449 dmmpoga ⊢ ∀ m ∈ V ∀ n ∈ V o ∈ dom ⁡ m ∩ dom ⁡ n ⟼ m ⁡ o + n ⁡ o ∈ V → dom ⁡ ∘ f ⁡ + = V × V
537 535 536 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
538 532 537 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 ⁡ +
539 524 529 531 538 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
540 17 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
541 elrabi ⊢ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → v ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin
542 18 psrbagf ⊢ v ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → v : I ⟶ ℕ 0
543 541 542 syl ⊢ v ∈ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d → v : I ⟶ ℕ 0
544 543 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
545 133 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
546 494 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
547 540 544 545 545 546 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
548 135 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
549 80 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
550 548 548 540 540 74 549 549 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
551 550 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
552 506 oveq2i ⊢ v + f i ∈ I ⟼ if i = X 1 0 − if i = X 1 0 = v + f I × 0
553 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
554 nn0cn ⊢ p ∈ ℕ 0 → p ∈ ℂ
555 554 addridd ⊢ p ∈ ℕ 0 → p + 0 = p
556 555 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
557 540 544 553 556 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
558 552 557 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
559 547 551 558 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
560 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
561 560 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
562 559 561 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
563 20 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
564 18 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
565 458 563 564 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
566 18 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
567 565 566 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
568 567 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
569 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
570 568 569 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
571 485 570 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
572 481 571 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
573 572 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
574 457 573 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
575 574 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
576 575 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
577 576 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
578 239 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
579 577 578 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
580 579 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
581 575 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
582 581 548 540 540 74 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
583 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
584 581 548 540 540 74 583 549 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
585 582 548 540 540 74 584 549 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
586 575 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
587 580 585 586 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
588 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
589 588 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
590 587 589 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
591 562 590 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
592 448 523 539 591 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
593 9 106 13 428 442 447 592 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
594 555 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
595 486 488 508 594 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
596 507 595 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
597 496 502 596 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
598 597 514 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
599 598 517 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
600 599 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
601 485 600 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
602 481 601 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
603 602 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
604 457 603 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
605 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
606 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
607 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
608 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
609 608 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
610 607 609 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
611 604 605 606 610 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
612 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 → X ∈ I
613 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 → 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
615 604 614 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 18 612 613 615 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
617 574 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
618 133 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 17 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
621 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
622 iftrue ⊢ y = X → if y = X 1 0 = 1
623 1ex ⊢ 1 ∈ V
624 622 70 623 fvmpt ⊢ X ∈ I → y ∈ I ⟼ if y = X 1 0 ⁡ X = 1
625 624 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
626 617 619 620 620 74 621 625 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
627 612 626 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
628 627 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
629 nn0sscn ⊢ ℕ 0 ⊆ ℂ
630 629 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
631 574 630 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
632 631 612 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
633 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
634 632 633 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
635 628 634 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
636 617 619 620 620 74 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
637 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
638 80 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
639 617 619 620 620 74 637 638 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
640 574 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
641 640 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
642 239 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
643 641 642 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
644 620 636 619 617 639 638 643 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
645 644 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
646 635 645 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
647 616 646 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
648 30 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
649 648 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
650 649 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
651 650 641 642 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
652 651 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
653 67 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
654 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
655 653 636 620 620 74 654 639 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
656 653 619 620 620 74 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
657 653 619 620 620 74 654 638 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
658 656 617 620 620 74 657 637 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
659 652 655 658 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
660 659 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
661 647 660 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
662 11 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
663 574 612 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
664 663 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
665 39 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
666 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
667 20 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
668 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
669 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
670 18 245 669 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
671 666 667 668 670 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
672 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
673 671 672 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
674 485 673 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
675 481 674 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
676 675 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
677 457 676 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
678 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
679 677 678 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
680 665 679 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
681 44 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
682 23 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
683 18 669 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
684 682 677 683 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
685 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
686 684 685 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
687 681 686 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
688 9 26 37 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
689 662 664 680 687 688 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
690 661 689 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
691 690 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
692 611 691 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
693 692 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
694 snex ⊢ y ∈ I ⟼ if y = X 1 0 ∈ V
695 357 694 xpex ⊢ k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∈ V
696 695 funimaex ⊢ Fun ⁡ ∘ f ⁡ + → ∘ f ⁡ + k ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin | k ≤ f d × y ∈ I ⟼ if y = X 1 0 ∈ V
697 530 696 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
698 28 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
699 9 37 662 680 687 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
700 9 26 698 663 699 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
701 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
702 361 701 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
703 702 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
704 703 25 115 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
705 462 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
706 469 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
707 472 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
708 475 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
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 ⁡ m → m ⁡ o = m ⁡ o
710 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
711 705 706 707 708 477 709 710 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
712 711 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
713 712 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
714 20 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
715 oveq2 ⊢ n = y ∈ I ⟼ if y = X 1 0 → m + f n = m + f y ∈ I ⟼ if y = X 1 0
716 715 eqeq2d ⊢ n = y ∈ I ⟼ if y = X 1 0 → u = m + f n ↔ u = m + f y ∈ I ⟼ if y = X 1 0
717 716 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
718 714 717 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
719 713 718 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
720 719 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
721 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
722 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
723 fveq1 ⊢ k = m + f y ∈ I ⟼ if y = X 1 0 → k ⁡ X = m + f y ∈ I ⟼ if y = X 1 0 ⁡ X
724 723 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
725 722 724 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
726 725 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
727 721 726 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
728 458 714 564 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
729 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
730 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
731 18 245 46 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
732 729 714 730 731 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
733 721 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
734 733 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
735 732 734 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
736 6 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
737 487 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
738 737 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
739 135 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
740 17 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
741 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
742 624 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
743 738 739 740 740 74 741 742 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
744 736 743 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
745 737 736 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
746 nn0p1nn ⊢ m ⁡ X ∈ ℕ 0 → m ⁡ X + 1 ∈ ℕ
747 745 746 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
748 744 747 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
749 748 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
750 749 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
751 750 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
752 735 751 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
753 727 728 752 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
754 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
755 753 754 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
756 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
757 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
758 757 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
759 133 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
760 757 125 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
761 760 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
762 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 + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → X ∈ I
763 761 762 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
764 341 notbid ⊢ k = u → ¬ k ≤ f d ∧ k ⁡ X = 0 ↔ ¬ u ≤ f d ∧ u ⁡ X = 0
765 120 764 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
766 765 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
767 766 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
768 767 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
769 768 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
770 769 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
771 757 126 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
772 771 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
773 772 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
774 23 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
775 90 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
776 774 775 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
777 776 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
778 17 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
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 → u ⁡ i = u ⁡ i
780 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
781 773 777 778 778 74 779 780 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
782 770 781 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
783 782 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
784 783 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
785 67 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
786 71 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
787 17 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
788 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
789 80 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
790 785 786 787 787 74 788 789 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
791 790 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
792 160 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
793 792 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
794 31 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
795 794 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
796 795 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
797 796 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
798 797 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
799 791 793 798 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
800 784 799 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
801 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
802 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 + f y ∈ I ⟼ if y = X 1 0 ∧ ¬ k ≤ f d ∧ k ⁡ X = 0 → d : I ⟶ ℕ 0
803 802 762 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
804 803 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
805 804 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
806 801 805 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
807 806 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
808 177 800 807 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
809 808 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
810 67 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
811 810 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
812 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
813 773 811 778 778 74 779 812 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
814 809 813 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
815 814 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
816 767 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
817 816 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
818 imnan ⊢ u ≤ f d → ¬ u ⁡ X = 0 ↔ ¬ u ≤ f d ∧ u ⁡ X = 0
819 817 818 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
820 819 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
821 815 820 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
822 821 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
823 763 822 193 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
824 823 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
825 824 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
826 175 breq2d ⊢ i = X → 1 ≤ u ⁡ i ↔ 1 ≤ u ⁡ X
827 825 826 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
828 827 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
829 761 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
830 829 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
831 830 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
832 828 831 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
833 brif1 ⊢ if i = X 1 0 ≤ u ⁡ i ↔ if- i = X 1 ≤ u ⁡ i 0 ≤ u ⁡ i
834 832 833 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
835 834 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
836 71 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
837 17 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
838 80 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
839 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
840 836 772 837 837 74 838 839 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
841 835 840 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
842 18 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
843 758 759 841 842 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
844 843 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
845 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
846 810 836 837 837 74 845 838 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
847 772 776 837 837 74 839 846 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
848 769 847 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
849 848 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
850 829 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
851 62 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
852 802 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
853 852 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
854 850 851 853 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
855 849 854 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
856 855 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
857 772 836 837 837 74 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
858 772 836 837 837 74 839 838 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
859 857 810 837 837 74 858 845 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 y ∈ I ⟼ if y = X 1 0 ≤ f d ↔ ∀ i ∈ I u ⁡ i − if i = X 1 0 ≤ d ⁡ i
860 856 859 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
861 756 844 860 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
862 829 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
863 239 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
864 862 863 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
865 864 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
866 857 836 837 837 74 858 838 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
867 761 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
868 865 866 867 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
869 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
870 869 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
871 755 861 868 870 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
872 456 720 871 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
873 872 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
874 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
875 873 874 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
876 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
877 875 876 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
878 704 877 115 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
879 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
880 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 = ∅
881 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 = ∅
882 879 880 881 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 = ∅
883 882 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 = ∅
884 883 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 = ∅
885 281 101 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
886 875 885 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
887 9 106 10 13 697 700 878 884 886 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
888 693 887 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
889 427 593 888 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
890 417 adantr ⊢ φ ∧ d ∈ h ∈ ℕ 0 I | h -1 ℕ ∈ Fin → I mPSDer R ⁡ X ⁡ G ∈ B
891 1 2 37 4 18 387 890 14 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
892 8 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 18 287 892 249 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
894 269 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
898 311 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 9 26 37 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
900 226 898 230 273 899 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 9 10 13 223 323 277 284 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
905 891 903 904 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
906 421 423 424 424 425 889 905 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
907 419 906 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
908 411 413 907 3eqtr4d ⊢ φ → I mPSDer R ⁡ X ⁡ F · ˙ G = I mPSDer R ⁡ X ⁡ F · ˙ G + ˙ F · ˙ I mPSDer R ⁡ X ⁡ G