Step |
Hyp |
Ref |
Expression |
1 |
|
psrring.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
2 |
|
psrring.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑉 ) |
3 |
|
psrring.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
4 |
|
psr1cl.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
5 |
|
psr1cl.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
6 |
|
psr1cl.o |
⊢ 1 = ( 1r ‘ 𝑅 ) |
7 |
|
psr1cl.u |
⊢ 𝑈 = ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) |
8 |
|
psr1cl.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
9 |
|
psrlidm.t |
⊢ · = ( .r ‘ 𝑆 ) |
10 |
|
psrlidm.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
12 |
1 2 3 4 5 6 7 8
|
psr1cl |
⊢ ( 𝜑 → 𝑈 ∈ 𝐵 ) |
13 |
1 8 9 3 10 12
|
psrmulcl |
⊢ ( 𝜑 → ( 𝑋 · 𝑈 ) ∈ 𝐵 ) |
14 |
1 11 4 8 13
|
psrelbas |
⊢ ( 𝜑 → ( 𝑋 · 𝑈 ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
15 |
14
|
ffnd |
⊢ ( 𝜑 → ( 𝑋 · 𝑈 ) Fn 𝐷 ) |
16 |
1 11 4 8 10
|
psrelbas |
⊢ ( 𝜑 → 𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
17 |
16
|
ffnd |
⊢ ( 𝜑 → 𝑋 Fn 𝐷 ) |
18 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
19 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑋 ∈ 𝐵 ) |
20 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑈 ∈ 𝐵 ) |
21 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑦 ∈ 𝐷 ) |
22 |
1 8 18 9 4 19 20 21
|
psrmulval |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 · 𝑈 ) ‘ 𝑦 ) = ( 𝑅 Σg ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) ) |
23 |
|
breq1 |
⊢ ( 𝑔 = 𝑦 → ( 𝑔 ∘r ≤ 𝑦 ↔ 𝑦 ∘r ≤ 𝑦 ) ) |
24 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝐼 ∈ 𝑉 ) |
25 |
4
|
psrbagf |
⊢ ( 𝑦 ∈ 𝐷 → 𝑦 : 𝐼 ⟶ ℕ0 ) |
26 |
25
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑦 : 𝐼 ⟶ ℕ0 ) |
27 |
|
nn0re |
⊢ ( 𝑧 ∈ ℕ0 → 𝑧 ∈ ℝ ) |
28 |
27
|
leidd |
⊢ ( 𝑧 ∈ ℕ0 → 𝑧 ≤ 𝑧 ) |
29 |
28
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ℕ0 ) → 𝑧 ≤ 𝑧 ) |
30 |
24 26 29
|
caofref |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑦 ∘r ≤ 𝑦 ) |
31 |
23 21 30
|
elrabd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑦 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) |
32 |
31
|
snssd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → { 𝑦 } ⊆ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) |
33 |
32
|
resmptd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ↾ { 𝑦 } ) = ( 𝑧 ∈ { 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) |
34 |
33
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑅 Σg ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ↾ { 𝑦 } ) ) = ( 𝑅 Σg ( 𝑧 ∈ { 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) ) |
35 |
|
ringcmn |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ CMnd ) |
36 |
3 35
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ CMnd ) |
37 |
36
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑅 ∈ CMnd ) |
38 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
39 |
4 38
|
rab2ex |
⊢ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∈ V |
40 |
39
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∈ V ) |
41 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑅 ∈ Ring ) |
42 |
16
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
43 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) |
44 |
|
breq1 |
⊢ ( 𝑔 = 𝑧 → ( 𝑔 ∘r ≤ 𝑦 ↔ 𝑧 ∘r ≤ 𝑦 ) ) |
45 |
44
|
elrab |
⊢ ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↔ ( 𝑧 ∈ 𝐷 ∧ 𝑧 ∘r ≤ 𝑦 ) ) |
46 |
43 45
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( 𝑧 ∈ 𝐷 ∧ 𝑧 ∘r ≤ 𝑦 ) ) |
47 |
46
|
simpld |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑧 ∈ 𝐷 ) |
48 |
42 47
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( 𝑋 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) ) |
49 |
1 11 4 8 20
|
psrelbas |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑈 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
50 |
49
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑈 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
51 |
21
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑦 ∈ 𝐷 ) |
52 |
4
|
psrbagf |
⊢ ( 𝑧 ∈ 𝐷 → 𝑧 : 𝐼 ⟶ ℕ0 ) |
53 |
47 52
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑧 : 𝐼 ⟶ ℕ0 ) |
54 |
46
|
simprd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑧 ∘r ≤ 𝑦 ) |
55 |
4
|
psrbagcon |
⊢ ( ( 𝑦 ∈ 𝐷 ∧ 𝑧 : 𝐼 ⟶ ℕ0 ∧ 𝑧 ∘r ≤ 𝑦 ) → ( ( 𝑦 ∘f − 𝑧 ) ∈ 𝐷 ∧ ( 𝑦 ∘f − 𝑧 ) ∘r ≤ 𝑦 ) ) |
56 |
51 53 54 55
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( ( 𝑦 ∘f − 𝑧 ) ∈ 𝐷 ∧ ( 𝑦 ∘f − 𝑧 ) ∘r ≤ 𝑦 ) ) |
57 |
56
|
simpld |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( 𝑦 ∘f − 𝑧 ) ∈ 𝐷 ) |
58 |
50 57
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ∈ ( Base ‘ 𝑅 ) ) |
59 |
11 18
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ∈ ( Base ‘ 𝑅 ) ) |
60 |
41 48 58 59
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ∈ ( Base ‘ 𝑅 ) ) |
61 |
60
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) : { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ⟶ ( Base ‘ 𝑅 ) ) |
62 |
|
eldifi |
⊢ ( 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) → 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) |
63 |
62 57
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( 𝑦 ∘f − 𝑧 ) ∈ 𝐷 ) |
64 |
|
eqeq1 |
⊢ ( 𝑥 = ( 𝑦 ∘f − 𝑧 ) → ( 𝑥 = ( 𝐼 × { 0 } ) ↔ ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) ) ) |
65 |
64
|
ifbid |
⊢ ( 𝑥 = ( 𝑦 ∘f − 𝑧 ) → if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) = if ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) ) |
66 |
6
|
fvexi |
⊢ 1 ∈ V |
67 |
5
|
fvexi |
⊢ 0 ∈ V |
68 |
66 67
|
ifex |
⊢ if ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) ∈ V |
69 |
65 7 68
|
fvmpt |
⊢ ( ( 𝑦 ∘f − 𝑧 ) ∈ 𝐷 → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) = if ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) ) |
70 |
63 69
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) = if ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) ) |
71 |
|
eldifsni |
⊢ ( 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) → 𝑧 ≠ 𝑦 ) |
72 |
71
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → 𝑧 ≠ 𝑦 ) |
73 |
72
|
necomd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → 𝑦 ≠ 𝑧 ) |
74 |
24
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝐼 ∈ 𝑉 ) |
75 |
|
nn0sscn |
⊢ ℕ0 ⊆ ℂ |
76 |
|
fss |
⊢ ( ( 𝑦 : 𝐼 ⟶ ℕ0 ∧ ℕ0 ⊆ ℂ ) → 𝑦 : 𝐼 ⟶ ℂ ) |
77 |
26 75 76
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑦 : 𝐼 ⟶ ℂ ) |
78 |
77
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑦 : 𝐼 ⟶ ℂ ) |
79 |
|
fss |
⊢ ( ( 𝑧 : 𝐼 ⟶ ℕ0 ∧ ℕ0 ⊆ ℂ ) → 𝑧 : 𝐼 ⟶ ℂ ) |
80 |
53 75 79
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑧 : 𝐼 ⟶ ℂ ) |
81 |
|
ofsubeq0 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑦 : 𝐼 ⟶ ℂ ∧ 𝑧 : 𝐼 ⟶ ℂ ) → ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑧 ) ) |
82 |
74 78 80 81
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑧 ) ) |
83 |
62 82
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑧 ) ) |
84 |
83
|
necon3bbid |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( ¬ ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 ≠ 𝑧 ) ) |
85 |
73 84
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ¬ ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) ) |
86 |
85
|
iffalsed |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → if ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) = 0 ) |
87 |
70 86
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) = 0 ) |
88 |
87
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) = ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) 0 ) ) |
89 |
11 18 5
|
ringrz |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) 0 ) = 0 ) |
90 |
41 48 89
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) 0 ) = 0 ) |
91 |
62 90
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) 0 ) = 0 ) |
92 |
88 91
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) = 0 ) |
93 |
92 40
|
suppss2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) supp 0 ) ⊆ { 𝑦 } ) |
94 |
40
|
mptexd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ∈ V ) |
95 |
|
funmpt |
⊢ Fun ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) |
96 |
95
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → Fun ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) |
97 |
67
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 0 ∈ V ) |
98 |
|
snfi |
⊢ { 𝑦 } ∈ Fin |
99 |
98
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → { 𝑦 } ∈ Fin ) |
100 |
|
suppssfifsupp |
⊢ ( ( ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ∈ V ∧ Fun ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ∧ 0 ∈ V ) ∧ ( { 𝑦 } ∈ Fin ∧ ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) supp 0 ) ⊆ { 𝑦 } ) ) → ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) finSupp 0 ) |
101 |
94 96 97 99 93 100
|
syl32anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) finSupp 0 ) |
102 |
11 5 37 40 61 93 101
|
gsumres |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑅 Σg ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ↾ { 𝑦 } ) ) = ( 𝑅 Σg ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) ) |
103 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑅 ∈ Ring ) |
104 |
|
ringmnd |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Mnd ) |
105 |
103 104
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑅 ∈ Mnd ) |
106 |
|
eqid |
⊢ 𝑦 = 𝑦 |
107 |
|
ofsubeq0 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑦 : 𝐼 ⟶ ℂ ∧ 𝑦 : 𝐼 ⟶ ℂ ) → ( ( 𝑦 ∘f − 𝑦 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑦 ) ) |
108 |
24 77 77 107
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑦 ∘f − 𝑦 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑦 ) ) |
109 |
106 108
|
mpbiri |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑦 ∘f − 𝑦 ) = ( 𝐼 × { 0 } ) ) |
110 |
109
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) = ( 𝑈 ‘ ( 𝐼 × { 0 } ) ) ) |
111 |
|
fconstmpt |
⊢ ( 𝐼 × { 0 } ) = ( 𝑤 ∈ 𝐼 ↦ 0 ) |
112 |
4
|
fczpsrbag |
⊢ ( 𝐼 ∈ 𝑉 → ( 𝑤 ∈ 𝐼 ↦ 0 ) ∈ 𝐷 ) |
113 |
2 112
|
syl |
⊢ ( 𝜑 → ( 𝑤 ∈ 𝐼 ↦ 0 ) ∈ 𝐷 ) |
114 |
111 113
|
eqeltrid |
⊢ ( 𝜑 → ( 𝐼 × { 0 } ) ∈ 𝐷 ) |
115 |
114
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝐼 × { 0 } ) ∈ 𝐷 ) |
116 |
|
iftrue |
⊢ ( 𝑥 = ( 𝐼 × { 0 } ) → if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) = 1 ) |
117 |
116 7 66
|
fvmpt |
⊢ ( ( 𝐼 × { 0 } ) ∈ 𝐷 → ( 𝑈 ‘ ( 𝐼 × { 0 } ) ) = 1 ) |
118 |
115 117
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑈 ‘ ( 𝐼 × { 0 } ) ) = 1 ) |
119 |
110 118
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) = 1 ) |
120 |
119
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) = ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) 1 ) ) |
121 |
16
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑋 ‘ 𝑦 ) ∈ ( Base ‘ 𝑅 ) ) |
122 |
11 18 6
|
ringridm |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ‘ 𝑦 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) 1 ) = ( 𝑋 ‘ 𝑦 ) ) |
123 |
103 121 122
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) 1 ) = ( 𝑋 ‘ 𝑦 ) ) |
124 |
120 123
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) = ( 𝑋 ‘ 𝑦 ) ) |
125 |
124 121
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) ∈ ( Base ‘ 𝑅 ) ) |
126 |
|
fveq2 |
⊢ ( 𝑧 = 𝑦 → ( 𝑋 ‘ 𝑧 ) = ( 𝑋 ‘ 𝑦 ) ) |
127 |
|
oveq2 |
⊢ ( 𝑧 = 𝑦 → ( 𝑦 ∘f − 𝑧 ) = ( 𝑦 ∘f − 𝑦 ) ) |
128 |
127
|
fveq2d |
⊢ ( 𝑧 = 𝑦 → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) = ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) |
129 |
126 128
|
oveq12d |
⊢ ( 𝑧 = 𝑦 → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) = ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) ) |
130 |
11 129
|
gsumsn |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝑦 ∈ 𝐷 ∧ ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) ∈ ( Base ‘ 𝑅 ) ) → ( 𝑅 Σg ( 𝑧 ∈ { 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) = ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) ) |
131 |
105 21 125 130
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑅 Σg ( 𝑧 ∈ { 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) = ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) ) |
132 |
34 102 131
|
3eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑅 Σg ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) = ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) ) |
133 |
22 132 124
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 · 𝑈 ) ‘ 𝑦 ) = ( 𝑋 ‘ 𝑦 ) ) |
134 |
15 17 133
|
eqfnfvd |
⊢ ( 𝜑 → ( 𝑋 · 𝑈 ) = 𝑋 ) |