Step |
Hyp |
Ref |
Expression |
1 |
|
evlslem2.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
2 |
|
evlslem2.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
3 |
|
evlslem2.m |
⊢ · = ( .r ‘ 𝑆 ) |
4 |
|
evlslem2.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
5 |
|
evlslem2.d |
⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
6 |
|
evlslem2.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
7 |
|
evlslem2.r |
⊢ ( 𝜑 → 𝑅 ∈ CRing ) |
8 |
|
evlslem2.s |
⊢ ( 𝜑 → 𝑆 ∈ CRing ) |
9 |
|
evlslem2.e1 |
⊢ ( 𝜑 → 𝐸 ∈ ( 𝑃 GrpHom 𝑆 ) ) |
10 |
|
evlslem2.e2 |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷 ) ) ) → ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = ( 𝑗 ∘f + 𝑖 ) , ( ( 𝑥 ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑦 ‘ 𝑖 ) ) , 0 ) ) ) = ( ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) · ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) |
11 |
|
eqid |
⊢ ( .r ‘ 𝑃 ) = ( .r ‘ 𝑃 ) |
12 |
|
eqid |
⊢ ( 0g ‘ 𝑃 ) = ( 0g ‘ 𝑃 ) |
13 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
14 |
5 13
|
rabex2 |
⊢ 𝐷 ∈ V |
15 |
14
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝐷 ∈ V ) |
16 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
17 |
7 16
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
18 |
1 6 17
|
mplringd |
⊢ ( 𝜑 → 𝑃 ∈ Ring ) |
19 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑃 ∈ Ring ) |
20 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
21 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑗 ∈ 𝐷 ) → 𝐼 ∈ 𝑊 ) |
22 |
17
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑗 ∈ 𝐷 ) → 𝑅 ∈ Ring ) |
23 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑥 ∈ 𝐵 ) |
24 |
1 20 2 5 23
|
mplelf |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑥 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
25 |
24
|
ffvelcdmda |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑗 ∈ 𝐷 ) → ( 𝑥 ‘ 𝑗 ) ∈ ( Base ‘ 𝑅 ) ) |
26 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑗 ∈ 𝐷 ) → 𝑗 ∈ 𝐷 ) |
27 |
1 5 4 20 21 22 2 25 26
|
mplmon2cl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑗 ∈ 𝐷 ) → ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ∈ 𝐵 ) |
28 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑖 ∈ 𝐷 ) → 𝐼 ∈ 𝑊 ) |
29 |
17
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑖 ∈ 𝐷 ) → 𝑅 ∈ Ring ) |
30 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑦 ∈ 𝐵 ) |
31 |
1 20 2 5 30
|
mplelf |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑦 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
32 |
31
|
ffvelcdmda |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑖 ∈ 𝐷 ) → ( 𝑦 ‘ 𝑖 ) ∈ ( Base ‘ 𝑅 ) ) |
33 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑖 ∈ 𝐷 ) → 𝑖 ∈ 𝐷 ) |
34 |
1 5 4 20 28 29 2 32 33
|
mplmon2cl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑖 ∈ 𝐷 ) → ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ∈ 𝐵 ) |
35 |
14
|
mptex |
⊢ ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) ∈ V |
36 |
|
funmpt |
⊢ Fun ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) |
37 |
|
fvex |
⊢ ( 0g ‘ 𝑃 ) ∈ V |
38 |
35 36 37
|
3pm3.2i |
⊢ ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) ∈ V ∧ Fun ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) ∧ ( 0g ‘ 𝑃 ) ∈ V ) |
39 |
38
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) ∈ V ∧ Fun ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) ∧ ( 0g ‘ 𝑃 ) ∈ V ) ) |
40 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ 𝐵 ) |
41 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝑅 ∈ CRing ) |
42 |
1 2 4 40 41
|
mplelsfi |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝑦 finSupp 0 ) |
43 |
42
|
fsuppimpd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑦 supp 0 ) ∈ Fin ) |
44 |
1 20 2 5 40
|
mplelf |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝑦 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
45 |
|
ssidd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑦 supp 0 ) ⊆ ( 𝑦 supp 0 ) ) |
46 |
14
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝐷 ∈ V ) |
47 |
4
|
fvexi |
⊢ 0 ∈ V |
48 |
47
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 0 ∈ V ) |
49 |
44 45 46 48
|
suppssr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑗 ∈ ( 𝐷 ∖ ( 𝑦 supp 0 ) ) ) → ( 𝑦 ‘ 𝑗 ) = 0 ) |
50 |
49
|
ifeq1d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑗 ∈ ( 𝐷 ∖ ( 𝑦 supp 0 ) ) ) → if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) = if ( 𝑘 = 𝑗 , 0 , 0 ) ) |
51 |
|
ifid |
⊢ if ( 𝑘 = 𝑗 , 0 , 0 ) = 0 |
52 |
50 51
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑗 ∈ ( 𝐷 ∖ ( 𝑦 supp 0 ) ) ) → if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) = 0 ) |
53 |
52
|
mpteq2dv |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑗 ∈ ( 𝐷 ∖ ( 𝑦 supp 0 ) ) ) → ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) = ( 𝑘 ∈ 𝐷 ↦ 0 ) ) |
54 |
|
ringgrp |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp ) |
55 |
17 54
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Grp ) |
56 |
1 5 4 12 6 55
|
mpl0 |
⊢ ( 𝜑 → ( 0g ‘ 𝑃 ) = ( 𝐷 × { 0 } ) ) |
57 |
|
fconstmpt |
⊢ ( 𝐷 × { 0 } ) = ( 𝑘 ∈ 𝐷 ↦ 0 ) |
58 |
56 57
|
eqtrdi |
⊢ ( 𝜑 → ( 0g ‘ 𝑃 ) = ( 𝑘 ∈ 𝐷 ↦ 0 ) ) |
59 |
58
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑗 ∈ ( 𝐷 ∖ ( 𝑦 supp 0 ) ) ) → ( 0g ‘ 𝑃 ) = ( 𝑘 ∈ 𝐷 ↦ 0 ) ) |
60 |
53 59
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑗 ∈ ( 𝐷 ∖ ( 𝑦 supp 0 ) ) ) → ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) = ( 0g ‘ 𝑃 ) ) |
61 |
60 46
|
suppss2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ⊆ ( 𝑦 supp 0 ) ) |
62 |
|
suppssfifsupp |
⊢ ( ( ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) ∈ V ∧ Fun ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) ∧ ( 0g ‘ 𝑃 ) ∈ V ) ∧ ( ( 𝑦 supp 0 ) ∈ Fin ∧ ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ⊆ ( 𝑦 supp 0 ) ) ) → ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) finSupp ( 0g ‘ 𝑃 ) ) |
63 |
39 43 61 62
|
syl12anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) finSupp ( 0g ‘ 𝑃 ) ) |
64 |
63
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐵 ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) finSupp ( 0g ‘ 𝑃 ) ) |
65 |
|
fveq1 |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ‘ 𝑗 ) = ( 𝑥 ‘ 𝑗 ) ) |
66 |
65
|
ifeq1d |
⊢ ( 𝑦 = 𝑥 → if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) = if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) |
67 |
66
|
mpteq2dv |
⊢ ( 𝑦 = 𝑥 → ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) = ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) |
68 |
67
|
mpteq2dv |
⊢ ( 𝑦 = 𝑥 → ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) = ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) |
69 |
68
|
breq1d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) finSupp ( 0g ‘ 𝑃 ) ↔ ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) finSupp ( 0g ‘ 𝑃 ) ) ) |
70 |
69
|
cbvralvw |
⊢ ( ∀ 𝑦 ∈ 𝐵 ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) finSupp ( 0g ‘ 𝑃 ) ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) finSupp ( 0g ‘ 𝑃 ) ) |
71 |
64 70
|
sylib |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) finSupp ( 0g ‘ 𝑃 ) ) |
72 |
71
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) finSupp ( 0g ‘ 𝑃 ) ) |
73 |
72
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) finSupp ( 0g ‘ 𝑃 ) ) |
74 |
|
equequ2 |
⊢ ( 𝑖 = 𝑗 → ( 𝑘 = 𝑖 ↔ 𝑘 = 𝑗 ) ) |
75 |
|
fveq2 |
⊢ ( 𝑖 = 𝑗 → ( 𝑦 ‘ 𝑖 ) = ( 𝑦 ‘ 𝑗 ) ) |
76 |
74 75
|
ifbieq1d |
⊢ ( 𝑖 = 𝑗 → if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) = if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) |
77 |
76
|
mpteq2dv |
⊢ ( 𝑖 = 𝑗 → ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) = ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) |
78 |
77
|
cbvmptv |
⊢ ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) = ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) |
79 |
63
|
adantrl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑦 ‘ 𝑗 ) , 0 ) ) ) finSupp ( 0g ‘ 𝑃 ) ) |
80 |
78 79
|
eqbrtrid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) finSupp ( 0g ‘ 𝑃 ) ) |
81 |
2 11 12 15 15 19 27 34 73 80
|
gsumdixp |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑃 Σg ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ( .r ‘ 𝑃 ) ( 𝑃 Σg ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) = ( 𝑃 Σg ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) |
82 |
81
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐸 ‘ ( ( 𝑃 Σg ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ( .r ‘ 𝑃 ) ( 𝑃 Σg ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) = ( 𝐸 ‘ ( 𝑃 Σg ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) ) |
83 |
|
ringcmn |
⊢ ( 𝑃 ∈ Ring → 𝑃 ∈ CMnd ) |
84 |
18 83
|
syl |
⊢ ( 𝜑 → 𝑃 ∈ CMnd ) |
85 |
84
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑃 ∈ CMnd ) |
86 |
|
crngring |
⊢ ( 𝑆 ∈ CRing → 𝑆 ∈ Ring ) |
87 |
8 86
|
syl |
⊢ ( 𝜑 → 𝑆 ∈ Ring ) |
88 |
87
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑆 ∈ Ring ) |
89 |
|
ringmnd |
⊢ ( 𝑆 ∈ Ring → 𝑆 ∈ Mnd ) |
90 |
88 89
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑆 ∈ Mnd ) |
91 |
14 14
|
xpex |
⊢ ( 𝐷 × 𝐷 ) ∈ V |
92 |
91
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐷 × 𝐷 ) ∈ V ) |
93 |
|
ghmmhm |
⊢ ( 𝐸 ∈ ( 𝑃 GrpHom 𝑆 ) → 𝐸 ∈ ( 𝑃 MndHom 𝑆 ) ) |
94 |
9 93
|
syl |
⊢ ( 𝜑 → 𝐸 ∈ ( 𝑃 MndHom 𝑆 ) ) |
95 |
94
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝐸 ∈ ( 𝑃 MndHom 𝑆 ) ) |
96 |
18
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ ( 𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷 ) ) → 𝑃 ∈ Ring ) |
97 |
27
|
adantrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ ( 𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷 ) ) → ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ∈ 𝐵 ) |
98 |
34
|
adantrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ ( 𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷 ) ) → ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ∈ 𝐵 ) |
99 |
2 11
|
ringcl |
⊢ ( ( 𝑃 ∈ Ring ∧ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ∈ 𝐵 ∧ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ∈ 𝐵 ) → ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ∈ 𝐵 ) |
100 |
96 97 98 99
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ ( 𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷 ) ) → ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ∈ 𝐵 ) |
101 |
100
|
ralrimivva |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ∀ 𝑗 ∈ 𝐷 ∀ 𝑖 ∈ 𝐷 ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ∈ 𝐵 ) |
102 |
|
eqid |
⊢ ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) = ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) |
103 |
102
|
fmpo |
⊢ ( ∀ 𝑗 ∈ 𝐷 ∀ 𝑖 ∈ 𝐷 ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ∈ 𝐵 ↔ ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) : ( 𝐷 × 𝐷 ) ⟶ 𝐵 ) |
104 |
101 103
|
sylib |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) : ( 𝐷 × 𝐷 ) ⟶ 𝐵 ) |
105 |
14 14
|
mpoex |
⊢ ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ∈ V |
106 |
102
|
mpofun |
⊢ Fun ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) |
107 |
105 106 37
|
3pm3.2i |
⊢ ( ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ∈ V ∧ Fun ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ∧ ( 0g ‘ 𝑃 ) ∈ V ) |
108 |
107
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ∈ V ∧ Fun ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ∧ ( 0g ‘ 𝑃 ) ∈ V ) ) |
109 |
73
|
fsuppimpd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ∈ Fin ) |
110 |
80
|
fsuppimpd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ∈ Fin ) |
111 |
|
xpfi |
⊢ ( ( ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ∈ Fin ∧ ( ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ∈ Fin ) → ( ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) × ( ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ) ∈ Fin ) |
112 |
109 110 111
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) × ( ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ) ∈ Fin ) |
113 |
2 12 11 19 27 34 15 15
|
evlslem4 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) supp ( 0g ‘ 𝑃 ) ) ⊆ ( ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) × ( ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ) ) |
114 |
|
suppssfifsupp |
⊢ ( ( ( ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ∈ V ∧ Fun ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ∧ ( 0g ‘ 𝑃 ) ∈ V ) ∧ ( ( ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) × ( ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ) ∈ Fin ∧ ( ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) supp ( 0g ‘ 𝑃 ) ) ⊆ ( ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) × ( ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ) ) ) → ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) finSupp ( 0g ‘ 𝑃 ) ) |
115 |
108 112 113 114
|
syl12anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) finSupp ( 0g ‘ 𝑃 ) ) |
116 |
2 12 85 90 92 95 104 115
|
gsummhm |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑆 Σg ( 𝐸 ∘ ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) = ( 𝐸 ‘ ( 𝑃 Σg ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) ) |
117 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ ( 𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷 ) ) → 𝐼 ∈ 𝑊 ) |
118 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ ( 𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷 ) ) → 𝑅 ∈ CRing ) |
119 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
120 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ ( 𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷 ) ) → 𝑗 ∈ 𝐷 ) |
121 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ ( 𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷 ) ) → 𝑖 ∈ 𝐷 ) |
122 |
25
|
adantrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ ( 𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷 ) ) → ( 𝑥 ‘ 𝑗 ) ∈ ( Base ‘ 𝑅 ) ) |
123 |
32
|
adantrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ ( 𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷 ) ) → ( 𝑦 ‘ 𝑖 ) ∈ ( Base ‘ 𝑅 ) ) |
124 |
1 5 4 20 117 118 11 119 120 121 122 123
|
mplmon2mul |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ ( 𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷 ) ) → ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) = ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = ( 𝑗 ∘f + 𝑖 ) , ( ( 𝑥 ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑦 ‘ 𝑖 ) ) , 0 ) ) ) |
125 |
124
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ ( 𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷 ) ) → ( 𝐸 ‘ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) = ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = ( 𝑗 ∘f + 𝑖 ) , ( ( 𝑥 ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑦 ‘ 𝑖 ) ) , 0 ) ) ) ) |
126 |
10
|
anassrs |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ ( 𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷 ) ) → ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = ( 𝑗 ∘f + 𝑖 ) , ( ( 𝑥 ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑦 ‘ 𝑖 ) ) , 0 ) ) ) = ( ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) · ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) |
127 |
125 126
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ ( 𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷 ) ) → ( 𝐸 ‘ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) = ( ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) · ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) |
128 |
127
|
3impb |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷 ) → ( 𝐸 ‘ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) = ( ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) · ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) |
129 |
128
|
mpoeq3dva |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) = ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) · ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) |
130 |
129
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑆 Σg ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) = ( 𝑆 Σg ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) · ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) ) |
131 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) = ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) |
132 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
133 |
2 132
|
ghmf |
⊢ ( 𝐸 ∈ ( 𝑃 GrpHom 𝑆 ) → 𝐸 : 𝐵 ⟶ ( Base ‘ 𝑆 ) ) |
134 |
9 133
|
syl |
⊢ ( 𝜑 → 𝐸 : 𝐵 ⟶ ( Base ‘ 𝑆 ) ) |
135 |
134
|
feqmptd |
⊢ ( 𝜑 → 𝐸 = ( 𝑧 ∈ 𝐵 ↦ ( 𝐸 ‘ 𝑧 ) ) ) |
136 |
135
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝐸 = ( 𝑧 ∈ 𝐵 ↦ ( 𝐸 ‘ 𝑧 ) ) ) |
137 |
|
fveq2 |
⊢ ( 𝑧 = ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) → ( 𝐸 ‘ 𝑧 ) = ( 𝐸 ‘ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) |
138 |
100 131 136 137
|
fmpoco |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐸 ∘ ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) = ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) |
139 |
138
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑆 Σg ( 𝐸 ∘ ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) = ( 𝑆 Σg ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) ) |
140 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) = ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) |
141 |
|
fveq2 |
⊢ ( 𝑧 = ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) → ( 𝐸 ‘ 𝑧 ) = ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) |
142 |
27 140 136 141
|
fmptco |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐸 ∘ ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) = ( 𝑗 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ) |
143 |
142
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑆 Σg ( 𝐸 ∘ ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ) = ( 𝑆 Σg ( 𝑗 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ) ) |
144 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) = ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) |
145 |
|
fveq2 |
⊢ ( 𝑧 = ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) → ( 𝐸 ‘ 𝑧 ) = ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) |
146 |
34 144 136 145
|
fmptco |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐸 ∘ ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) = ( 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) |
147 |
146
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑆 Σg ( 𝐸 ∘ ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) = ( 𝑆 Σg ( 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) |
148 |
143 147
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑆 Σg ( 𝐸 ∘ ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ) · ( 𝑆 Σg ( 𝐸 ∘ ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) = ( ( 𝑆 Σg ( 𝑗 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ) · ( 𝑆 Σg ( 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) ) |
149 |
|
eqid |
⊢ ( 0g ‘ 𝑆 ) = ( 0g ‘ 𝑆 ) |
150 |
134
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑗 ∈ 𝐷 ) → 𝐸 : 𝐵 ⟶ ( Base ‘ 𝑆 ) ) |
151 |
150 27
|
ffvelcdmd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑗 ∈ 𝐷 ) → ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ∈ ( Base ‘ 𝑆 ) ) |
152 |
134
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑖 ∈ 𝐷 ) → 𝐸 : 𝐵 ⟶ ( Base ‘ 𝑆 ) ) |
153 |
152 34
|
ffvelcdmd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑖 ∈ 𝐷 ) → ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ∈ ( Base ‘ 𝑆 ) ) |
154 |
14
|
mptex |
⊢ ( 𝑗 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ∈ V |
155 |
|
funmpt |
⊢ Fun ( 𝑗 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) |
156 |
|
fvex |
⊢ ( 0g ‘ 𝑆 ) ∈ V |
157 |
154 155 156
|
3pm3.2i |
⊢ ( ( 𝑗 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ∈ V ∧ Fun ( 𝑗 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ∧ ( 0g ‘ 𝑆 ) ∈ V ) |
158 |
157
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑗 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ∈ V ∧ Fun ( 𝑗 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ∧ ( 0g ‘ 𝑆 ) ∈ V ) ) |
159 |
|
ssidd |
⊢ ( 𝜑 → ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ⊆ ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ) |
160 |
12 149
|
ghmid |
⊢ ( 𝐸 ∈ ( 𝑃 GrpHom 𝑆 ) → ( 𝐸 ‘ ( 0g ‘ 𝑃 ) ) = ( 0g ‘ 𝑆 ) ) |
161 |
9 160
|
syl |
⊢ ( 𝜑 → ( 𝐸 ‘ ( 0g ‘ 𝑃 ) ) = ( 0g ‘ 𝑆 ) ) |
162 |
14
|
mptex |
⊢ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ∈ V |
163 |
162
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐷 ) → ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ∈ V ) |
164 |
37
|
a1i |
⊢ ( 𝜑 → ( 0g ‘ 𝑃 ) ∈ V ) |
165 |
159 161 163 164
|
suppssfv |
⊢ ( 𝜑 → ( ( 𝑗 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) supp ( 0g ‘ 𝑆 ) ) ⊆ ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ) |
166 |
165
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑗 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) supp ( 0g ‘ 𝑆 ) ) ⊆ ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ) |
167 |
|
suppssfifsupp |
⊢ ( ( ( ( 𝑗 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ∈ V ∧ Fun ( 𝑗 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ∧ ( 0g ‘ 𝑆 ) ∈ V ) ∧ ( ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ∈ Fin ∧ ( ( 𝑗 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) supp ( 0g ‘ 𝑆 ) ) ⊆ ( ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ) ) → ( 𝑗 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) finSupp ( 0g ‘ 𝑆 ) ) |
168 |
158 109 166 167
|
syl12anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑗 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) finSupp ( 0g ‘ 𝑆 ) ) |
169 |
14
|
mptex |
⊢ ( 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ∈ V |
170 |
|
funmpt |
⊢ Fun ( 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) |
171 |
169 170 156
|
3pm3.2i |
⊢ ( ( 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ∈ V ∧ Fun ( 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ∧ ( 0g ‘ 𝑆 ) ∈ V ) |
172 |
171
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ∈ V ∧ Fun ( 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ∧ ( 0g ‘ 𝑆 ) ∈ V ) ) |
173 |
|
ssidd |
⊢ ( 𝜑 → ( ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ⊆ ( ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ) |
174 |
14
|
mptex |
⊢ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ∈ V |
175 |
174
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐷 ) → ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ∈ V ) |
176 |
173 161 175 164
|
suppssfv |
⊢ ( 𝜑 → ( ( 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) supp ( 0g ‘ 𝑆 ) ) ⊆ ( ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ) |
177 |
176
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) supp ( 0g ‘ 𝑆 ) ) ⊆ ( ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ) |
178 |
|
suppssfifsupp |
⊢ ( ( ( ( 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ∈ V ∧ Fun ( 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ∧ ( 0g ‘ 𝑆 ) ∈ V ) ∧ ( ( ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ∈ Fin ∧ ( ( 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) supp ( 0g ‘ 𝑆 ) ) ⊆ ( ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) supp ( 0g ‘ 𝑃 ) ) ) ) → ( 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) finSupp ( 0g ‘ 𝑆 ) ) |
179 |
172 110 177 178
|
syl12anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) finSupp ( 0g ‘ 𝑆 ) ) |
180 |
132 3 149 15 15 88 151 153 168 179
|
gsumdixp |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑆 Σg ( 𝑗 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ) · ( 𝑆 Σg ( 𝑖 ∈ 𝐷 ↦ ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) = ( 𝑆 Σg ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) · ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) ) |
181 |
148 180
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑆 Σg ( 𝐸 ∘ ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ) · ( 𝑆 Σg ( 𝐸 ∘ ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) = ( 𝑆 Σg ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) · ( 𝐸 ‘ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) ) |
182 |
130 139 181
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑆 Σg ( 𝐸 ∘ ( 𝑗 ∈ 𝐷 , 𝑖 ∈ 𝐷 ↦ ( ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ( .r ‘ 𝑃 ) ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) = ( ( 𝑆 Σg ( 𝐸 ∘ ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ) · ( 𝑆 Σg ( 𝐸 ∘ ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) ) |
183 |
82 116 182
|
3eqtr2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐸 ‘ ( ( 𝑃 Σg ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ( .r ‘ 𝑃 ) ( 𝑃 Σg ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) = ( ( 𝑆 Σg ( 𝐸 ∘ ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ) · ( 𝑆 Σg ( 𝐸 ∘ ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) ) |
184 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝐼 ∈ 𝑊 ) |
185 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑅 ∈ Ring ) |
186 |
1 5 4 2 184 185 23
|
mplcoe4 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑥 = ( 𝑃 Σg ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ) |
187 |
1 5 4 2 184 185 30
|
mplcoe4 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑦 = ( 𝑃 Σg ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) |
188 |
186 187
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) = ( ( 𝑃 Σg ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ( .r ‘ 𝑃 ) ( 𝑃 Σg ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) |
189 |
188
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐸 ‘ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ) = ( 𝐸 ‘ ( ( 𝑃 Σg ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ( .r ‘ 𝑃 ) ( 𝑃 Σg ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) ) |
190 |
186
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐸 ‘ 𝑥 ) = ( 𝐸 ‘ ( 𝑃 Σg ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ) ) |
191 |
27
|
fmpttd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) : 𝐷 ⟶ 𝐵 ) |
192 |
2 12 85 90 15 95 191 73
|
gsummhm |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑆 Σg ( 𝐸 ∘ ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ) = ( 𝐸 ‘ ( 𝑃 Σg ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ) ) |
193 |
190 192
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐸 ‘ 𝑥 ) = ( 𝑆 Σg ( 𝐸 ∘ ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ) ) |
194 |
187
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐸 ‘ 𝑦 ) = ( 𝐸 ‘ ( 𝑃 Σg ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) |
195 |
34
|
fmpttd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) : 𝐷 ⟶ 𝐵 ) |
196 |
2 12 85 90 15 95 195 80
|
gsummhm |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑆 Σg ( 𝐸 ∘ ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) = ( 𝐸 ‘ ( 𝑃 Σg ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) |
197 |
194 196
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐸 ‘ 𝑦 ) = ( 𝑆 Σg ( 𝐸 ∘ ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) |
198 |
193 197
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝐸 ‘ 𝑥 ) · ( 𝐸 ‘ 𝑦 ) ) = ( ( 𝑆 Σg ( 𝐸 ∘ ( 𝑗 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑗 , ( 𝑥 ‘ 𝑗 ) , 0 ) ) ) ) ) · ( 𝑆 Σg ( 𝐸 ∘ ( 𝑖 ∈ 𝐷 ↦ ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = 𝑖 , ( 𝑦 ‘ 𝑖 ) , 0 ) ) ) ) ) ) ) |
199 |
183 189 198
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐸 ‘ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ) = ( ( 𝐸 ‘ 𝑥 ) · ( 𝐸 ‘ 𝑦 ) ) ) |