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