Step | Hyp | Ref
| Expression |
1 | | evlslem1.b |
. . 3
⊢ 𝐵 = (Base‘𝑃) |
2 | | eqid 2622 |
. . 3
⊢
(1r‘𝑃) = (1r‘𝑃) |
3 | | eqid 2622 |
. . 3
⊢
(1r‘𝑆) = (1r‘𝑆) |
4 | | eqid 2622 |
. . 3
⊢
(.r‘𝑃) = (.r‘𝑃) |
5 | | evlslem1.m |
. . 3
⊢ · =
(.r‘𝑆) |
6 | | evlslem1.i |
. . . 4
⊢ (𝜑 → 𝐼 ∈ V) |
7 | | evlslem1.r |
. . . . 5
⊢ (𝜑 → 𝑅 ∈ CRing) |
8 | | crngring 18558 |
. . . . 5
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) |
9 | 7, 8 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑅 ∈ Ring) |
10 | | evlslem1.p |
. . . . 5
⊢ 𝑃 = (𝐼 mPoly 𝑅) |
11 | 10 | mplring 19452 |
. . . 4
⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring) |
12 | 6, 9, 11 | syl2anc 693 |
. . 3
⊢ (𝜑 → 𝑃 ∈ Ring) |
13 | | evlslem1.s |
. . . 4
⊢ (𝜑 → 𝑆 ∈ CRing) |
14 | | crngring 18558 |
. . . 4
⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) |
15 | 13, 14 | syl 17 |
. . 3
⊢ (𝜑 → 𝑆 ∈ Ring) |
16 | | fveq2 6191 |
. . . . . . 7
⊢ (𝑥 = (1r‘𝑅) → (𝐴‘𝑥) = (𝐴‘(1r‘𝑅))) |
17 | 16 | fveq2d 6195 |
. . . . . 6
⊢ (𝑥 = (1r‘𝑅) → (𝐸‘(𝐴‘𝑥)) = (𝐸‘(𝐴‘(1r‘𝑅)))) |
18 | | fveq2 6191 |
. . . . . 6
⊢ (𝑥 = (1r‘𝑅) → (𝐹‘𝑥) = (𝐹‘(1r‘𝑅))) |
19 | 17, 18 | eqeq12d 2637 |
. . . . 5
⊢ (𝑥 = (1r‘𝑅) → ((𝐸‘(𝐴‘𝑥)) = (𝐹‘𝑥) ↔ (𝐸‘(𝐴‘(1r‘𝑅))) = (𝐹‘(1r‘𝑅)))) |
20 | | evlslem1.d |
. . . . . . . . 9
⊢ 𝐷 = {ℎ ∈ (ℕ0
↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
21 | | eqid 2622 |
. . . . . . . . 9
⊢
(0g‘𝑅) = (0g‘𝑅) |
22 | | evlslem1.k |
. . . . . . . . 9
⊢ 𝐾 = (Base‘𝑅) |
23 | | evlslem1.a |
. . . . . . . . 9
⊢ 𝐴 = (algSc‘𝑃) |
24 | 6 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝐼 ∈ V) |
25 | 9 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑅 ∈ Ring) |
26 | | simpr 477 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ 𝐾) |
27 | 10, 20, 21, 22, 23, 24, 25, 26 | mplascl 19496 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐴‘𝑥) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝑅)))) |
28 | 27 | fveq2d 6195 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐸‘(𝐴‘𝑥)) = (𝐸‘(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝑅))))) |
29 | | evlslem1.c |
. . . . . . . 8
⊢ 𝐶 = (Base‘𝑆) |
30 | | evlslem1.t |
. . . . . . . 8
⊢ 𝑇 = (mulGrp‘𝑆) |
31 | | evlslem1.x |
. . . . . . . 8
⊢ ↑ =
(.g‘𝑇) |
32 | | evlslem1.v |
. . . . . . . 8
⊢ 𝑉 = (𝐼 mVar 𝑅) |
33 | | evlslem1.e |
. . . . . . . 8
⊢ 𝐸 = (𝑝 ∈ 𝐵 ↦ (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))))) |
34 | 7 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑅 ∈ CRing) |
35 | 13 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑆 ∈ CRing) |
36 | | evlslem1.f |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
37 | 36 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
38 | | evlslem1.g |
. . . . . . . . 9
⊢ (𝜑 → 𝐺:𝐼⟶𝐶) |
39 | 38 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝐺:𝐼⟶𝐶) |
40 | 20 | psrbag0 19494 |
. . . . . . . . . 10
⊢ (𝐼 ∈ V → (𝐼 × {0}) ∈ 𝐷) |
41 | 6, 40 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝐼 × {0}) ∈ 𝐷) |
42 | 41 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐼 × {0}) ∈ 𝐷) |
43 | 10, 1, 29, 22, 20, 30, 31, 5, 32, 33, 24, 34, 35, 37, 39, 21, 42, 26 | evlslem3 19514 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐸‘(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝑅)))) = ((𝐹‘𝑥) · (𝑇 Σg ((𝐼 × {0})
∘𝑓 ↑ 𝐺)))) |
44 | | 0zd 11389 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 0 ∈ ℤ) |
45 | | fvexd 6203 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ V) |
46 | | fconstmpt 5163 |
. . . . . . . . . . . . . . 15
⊢ (𝐼 × {0}) = (𝑥 ∈ 𝐼 ↦ 0) |
47 | 46 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐼 × {0}) = (𝑥 ∈ 𝐼 ↦ 0)) |
48 | 38 | feqmptd 6249 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐼 ↦ (𝐺‘𝑥))) |
49 | 6, 44, 45, 47, 48 | offval2 6914 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐼 × {0}) ∘𝑓
↑
𝐺) = (𝑥 ∈ 𝐼 ↦ (0 ↑ (𝐺‘𝑥)))) |
50 | 38 | ffvelrnda 6359 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ 𝐶) |
51 | 30, 29 | mgpbas 18495 |
. . . . . . . . . . . . . . . 16
⊢ 𝐶 = (Base‘𝑇) |
52 | 30, 3 | ringidval 18503 |
. . . . . . . . . . . . . . . 16
⊢
(1r‘𝑆) = (0g‘𝑇) |
53 | 51, 52, 31 | mulg0 17546 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺‘𝑥) ∈ 𝐶 → (0 ↑ (𝐺‘𝑥)) = (1r‘𝑆)) |
54 | 50, 53 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (0 ↑ (𝐺‘𝑥)) = (1r‘𝑆)) |
55 | 54 | mpteq2dva 4744 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (0 ↑ (𝐺‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (1r‘𝑆))) |
56 | 49, 55 | eqtrd 2656 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐼 × {0}) ∘𝑓
↑
𝐺) = (𝑥 ∈ 𝐼 ↦ (1r‘𝑆))) |
57 | 56 | oveq2d 6666 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑇 Σg ((𝐼 × {0})
∘𝑓 ↑ 𝐺)) = (𝑇 Σg (𝑥 ∈ 𝐼 ↦ (1r‘𝑆)))) |
58 | 30 | crngmgp 18555 |
. . . . . . . . . . . . . 14
⊢ (𝑆 ∈ CRing → 𝑇 ∈ CMnd) |
59 | 13, 58 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑇 ∈ CMnd) |
60 | | cmnmnd 18208 |
. . . . . . . . . . . . 13
⊢ (𝑇 ∈ CMnd → 𝑇 ∈ Mnd) |
61 | 59, 60 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑇 ∈ Mnd) |
62 | 52 | gsumz 17374 |
. . . . . . . . . . . 12
⊢ ((𝑇 ∈ Mnd ∧ 𝐼 ∈ V) → (𝑇 Σg
(𝑥 ∈ 𝐼 ↦ (1r‘𝑆))) = (1r‘𝑆)) |
63 | 61, 6, 62 | syl2anc 693 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑇 Σg (𝑥 ∈ 𝐼 ↦ (1r‘𝑆))) = (1r‘𝑆)) |
64 | 57, 63 | eqtrd 2656 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑇 Σg ((𝐼 × {0})
∘𝑓 ↑ 𝐺)) = (1r‘𝑆)) |
65 | 64 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝑇 Σg ((𝐼 × {0})
∘𝑓 ↑ 𝐺)) = (1r‘𝑆)) |
66 | 65 | oveq2d 6666 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝐹‘𝑥) · (𝑇 Σg ((𝐼 × {0})
∘𝑓 ↑ 𝐺))) = ((𝐹‘𝑥) ·
(1r‘𝑆))) |
67 | 22, 29 | rhmf 18726 |
. . . . . . . . . . 11
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:𝐾⟶𝐶) |
68 | 36, 67 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:𝐾⟶𝐶) |
69 | 68 | ffvelrnda 6359 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐹‘𝑥) ∈ 𝐶) |
70 | 29, 5, 3 | ringridm 18572 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Ring ∧ (𝐹‘𝑥) ∈ 𝐶) → ((𝐹‘𝑥) ·
(1r‘𝑆)) =
(𝐹‘𝑥)) |
71 | 15, 69, 70 | syl2an2r 876 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝐹‘𝑥) ·
(1r‘𝑆)) =
(𝐹‘𝑥)) |
72 | 66, 71 | eqtrd 2656 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝐹‘𝑥) · (𝑇 Σg ((𝐼 × {0})
∘𝑓 ↑ 𝐺))) = (𝐹‘𝑥)) |
73 | 28, 43, 72 | 3eqtrd 2660 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐸‘(𝐴‘𝑥)) = (𝐹‘𝑥)) |
74 | 73 | ralrimiva 2966 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ 𝐾 (𝐸‘(𝐴‘𝑥)) = (𝐹‘𝑥)) |
75 | | eqid 2622 |
. . . . . . 7
⊢
(1r‘𝑅) = (1r‘𝑅) |
76 | 22, 75 | ringidcl 18568 |
. . . . . 6
⊢ (𝑅 ∈ Ring →
(1r‘𝑅)
∈ 𝐾) |
77 | 9, 76 | syl 17 |
. . . . 5
⊢ (𝜑 → (1r‘𝑅) ∈ 𝐾) |
78 | 19, 74, 77 | rspcdva 3316 |
. . . 4
⊢ (𝜑 → (𝐸‘(𝐴‘(1r‘𝑅))) = (𝐹‘(1r‘𝑅))) |
79 | 10 | mplassa 19454 |
. . . . . . . . 9
⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ CRing) → 𝑃 ∈ AssAlg) |
80 | 6, 7, 79 | syl2anc 693 |
. . . . . . . 8
⊢ (𝜑 → 𝑃 ∈ AssAlg) |
81 | | eqid 2622 |
. . . . . . . . 9
⊢
(Scalar‘𝑃) =
(Scalar‘𝑃) |
82 | 23, 81 | asclrhm 19342 |
. . . . . . . 8
⊢ (𝑃 ∈ AssAlg → 𝐴 ∈ ((Scalar‘𝑃) RingHom 𝑃)) |
83 | 80, 82 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ((Scalar‘𝑃) RingHom 𝑃)) |
84 | 10, 6, 7 | mplsca 19445 |
. . . . . . . 8
⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) |
85 | 84 | oveq1d 6665 |
. . . . . . 7
⊢ (𝜑 → (𝑅 RingHom 𝑃) = ((Scalar‘𝑃) RingHom 𝑃)) |
86 | 83, 85 | eleqtrrd 2704 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ (𝑅 RingHom 𝑃)) |
87 | 75, 2 | rhm1 18730 |
. . . . . 6
⊢ (𝐴 ∈ (𝑅 RingHom 𝑃) → (𝐴‘(1r‘𝑅)) = (1r‘𝑃)) |
88 | 86, 87 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝐴‘(1r‘𝑅)) = (1r‘𝑃)) |
89 | 88 | fveq2d 6195 |
. . . 4
⊢ (𝜑 → (𝐸‘(𝐴‘(1r‘𝑅))) = (𝐸‘(1r‘𝑃))) |
90 | 75, 3 | rhm1 18730 |
. . . . 5
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆)) |
91 | 36, 90 | syl 17 |
. . . 4
⊢ (𝜑 → (𝐹‘(1r‘𝑅)) = (1r‘𝑆)) |
92 | 78, 89, 91 | 3eqtr3d 2664 |
. . 3
⊢ (𝜑 → (𝐸‘(1r‘𝑃)) = (1r‘𝑆)) |
93 | | eqid 2622 |
. . . . 5
⊢
(+g‘𝑃) = (+g‘𝑃) |
94 | | eqid 2622 |
. . . . 5
⊢
(+g‘𝑆) = (+g‘𝑆) |
95 | | ringgrp 18552 |
. . . . . 6
⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp) |
96 | 12, 95 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑃 ∈ Grp) |
97 | | ringgrp 18552 |
. . . . . 6
⊢ (𝑆 ∈ Ring → 𝑆 ∈ Grp) |
98 | 15, 97 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑆 ∈ Grp) |
99 | | eqid 2622 |
. . . . . . 7
⊢
(0g‘𝑆) = (0g‘𝑆) |
100 | | ringcmn 18581 |
. . . . . . . . 9
⊢ (𝑆 ∈ Ring → 𝑆 ∈ CMnd) |
101 | 15, 100 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑆 ∈ CMnd) |
102 | 101 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝑆 ∈ CMnd) |
103 | | ovex 6678 |
. . . . . . . . 9
⊢
(ℕ0 ↑𝑚 𝐼) ∈ V |
104 | 20, 103 | rabex2 4815 |
. . . . . . . 8
⊢ 𝐷 ∈ V |
105 | 104 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐷 ∈ V) |
106 | 6 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐼 ∈ V) |
107 | 7 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝑅 ∈ CRing) |
108 | 13 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝑆 ∈ CRing) |
109 | 36 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
110 | 38 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐺:𝐼⟶𝐶) |
111 | | simpr 477 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝑝 ∈ 𝐵) |
112 | 10, 1, 29, 22, 20, 30, 31, 5, 32, 33, 106, 107, 108, 109, 110, 111 | evlslem6 19513 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))):𝐷⟶𝐶 ∧ (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))) finSupp
(0g‘𝑆))) |
113 | 112 | simpld 475 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))):𝐷⟶𝐶) |
114 | 112 | simprd 479 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))) finSupp
(0g‘𝑆)) |
115 | 29, 99, 102, 105, 113, 114 | gsumcl 18316 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) ∈ 𝐶) |
116 | 115, 33 | fmptd 6385 |
. . . . 5
⊢ (𝜑 → 𝐸:𝐵⟶𝐶) |
117 | | eqid 2622 |
. . . . . . . . . . . . . . . . 17
⊢
(+g‘𝑅) = (+g‘𝑅) |
118 | | simplrl 800 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → 𝑥 ∈ 𝐵) |
119 | | simplrr 801 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → 𝑦 ∈ 𝐵) |
120 | 10, 1, 117, 93, 118, 119 | mpladd 19442 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → (𝑥(+g‘𝑃)𝑦) = (𝑥 ∘𝑓
(+g‘𝑅)𝑦)) |
121 | 120 | fveq1d 6193 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → ((𝑥(+g‘𝑃)𝑦)‘𝑏) = ((𝑥 ∘𝑓
(+g‘𝑅)𝑦)‘𝑏)) |
122 | | simprl 794 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵) |
123 | 10, 22, 1, 20, 122 | mplelf 19433 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥:𝐷⟶𝐾) |
124 | 123 | ffnd 6046 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 Fn 𝐷) |
125 | 124 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → 𝑥 Fn 𝐷) |
126 | | simprr 796 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵) |
127 | 10, 22, 1, 20, 126 | mplelf 19433 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦:𝐷⟶𝐾) |
128 | 127 | ffnd 6046 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 Fn 𝐷) |
129 | 128 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → 𝑦 Fn 𝐷) |
130 | 104 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → 𝐷 ∈ V) |
131 | | simpr 477 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ 𝐷) |
132 | | fnfvof 6911 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 Fn 𝐷 ∧ 𝑦 Fn 𝐷) ∧ (𝐷 ∈ V ∧ 𝑏 ∈ 𝐷)) → ((𝑥 ∘𝑓
(+g‘𝑅)𝑦)‘𝑏) = ((𝑥‘𝑏)(+g‘𝑅)(𝑦‘𝑏))) |
133 | 125, 129,
130, 131, 132 | syl22anc 1327 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → ((𝑥 ∘𝑓
(+g‘𝑅)𝑦)‘𝑏) = ((𝑥‘𝑏)(+g‘𝑅)(𝑦‘𝑏))) |
134 | 121, 133 | eqtrd 2656 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → ((𝑥(+g‘𝑃)𝑦)‘𝑏) = ((𝑥‘𝑏)(+g‘𝑅)(𝑦‘𝑏))) |
135 | 134 | fveq2d 6195 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → (𝐹‘((𝑥(+g‘𝑃)𝑦)‘𝑏)) = (𝐹‘((𝑥‘𝑏)(+g‘𝑅)(𝑦‘𝑏)))) |
136 | | rhmghm 18725 |
. . . . . . . . . . . . . . . 16
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) |
137 | 36, 136 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆)) |
138 | 137 | ad2antrr 762 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) |
139 | 123 | ffvelrnda 6359 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → (𝑥‘𝑏) ∈ 𝐾) |
140 | 127 | ffvelrnda 6359 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → (𝑦‘𝑏) ∈ 𝐾) |
141 | 22, 117, 94 | ghmlin 17665 |
. . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (𝑥‘𝑏) ∈ 𝐾 ∧ (𝑦‘𝑏) ∈ 𝐾) → (𝐹‘((𝑥‘𝑏)(+g‘𝑅)(𝑦‘𝑏))) = ((𝐹‘(𝑥‘𝑏))(+g‘𝑆)(𝐹‘(𝑦‘𝑏)))) |
142 | 138, 139,
140, 141 | syl3anc 1326 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → (𝐹‘((𝑥‘𝑏)(+g‘𝑅)(𝑦‘𝑏))) = ((𝐹‘(𝑥‘𝑏))(+g‘𝑆)(𝐹‘(𝑦‘𝑏)))) |
143 | 135, 142 | eqtrd 2656 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → (𝐹‘((𝑥(+g‘𝑃)𝑦)‘𝑏)) = ((𝐹‘(𝑥‘𝑏))(+g‘𝑆)(𝐹‘(𝑦‘𝑏)))) |
144 | 143 | oveq1d 6665 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → ((𝐹‘((𝑥(+g‘𝑃)𝑦)‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))) = (((𝐹‘(𝑥‘𝑏))(+g‘𝑆)(𝐹‘(𝑦‘𝑏))) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))) |
145 | 15 | ad2antrr 762 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → 𝑆 ∈ Ring) |
146 | 68 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → 𝐹:𝐾⟶𝐶) |
147 | 146, 139 | ffvelrnd 6360 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑥‘𝑏)) ∈ 𝐶) |
148 | 146, 140 | ffvelrnd 6360 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑦‘𝑏)) ∈ 𝐶) |
149 | 59 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → 𝑇 ∈ CMnd) |
150 | 38 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → 𝐺:𝐼⟶𝐶) |
151 | 6 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → 𝐼 ∈ V) |
152 | 20, 51, 31, 52, 149, 131, 150, 151 | psrbagev2 19511 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)) ∈ 𝐶) |
153 | 29, 94, 5 | ringdir 18567 |
. . . . . . . . . . . 12
⊢ ((𝑆 ∈ Ring ∧ ((𝐹‘(𝑥‘𝑏)) ∈ 𝐶 ∧ (𝐹‘(𝑦‘𝑏)) ∈ 𝐶 ∧ (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)) ∈ 𝐶)) → (((𝐹‘(𝑥‘𝑏))(+g‘𝑆)(𝐹‘(𝑦‘𝑏))) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))) = (((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))(+g‘𝑆)((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) |
154 | 145, 147,
148, 152, 153 | syl13anc 1328 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → (((𝐹‘(𝑥‘𝑏))(+g‘𝑆)(𝐹‘(𝑦‘𝑏))) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))) = (((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))(+g‘𝑆)((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) |
155 | 144, 154 | eqtrd 2656 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → ((𝐹‘((𝑥(+g‘𝑃)𝑦)‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))) = (((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))(+g‘𝑆)((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) |
156 | 155 | mpteq2dva 4744 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑏 ∈ 𝐷 ↦ ((𝐹‘((𝑥(+g‘𝑃)𝑦)‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))) = (𝑏 ∈ 𝐷 ↦ (((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))(+g‘𝑆)((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))))) |
157 | 104 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐷 ∈ V) |
158 | | ovexd 6680 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))) ∈
V) |
159 | | ovexd 6680 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐷) → ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))) ∈
V) |
160 | | eqidd 2623 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))) = (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) |
161 | | eqidd 2623 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))) = (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) |
162 | 157, 158,
159, 160, 161 | offval2 6914 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))
∘𝑓 (+g‘𝑆)(𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) = (𝑏 ∈ 𝐷 ↦ (((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))(+g‘𝑆)((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))))) |
163 | 156, 162 | eqtr4d 2659 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑏 ∈ 𝐷 ↦ ((𝐹‘((𝑥(+g‘𝑃)𝑦)‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))) = ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))
∘𝑓 (+g‘𝑆)(𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))))) |
164 | 163 | oveq2d 6666 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘((𝑥(+g‘𝑃)𝑦)‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) = (𝑆 Σg ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))
∘𝑓 (+g‘𝑆)(𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))))) |
165 | 101 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑆 ∈ CMnd) |
166 | 6 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐼 ∈ V) |
167 | 7 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ CRing) |
168 | 13 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑆 ∈ CRing) |
169 | 36 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
170 | 38 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺:𝐼⟶𝐶) |
171 | 10, 1, 29, 22, 20, 30, 31, 5, 32, 33, 166, 167, 168, 169, 170, 122 | evlslem6 19513 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))):𝐷⟶𝐶 ∧ (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))) finSupp
(0g‘𝑆))) |
172 | 171 | simpld 475 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))):𝐷⟶𝐶) |
173 | 10, 1, 29, 22, 20, 30, 31, 5, 32, 33, 166, 167, 168, 169, 170, 126 | evlslem6 19513 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))):𝐷⟶𝐶 ∧ (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))) finSupp
(0g‘𝑆))) |
174 | 173 | simpld 475 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))):𝐷⟶𝐶) |
175 | 171 | simprd 479 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))) finSupp
(0g‘𝑆)) |
176 | 173 | simprd 479 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))) finSupp
(0g‘𝑆)) |
177 | 29, 99, 94, 165, 157, 172, 174, 175, 176 | gsumadd 18323 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σg ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))
∘𝑓 (+g‘𝑆)(𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))))) = ((𝑆 Σg
(𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))))(+g‘𝑆)(𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))))) |
178 | 164, 177 | eqtrd 2656 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘((𝑥(+g‘𝑃)𝑦)‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) = ((𝑆 Σg
(𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))))(+g‘𝑆)(𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))))) |
179 | 96 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃 ∈ Grp) |
180 | 1, 93 | grpcl 17430 |
. . . . . . . 8
⊢ ((𝑃 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝑃)𝑦) ∈ 𝐵) |
181 | 179, 122,
126, 180 | syl3anc 1326 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑃)𝑦) ∈ 𝐵) |
182 | | fveq1 6190 |
. . . . . . . . . . . 12
⊢ (𝑝 = (𝑥(+g‘𝑃)𝑦) → (𝑝‘𝑏) = ((𝑥(+g‘𝑃)𝑦)‘𝑏)) |
183 | 182 | fveq2d 6195 |
. . . . . . . . . . 11
⊢ (𝑝 = (𝑥(+g‘𝑃)𝑦) → (𝐹‘(𝑝‘𝑏)) = (𝐹‘((𝑥(+g‘𝑃)𝑦)‘𝑏))) |
184 | 183 | oveq1d 6665 |
. . . . . . . . . 10
⊢ (𝑝 = (𝑥(+g‘𝑃)𝑦) → ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))) = ((𝐹‘((𝑥(+g‘𝑃)𝑦)‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))) |
185 | 184 | mpteq2dv 4745 |
. . . . . . . . 9
⊢ (𝑝 = (𝑥(+g‘𝑃)𝑦) → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))) = (𝑏 ∈ 𝐷 ↦ ((𝐹‘((𝑥(+g‘𝑃)𝑦)‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) |
186 | 185 | oveq2d 6666 |
. . . . . . . 8
⊢ (𝑝 = (𝑥(+g‘𝑃)𝑦) → (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) = (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘((𝑥(+g‘𝑃)𝑦)‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))))) |
187 | | ovex 6678 |
. . . . . . . 8
⊢ (𝑆 Σg
(𝑏 ∈ 𝐷 ↦ ((𝐹‘((𝑥(+g‘𝑃)𝑦)‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) ∈
V |
188 | 186, 33, 187 | fvmpt 6282 |
. . . . . . 7
⊢ ((𝑥(+g‘𝑃)𝑦) ∈ 𝐵 → (𝐸‘(𝑥(+g‘𝑃)𝑦)) = (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘((𝑥(+g‘𝑃)𝑦)‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))))) |
189 | 181, 188 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘(𝑥(+g‘𝑃)𝑦)) = (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘((𝑥(+g‘𝑃)𝑦)‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))))) |
190 | | fveq1 6190 |
. . . . . . . . . . . . 13
⊢ (𝑝 = 𝑥 → (𝑝‘𝑏) = (𝑥‘𝑏)) |
191 | 190 | fveq2d 6195 |
. . . . . . . . . . . 12
⊢ (𝑝 = 𝑥 → (𝐹‘(𝑝‘𝑏)) = (𝐹‘(𝑥‘𝑏))) |
192 | 191 | oveq1d 6665 |
. . . . . . . . . . 11
⊢ (𝑝 = 𝑥 → ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))) = ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))) |
193 | 192 | mpteq2dv 4745 |
. . . . . . . . . 10
⊢ (𝑝 = 𝑥 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))) = (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) |
194 | 193 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝑝 = 𝑥 → (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) = (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))))) |
195 | | ovex 6678 |
. . . . . . . . 9
⊢ (𝑆 Σg
(𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) ∈
V |
196 | 194, 33, 195 | fvmpt 6282 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐵 → (𝐸‘𝑥) = (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))))) |
197 | 122, 196 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘𝑥) = (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))))) |
198 | | fveq1 6190 |
. . . . . . . . . . . . 13
⊢ (𝑝 = 𝑦 → (𝑝‘𝑏) = (𝑦‘𝑏)) |
199 | 198 | fveq2d 6195 |
. . . . . . . . . . . 12
⊢ (𝑝 = 𝑦 → (𝐹‘(𝑝‘𝑏)) = (𝐹‘(𝑦‘𝑏))) |
200 | 199 | oveq1d 6665 |
. . . . . . . . . . 11
⊢ (𝑝 = 𝑦 → ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))) = ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))) |
201 | 200 | mpteq2dv 4745 |
. . . . . . . . . 10
⊢ (𝑝 = 𝑦 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))) = (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) |
202 | 201 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝑝 = 𝑦 → (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) = (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))))) |
203 | | ovex 6678 |
. . . . . . . . 9
⊢ (𝑆 Σg
(𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) ∈
V |
204 | 202, 33, 203 | fvmpt 6282 |
. . . . . . . 8
⊢ (𝑦 ∈ 𝐵 → (𝐸‘𝑦) = (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))))) |
205 | 204 | ad2antll 765 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘𝑦) = (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))))) |
206 | 197, 205 | oveq12d 6668 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐸‘𝑥)(+g‘𝑆)(𝐸‘𝑦)) = ((𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑥‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺)))))(+g‘𝑆)(𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑦‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))))) |
207 | 178, 189,
206 | 3eqtr4d 2666 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘(𝑥(+g‘𝑃)𝑦)) = ((𝐸‘𝑥)(+g‘𝑆)(𝐸‘𝑦))) |
208 | 1, 29, 93, 94, 96, 98, 116, 207 | isghmd 17669 |
. . . 4
⊢ (𝜑 → 𝐸 ∈ (𝑃 GrpHom 𝑆)) |
209 | | eqid 2622 |
. . . . . . . . . . 11
⊢
(mulGrp‘𝑅) =
(mulGrp‘𝑅) |
210 | 209, 30 | rhmmhm 18722 |
. . . . . . . . . 10
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((mulGrp‘𝑅) MndHom 𝑇)) |
211 | 36, 210 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 ∈ ((mulGrp‘𝑅) MndHom 𝑇)) |
212 | 211 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → 𝐹 ∈ ((mulGrp‘𝑅) MndHom 𝑇)) |
213 | | simprll 802 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → 𝑥 ∈ 𝐵) |
214 | 10, 22, 1, 20, 213 | mplelf 19433 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → 𝑥:𝐷⟶𝐾) |
215 | | simprrl 804 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → 𝑧 ∈ 𝐷) |
216 | 214, 215 | ffvelrnd 6360 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → (𝑥‘𝑧) ∈ 𝐾) |
217 | | simprlr 803 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → 𝑦 ∈ 𝐵) |
218 | 10, 22, 1, 20, 217 | mplelf 19433 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → 𝑦:𝐷⟶𝐾) |
219 | | simprrr 805 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → 𝑤 ∈ 𝐷) |
220 | 218, 219 | ffvelrnd 6360 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → (𝑦‘𝑤) ∈ 𝐾) |
221 | 209, 22 | mgpbas 18495 |
. . . . . . . . 9
⊢ 𝐾 =
(Base‘(mulGrp‘𝑅)) |
222 | | eqid 2622 |
. . . . . . . . . 10
⊢
(.r‘𝑅) = (.r‘𝑅) |
223 | 209, 222 | mgpplusg 18493 |
. . . . . . . . 9
⊢
(.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
224 | 30, 5 | mgpplusg 18493 |
. . . . . . . . 9
⊢ · =
(+g‘𝑇) |
225 | 221, 223,
224 | mhmlin 17342 |
. . . . . . . 8
⊢ ((𝐹 ∈ ((mulGrp‘𝑅) MndHom 𝑇) ∧ (𝑥‘𝑧) ∈ 𝐾 ∧ (𝑦‘𝑤) ∈ 𝐾) → (𝐹‘((𝑥‘𝑧)(.r‘𝑅)(𝑦‘𝑤))) = ((𝐹‘(𝑥‘𝑧)) · (𝐹‘(𝑦‘𝑤)))) |
226 | 212, 216,
220, 225 | syl3anc 1326 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → (𝐹‘((𝑥‘𝑧)(.r‘𝑅)(𝑦‘𝑤))) = ((𝐹‘(𝑥‘𝑧)) · (𝐹‘(𝑦‘𝑤)))) |
227 | 61 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑣 ∈ 𝐼) → 𝑇 ∈ Mnd) |
228 | | simprl 794 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → 𝑧 ∈ 𝐷) |
229 | 20 | psrbagf 19365 |
. . . . . . . . . . . . . . 15
⊢ ((𝐼 ∈ V ∧ 𝑧 ∈ 𝐷) → 𝑧:𝐼⟶ℕ0) |
230 | 6, 228, 229 | syl2an2r 876 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → 𝑧:𝐼⟶ℕ0) |
231 | 230 | ffvelrnda 6359 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑣 ∈ 𝐼) → (𝑧‘𝑣) ∈
ℕ0) |
232 | | simprr 796 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → 𝑤 ∈ 𝐷) |
233 | 20 | psrbagf 19365 |
. . . . . . . . . . . . . . 15
⊢ ((𝐼 ∈ V ∧ 𝑤 ∈ 𝐷) → 𝑤:𝐼⟶ℕ0) |
234 | 6, 232, 233 | syl2an2r 876 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → 𝑤:𝐼⟶ℕ0) |
235 | 234 | ffvelrnda 6359 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑣 ∈ 𝐼) → (𝑤‘𝑣) ∈
ℕ0) |
236 | 38 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → 𝐺:𝐼⟶𝐶) |
237 | 236 | ffvelrnda 6359 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑣 ∈ 𝐼) → (𝐺‘𝑣) ∈ 𝐶) |
238 | 51, 31, 224 | mulgnn0dir 17571 |
. . . . . . . . . . . . 13
⊢ ((𝑇 ∈ Mnd ∧ ((𝑧‘𝑣) ∈ ℕ0 ∧ (𝑤‘𝑣) ∈ ℕ0 ∧ (𝐺‘𝑣) ∈ 𝐶)) → (((𝑧‘𝑣) + (𝑤‘𝑣)) ↑ (𝐺‘𝑣)) = (((𝑧‘𝑣) ↑ (𝐺‘𝑣)) · ((𝑤‘𝑣) ↑ (𝐺‘𝑣)))) |
239 | 227, 231,
235, 237, 238 | syl13anc 1328 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑣 ∈ 𝐼) → (((𝑧‘𝑣) + (𝑤‘𝑣)) ↑ (𝐺‘𝑣)) = (((𝑧‘𝑣) ↑ (𝐺‘𝑣)) · ((𝑤‘𝑣) ↑ (𝐺‘𝑣)))) |
240 | 239 | mpteq2dva 4744 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑣 ∈ 𝐼 ↦ (((𝑧‘𝑣) + (𝑤‘𝑣)) ↑ (𝐺‘𝑣))) = (𝑣 ∈ 𝐼 ↦ (((𝑧‘𝑣) ↑ (𝐺‘𝑣)) · ((𝑤‘𝑣) ↑ (𝐺‘𝑣))))) |
241 | 6 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → 𝐼 ∈ V) |
242 | | ovexd 6680 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑣 ∈ 𝐼) → ((𝑧‘𝑣) + (𝑤‘𝑣)) ∈ V) |
243 | | fvexd 6203 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑣 ∈ 𝐼) → (𝐺‘𝑣) ∈ V) |
244 | 230 | ffnd 6046 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → 𝑧 Fn 𝐼) |
245 | 234 | ffnd 6046 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → 𝑤 Fn 𝐼) |
246 | | inidm 3822 |
. . . . . . . . . . . . 13
⊢ (𝐼 ∩ 𝐼) = 𝐼 |
247 | | eqidd 2623 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑣 ∈ 𝐼) → (𝑧‘𝑣) = (𝑧‘𝑣)) |
248 | | eqidd 2623 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑣 ∈ 𝐼) → (𝑤‘𝑣) = (𝑤‘𝑣)) |
249 | 244, 245,
241, 241, 246, 247, 248 | offval 6904 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑧 ∘𝑓 + 𝑤) = (𝑣 ∈ 𝐼 ↦ ((𝑧‘𝑣) + (𝑤‘𝑣)))) |
250 | 38 | feqmptd 6249 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝐼 ↦ (𝐺‘𝑣))) |
251 | 250 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → 𝐺 = (𝑣 ∈ 𝐼 ↦ (𝐺‘𝑣))) |
252 | 241, 242,
243, 249, 251 | offval2 6914 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → ((𝑧 ∘𝑓 + 𝑤) ∘𝑓
↑
𝐺) = (𝑣 ∈ 𝐼 ↦ (((𝑧‘𝑣) + (𝑤‘𝑣)) ↑ (𝐺‘𝑣)))) |
253 | | ovexd 6680 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑣 ∈ 𝐼) → ((𝑧‘𝑣) ↑ (𝐺‘𝑣)) ∈ V) |
254 | | ovexd 6680 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑣 ∈ 𝐼) → ((𝑤‘𝑣) ↑ (𝐺‘𝑣)) ∈ V) |
255 | 38 | ffnd 6046 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐺 Fn 𝐼) |
256 | 255 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → 𝐺 Fn 𝐼) |
257 | | eqidd 2623 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑣 ∈ 𝐼) → (𝐺‘𝑣) = (𝐺‘𝑣)) |
258 | 244, 256,
241, 241, 246, 247, 257 | offval 6904 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑧 ∘𝑓 ↑ 𝐺) = (𝑣 ∈ 𝐼 ↦ ((𝑧‘𝑣) ↑ (𝐺‘𝑣)))) |
259 | 245, 256,
241, 241, 246, 248, 257 | offval 6904 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑤 ∘𝑓 ↑ 𝐺) = (𝑣 ∈ 𝐼 ↦ ((𝑤‘𝑣) ↑ (𝐺‘𝑣)))) |
260 | 241, 253,
254, 258, 259 | offval2 6914 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → ((𝑧 ∘𝑓 ↑ 𝐺) ∘𝑓
·
(𝑤
∘𝑓 ↑ 𝐺)) = (𝑣 ∈ 𝐼 ↦ (((𝑧‘𝑣) ↑ (𝐺‘𝑣)) · ((𝑤‘𝑣) ↑ (𝐺‘𝑣))))) |
261 | 240, 252,
260 | 3eqtr4d 2666 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → ((𝑧 ∘𝑓 + 𝑤) ∘𝑓
↑
𝐺) = ((𝑧 ∘𝑓 ↑ 𝐺) ∘𝑓
·
(𝑤
∘𝑓 ↑ 𝐺))) |
262 | 261 | oveq2d 6666 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑇 Σg ((𝑧 ∘𝑓 +
𝑤)
∘𝑓 ↑ 𝐺)) = (𝑇 Σg ((𝑧 ∘𝑓
↑
𝐺)
∘𝑓 · (𝑤 ∘𝑓 ↑ 𝐺)))) |
263 | 59 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → 𝑇 ∈ CMnd) |
264 | 20, 51, 31, 52, 263, 228, 236, 241 | psrbagev1 19510 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → ((𝑧 ∘𝑓 ↑ 𝐺):𝐼⟶𝐶 ∧ (𝑧 ∘𝑓 ↑ 𝐺) finSupp
(1r‘𝑆))) |
265 | 264 | simpld 475 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑧 ∘𝑓 ↑ 𝐺):𝐼⟶𝐶) |
266 | 20, 51, 31, 52, 263, 232, 236, 241 | psrbagev1 19510 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → ((𝑤 ∘𝑓 ↑ 𝐺):𝐼⟶𝐶 ∧ (𝑤 ∘𝑓 ↑ 𝐺) finSupp
(1r‘𝑆))) |
267 | 266 | simpld 475 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑤 ∘𝑓 ↑ 𝐺):𝐼⟶𝐶) |
268 | 264 | simprd 479 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑧 ∘𝑓 ↑ 𝐺) finSupp
(1r‘𝑆)) |
269 | 266 | simprd 479 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑤 ∘𝑓 ↑ 𝐺) finSupp
(1r‘𝑆)) |
270 | 51, 52, 224, 263, 241, 265, 267, 268, 269 | gsumadd 18323 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑇 Σg ((𝑧 ∘𝑓
↑
𝐺)
∘𝑓 · (𝑤 ∘𝑓 ↑ 𝐺))) = ((𝑇 Σg (𝑧 ∘𝑓
↑
𝐺)) · (𝑇 Σg (𝑤 ∘𝑓
↑
𝐺)))) |
271 | 262, 270 | eqtrd 2656 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑇 Σg ((𝑧 ∘𝑓 +
𝑤)
∘𝑓 ↑ 𝐺)) = ((𝑇 Σg (𝑧 ∘𝑓
↑
𝐺)) · (𝑇 Σg (𝑤 ∘𝑓
↑
𝐺)))) |
272 | 271 | adantrl 752 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → (𝑇 Σg ((𝑧 ∘𝑓 +
𝑤)
∘𝑓 ↑ 𝐺)) = ((𝑇 Σg (𝑧 ∘𝑓
↑
𝐺)) · (𝑇 Σg (𝑤 ∘𝑓
↑
𝐺)))) |
273 | 226, 272 | oveq12d 6668 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → ((𝐹‘((𝑥‘𝑧)(.r‘𝑅)(𝑦‘𝑤))) · (𝑇 Σg ((𝑧 ∘𝑓 +
𝑤)
∘𝑓 ↑ 𝐺))) = (((𝐹‘(𝑥‘𝑧)) · (𝐹‘(𝑦‘𝑤))) · ((𝑇 Σg (𝑧 ∘𝑓
↑
𝐺)) · (𝑇 Σg (𝑤 ∘𝑓
↑
𝐺))))) |
274 | 59 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → 𝑇 ∈ CMnd) |
275 | 68 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → 𝐹:𝐾⟶𝐶) |
276 | 275, 216 | ffvelrnd 6360 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → (𝐹‘(𝑥‘𝑧)) ∈ 𝐶) |
277 | 275, 220 | ffvelrnd 6360 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → (𝐹‘(𝑦‘𝑤)) ∈ 𝐶) |
278 | 20, 51, 31, 52, 263, 228, 236, 241 | psrbagev2 19511 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑇 Σg (𝑧 ∘𝑓
↑
𝐺)) ∈ 𝐶) |
279 | 278 | adantrl 752 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → (𝑇 Σg (𝑧 ∘𝑓
↑
𝐺)) ∈ 𝐶) |
280 | 20, 51, 31, 52, 263, 232, 236, 241 | psrbagev2 19511 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑇 Σg (𝑤 ∘𝑓
↑
𝐺)) ∈ 𝐶) |
281 | 280 | adantrl 752 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → (𝑇 Σg (𝑤 ∘𝑓
↑
𝐺)) ∈ 𝐶) |
282 | 51, 224 | cmn4 18212 |
. . . . . . 7
⊢ ((𝑇 ∈ CMnd ∧ ((𝐹‘(𝑥‘𝑧)) ∈ 𝐶 ∧ (𝐹‘(𝑦‘𝑤)) ∈ 𝐶) ∧ ((𝑇 Σg (𝑧 ∘𝑓
↑
𝐺)) ∈ 𝐶 ∧ (𝑇 Σg (𝑤 ∘𝑓
↑
𝐺)) ∈ 𝐶)) → (((𝐹‘(𝑥‘𝑧)) · (𝐹‘(𝑦‘𝑤))) · ((𝑇 Σg (𝑧 ∘𝑓
↑
𝐺)) · (𝑇 Σg (𝑤 ∘𝑓
↑
𝐺)))) = (((𝐹‘(𝑥‘𝑧)) · (𝑇 Σg (𝑧 ∘𝑓
↑
𝐺))) · ((𝐹‘(𝑦‘𝑤)) · (𝑇 Σg (𝑤 ∘𝑓
↑
𝐺))))) |
283 | 274, 276,
277, 279, 281, 282 | syl122anc 1335 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → (((𝐹‘(𝑥‘𝑧)) · (𝐹‘(𝑦‘𝑤))) · ((𝑇 Σg (𝑧 ∘𝑓
↑
𝐺)) · (𝑇 Σg (𝑤 ∘𝑓
↑
𝐺)))) = (((𝐹‘(𝑥‘𝑧)) · (𝑇 Σg (𝑧 ∘𝑓
↑
𝐺))) · ((𝐹‘(𝑦‘𝑤)) · (𝑇 Σg (𝑤 ∘𝑓
↑
𝐺))))) |
284 | 273, 283 | eqtrd 2656 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → ((𝐹‘((𝑥‘𝑧)(.r‘𝑅)(𝑦‘𝑤))) · (𝑇 Σg ((𝑧 ∘𝑓 +
𝑤)
∘𝑓 ↑ 𝐺))) = (((𝐹‘(𝑥‘𝑧)) · (𝑇 Σg (𝑧 ∘𝑓
↑
𝐺))) · ((𝐹‘(𝑦‘𝑤)) · (𝑇 Σg (𝑤 ∘𝑓
↑
𝐺))))) |
285 | 6 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → 𝐼 ∈ V) |
286 | 7 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → 𝑅 ∈ CRing) |
287 | 13 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → 𝑆 ∈ CRing) |
288 | 36 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
289 | 38 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → 𝐺:𝐼⟶𝐶) |
290 | 20 | psrbagaddcl 19370 |
. . . . . . 7
⊢ ((𝐼 ∈ V ∧ 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) → (𝑧 ∘𝑓 + 𝑤) ∈ 𝐷) |
291 | 285, 215,
219, 290 | syl3anc 1326 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → (𝑧 ∘𝑓 + 𝑤) ∈ 𝐷) |
292 | 9 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → 𝑅 ∈ Ring) |
293 | 22, 222 | ringcl 18561 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ (𝑥‘𝑧) ∈ 𝐾 ∧ (𝑦‘𝑤) ∈ 𝐾) → ((𝑥‘𝑧)(.r‘𝑅)(𝑦‘𝑤)) ∈ 𝐾) |
294 | 292, 216,
220, 293 | syl3anc 1326 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → ((𝑥‘𝑧)(.r‘𝑅)(𝑦‘𝑤)) ∈ 𝐾) |
295 | 10, 1, 29, 22, 20, 30, 31, 5, 32, 33, 285, 286, 287, 288, 289, 21, 291, 294 | evlslem3 19514 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → (𝐸‘(𝑣 ∈ 𝐷 ↦ if(𝑣 = (𝑧 ∘𝑓 + 𝑤), ((𝑥‘𝑧)(.r‘𝑅)(𝑦‘𝑤)), (0g‘𝑅)))) = ((𝐹‘((𝑥‘𝑧)(.r‘𝑅)(𝑦‘𝑤))) · (𝑇 Σg ((𝑧 ∘𝑓 +
𝑤)
∘𝑓 ↑ 𝐺)))) |
296 | 10, 1, 29, 22, 20, 30, 31, 5, 32, 33, 285, 286, 287, 288, 289, 21, 215, 216 | evlslem3 19514 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → (𝐸‘(𝑣 ∈ 𝐷 ↦ if(𝑣 = 𝑧, (𝑥‘𝑧), (0g‘𝑅)))) = ((𝐹‘(𝑥‘𝑧)) · (𝑇 Σg (𝑧 ∘𝑓
↑
𝐺)))) |
297 | 10, 1, 29, 22, 20, 30, 31, 5, 32, 33, 285, 286, 287, 288, 289, 21, 219, 220 | evlslem3 19514 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → (𝐸‘(𝑣 ∈ 𝐷 ↦ if(𝑣 = 𝑤, (𝑦‘𝑤), (0g‘𝑅)))) = ((𝐹‘(𝑦‘𝑤)) · (𝑇 Σg (𝑤 ∘𝑓
↑
𝐺)))) |
298 | 296, 297 | oveq12d 6668 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → ((𝐸‘(𝑣 ∈ 𝐷 ↦ if(𝑣 = 𝑧, (𝑥‘𝑧), (0g‘𝑅)))) · (𝐸‘(𝑣 ∈ 𝐷 ↦ if(𝑣 = 𝑤, (𝑦‘𝑤), (0g‘𝑅))))) = (((𝐹‘(𝑥‘𝑧)) · (𝑇 Σg (𝑧 ∘𝑓
↑
𝐺))) · ((𝐹‘(𝑦‘𝑤)) · (𝑇 Σg (𝑤 ∘𝑓
↑
𝐺))))) |
299 | 284, 295,
298 | 3eqtr4d 2666 |
. . . 4
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷))) → (𝐸‘(𝑣 ∈ 𝐷 ↦ if(𝑣 = (𝑧 ∘𝑓 + 𝑤), ((𝑥‘𝑧)(.r‘𝑅)(𝑦‘𝑤)), (0g‘𝑅)))) = ((𝐸‘(𝑣 ∈ 𝐷 ↦ if(𝑣 = 𝑧, (𝑥‘𝑧), (0g‘𝑅)))) · (𝐸‘(𝑣 ∈ 𝐷 ↦ if(𝑣 = 𝑤, (𝑦‘𝑤), (0g‘𝑅)))))) |
300 | 10, 1, 5, 21, 20, 6, 7, 13, 208, 299 | evlslem2 19512 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘(𝑥(.r‘𝑃)𝑦)) = ((𝐸‘𝑥) · (𝐸‘𝑦))) |
301 | 1, 2, 3, 4, 5, 12,
15, 92, 300, 29, 93, 94, 116, 207 | isrhmd 18729 |
. 2
⊢ (𝜑 → 𝐸 ∈ (𝑃 RingHom 𝑆)) |
302 | | ovex 6678 |
. . . . . 6
⊢ (𝑆 Σg
(𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓
↑
𝐺))))) ∈
V |
303 | 302, 33 | fnmpti 6022 |
. . . . 5
⊢ 𝐸 Fn 𝐵 |
304 | 303 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝐸 Fn 𝐵) |
305 | 22, 1 | rhmf 18726 |
. . . . . 6
⊢ (𝐴 ∈ (𝑅 RingHom 𝑃) → 𝐴:𝐾⟶𝐵) |
306 | 86, 305 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐴:𝐾⟶𝐵) |
307 | 306 | ffnd 6046 |
. . . 4
⊢ (𝜑 → 𝐴 Fn 𝐾) |
308 | | frn 6053 |
. . . . 5
⊢ (𝐴:𝐾⟶𝐵 → ran 𝐴 ⊆ 𝐵) |
309 | 306, 308 | syl 17 |
. . . 4
⊢ (𝜑 → ran 𝐴 ⊆ 𝐵) |
310 | | fnco 5999 |
. . . 4
⊢ ((𝐸 Fn 𝐵 ∧ 𝐴 Fn 𝐾 ∧ ran 𝐴 ⊆ 𝐵) → (𝐸 ∘ 𝐴) Fn 𝐾) |
311 | 304, 307,
309, 310 | syl3anc 1326 |
. . 3
⊢ (𝜑 → (𝐸 ∘ 𝐴) Fn 𝐾) |
312 | 68 | ffnd 6046 |
. . 3
⊢ (𝜑 → 𝐹 Fn 𝐾) |
313 | | fvco2 6273 |
. . . . 5
⊢ ((𝐴 Fn 𝐾 ∧ 𝑥 ∈ 𝐾) → ((𝐸 ∘ 𝐴)‘𝑥) = (𝐸‘(𝐴‘𝑥))) |
314 | 307, 313 | sylan 488 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝐸 ∘ 𝐴)‘𝑥) = (𝐸‘(𝐴‘𝑥))) |
315 | 314, 73 | eqtrd 2656 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝐸 ∘ 𝐴)‘𝑥) = (𝐹‘𝑥)) |
316 | 311, 312,
315 | eqfnfvd 6314 |
. 2
⊢ (𝜑 → (𝐸 ∘ 𝐴) = 𝐹) |
317 | 10, 32, 1, 6, 9 | mvrf2 19492 |
. . . . 5
⊢ (𝜑 → 𝑉:𝐼⟶𝐵) |
318 | 317 | ffnd 6046 |
. . . 4
⊢ (𝜑 → 𝑉 Fn 𝐼) |
319 | | frn 6053 |
. . . . 5
⊢ (𝑉:𝐼⟶𝐵 → ran 𝑉 ⊆ 𝐵) |
320 | 317, 319 | syl 17 |
. . . 4
⊢ (𝜑 → ran 𝑉 ⊆ 𝐵) |
321 | | fnco 5999 |
. . . 4
⊢ ((𝐸 Fn 𝐵 ∧ 𝑉 Fn 𝐼 ∧ ran 𝑉 ⊆ 𝐵) → (𝐸 ∘ 𝑉) Fn 𝐼) |
322 | 304, 318,
320, 321 | syl3anc 1326 |
. . 3
⊢ (𝜑 → (𝐸 ∘ 𝑉) Fn 𝐼) |
323 | | fvco2 6273 |
. . . . 5
⊢ ((𝑉 Fn 𝐼 ∧ 𝑥 ∈ 𝐼) → ((𝐸 ∘ 𝑉)‘𝑥) = (𝐸‘(𝑉‘𝑥))) |
324 | 318, 323 | sylan 488 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐸 ∘ 𝑉)‘𝑥) = (𝐸‘(𝑉‘𝑥))) |
325 | 6 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ V) |
326 | 7 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ CRing) |
327 | | simpr 477 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) |
328 | 32, 20, 21, 75, 325, 326, 327 | mvrval 19421 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)), (1r‘𝑅), (0g‘𝑅)))) |
329 | 328 | fveq2d 6195 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐸‘(𝑉‘𝑥)) = (𝐸‘(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)), (1r‘𝑅), (0g‘𝑅))))) |
330 | 13 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ CRing) |
331 | 36 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
332 | 38 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺:𝐼⟶𝐶) |
333 | 20 | psrbagsn 19495 |
. . . . . . . 8
⊢ (𝐼 ∈ V → (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)) ∈ 𝐷) |
334 | 6, 333 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)) ∈ 𝐷) |
335 | 334 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)) ∈ 𝐷) |
336 | 77 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (1r‘𝑅) ∈ 𝐾) |
337 | 10, 1, 29, 22, 20, 30, 31, 5, 32, 33, 325, 326, 330, 331, 332, 21, 335, 336 | evlslem3 19514 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐸‘(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)), (1r‘𝑅), (0g‘𝑅)))) = ((𝐹‘(1r‘𝑅)) · (𝑇 Σg ((𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)) ∘𝑓 ↑ 𝐺)))) |
338 | 91 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆)) |
339 | | 1nn0 11308 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℕ0 |
340 | | 0nn0 11307 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
ℕ0 |
341 | 339, 340 | keepel 4155 |
. . . . . . . . . . . . 13
⊢ if(𝑧 = 𝑥, 1, 0) ∈
ℕ0 |
342 | 341 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → if(𝑧 = 𝑥, 1, 0) ∈
ℕ0) |
343 | 38 | ffvelrnda 6359 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐺‘𝑧) ∈ 𝐶) |
344 | | eqidd 2623 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)) = (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0))) |
345 | 38 | feqmptd 6249 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝐼 ↦ (𝐺‘𝑧))) |
346 | 6, 342, 343, 344, 345 | offval2 6914 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)) ∘𝑓 ↑ 𝐺) = (𝑧 ∈ 𝐼 ↦ (if(𝑧 = 𝑥, 1, 0) ↑ (𝐺‘𝑧)))) |
347 | | oveq1 6657 |
. . . . . . . . . . . . . 14
⊢ (1 =
if(𝑧 = 𝑥, 1, 0) → (1 ↑ (𝐺‘𝑧)) = (if(𝑧 = 𝑥, 1, 0) ↑ (𝐺‘𝑧))) |
348 | 347 | eqeq1d 2624 |
. . . . . . . . . . . . 13
⊢ (1 =
if(𝑧 = 𝑥, 1, 0) → ((1 ↑ (𝐺‘𝑧)) = if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)) ↔ (if(𝑧 = 𝑥, 1, 0) ↑ (𝐺‘𝑧)) = if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)))) |
349 | | oveq1 6657 |
. . . . . . . . . . . . . 14
⊢ (0 =
if(𝑧 = 𝑥, 1, 0) → (0 ↑ (𝐺‘𝑧)) = (if(𝑧 = 𝑥, 1, 0) ↑ (𝐺‘𝑧))) |
350 | 349 | eqeq1d 2624 |
. . . . . . . . . . . . 13
⊢ (0 =
if(𝑧 = 𝑥, 1, 0) → ((0 ↑ (𝐺‘𝑧)) = if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)) ↔ (if(𝑧 = 𝑥, 1, 0) ↑ (𝐺‘𝑧)) = if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)))) |
351 | 343 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑧 = 𝑥) → (𝐺‘𝑧) ∈ 𝐶) |
352 | 51, 31 | mulg1 17548 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺‘𝑧) ∈ 𝐶 → (1 ↑ (𝐺‘𝑧)) = (𝐺‘𝑧)) |
353 | 351, 352 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑧 = 𝑥) → (1 ↑ (𝐺‘𝑧)) = (𝐺‘𝑧)) |
354 | | iftrue 4092 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = 𝑥 → if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)) = (𝐺‘𝑧)) |
355 | 354 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑧 = 𝑥) → if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)) = (𝐺‘𝑧)) |
356 | 353, 355 | eqtr4d 2659 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑧 = 𝑥) → (1 ↑ (𝐺‘𝑧)) = if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆))) |
357 | 51, 52, 31 | mulg0 17546 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺‘𝑧) ∈ 𝐶 → (0 ↑ (𝐺‘𝑧)) = (1r‘𝑆)) |
358 | 343, 357 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (0 ↑ (𝐺‘𝑧)) = (1r‘𝑆)) |
359 | 358 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ ¬ 𝑧 = 𝑥) → (0 ↑ (𝐺‘𝑧)) = (1r‘𝑆)) |
360 | | iffalse 4095 |
. . . . . . . . . . . . . . 15
⊢ (¬
𝑧 = 𝑥 → if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)) = (1r‘𝑆)) |
361 | 360 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ ¬ 𝑧 = 𝑥) → if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)) = (1r‘𝑆)) |
362 | 359, 361 | eqtr4d 2659 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ ¬ 𝑧 = 𝑥) → (0 ↑ (𝐺‘𝑧)) = if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆))) |
363 | 348, 350,
356, 362 | ifbothda 4123 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (if(𝑧 = 𝑥, 1, 0) ↑ (𝐺‘𝑧)) = if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆))) |
364 | 363 | mpteq2dva 4744 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑧 ∈ 𝐼 ↦ (if(𝑧 = 𝑥, 1, 0) ↑ (𝐺‘𝑧))) = (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)))) |
365 | 346, 364 | eqtrd 2656 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)) ∘𝑓 ↑ 𝐺) = (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)))) |
366 | 365 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)) ∘𝑓 ↑ 𝐺) = (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)))) |
367 | 366 | oveq2d 6666 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑇 Σg ((𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)) ∘𝑓 ↑ 𝐺)) = (𝑇 Σg (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆))))) |
368 | 61 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑇 ∈ Mnd) |
369 | 343 | adantlr 751 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) → (𝐺‘𝑧) ∈ 𝐶) |
370 | 29, 3 | ringidcl 18568 |
. . . . . . . . . . . . 13
⊢ (𝑆 ∈ Ring →
(1r‘𝑆)
∈ 𝐶) |
371 | 15, 370 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1r‘𝑆) ∈ 𝐶) |
372 | 371 | ad2antrr 762 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) → (1r‘𝑆) ∈ 𝐶) |
373 | 369, 372 | ifcld 4131 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑧 ∈ 𝐼) → if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)) ∈ 𝐶) |
374 | | eqid 2622 |
. . . . . . . . . 10
⊢ (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆))) = (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆))) |
375 | 373, 374 | fmptd 6385 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆))):𝐼⟶𝐶) |
376 | | eldifn 3733 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ (𝐼 ∖ {𝑥}) → ¬ 𝑧 ∈ {𝑥}) |
377 | | velsn 4193 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ {𝑥} ↔ 𝑧 = 𝑥) |
378 | 376, 377 | sylnib 318 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ (𝐼 ∖ {𝑥}) → ¬ 𝑧 = 𝑥) |
379 | 378, 360 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ (𝐼 ∖ {𝑥}) → if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)) = (1r‘𝑆)) |
380 | 379 | adantl 482 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑧 ∈ (𝐼 ∖ {𝑥})) → if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)) = (1r‘𝑆)) |
381 | 380, 325 | suppss2 7329 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆))) supp (1r‘𝑆)) ⊆ {𝑥}) |
382 | 51, 52, 368, 325, 327, 375, 381 | gsumpt 18361 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑇 Σg (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)))) = ((𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)))‘𝑥)) |
383 | | fveq2 6191 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑥 → (𝐺‘𝑧) = (𝐺‘𝑥)) |
384 | 354, 383 | eqtrd 2656 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑥 → if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)) = (𝐺‘𝑥)) |
385 | | fvex 6201 |
. . . . . . . . . 10
⊢ (𝐺‘𝑥) ∈ V |
386 | 384, 374,
385 | fvmpt 6282 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐼 → ((𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)))‘𝑥) = (𝐺‘𝑥)) |
387 | 386 | adantl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, (𝐺‘𝑧), (1r‘𝑆)))‘𝑥) = (𝐺‘𝑥)) |
388 | 367, 382,
387 | 3eqtrd 2660 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑇 Σg ((𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)) ∘𝑓 ↑ 𝐺)) = (𝐺‘𝑥)) |
389 | 338, 388 | oveq12d 6668 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐹‘(1r‘𝑅)) · (𝑇 Σg ((𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)) ∘𝑓 ↑ 𝐺))) =
((1r‘𝑆)
·
(𝐺‘𝑥))) |
390 | 29, 5, 3 | ringlidm 18571 |
. . . . . . 7
⊢ ((𝑆 ∈ Ring ∧ (𝐺‘𝑥) ∈ 𝐶) → ((1r‘𝑆) · (𝐺‘𝑥)) = (𝐺‘𝑥)) |
391 | 15, 50, 390 | syl2an2r 876 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((1r‘𝑆) · (𝐺‘𝑥)) = (𝐺‘𝑥)) |
392 | 389, 391 | eqtrd 2656 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐹‘(1r‘𝑅)) · (𝑇 Σg ((𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)) ∘𝑓 ↑ 𝐺))) = (𝐺‘𝑥)) |
393 | 329, 337,
392 | 3eqtrd 2660 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐸‘(𝑉‘𝑥)) = (𝐺‘𝑥)) |
394 | 324, 393 | eqtrd 2656 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐸 ∘ 𝑉)‘𝑥) = (𝐺‘𝑥)) |
395 | 322, 255,
394 | eqfnfvd 6314 |
. 2
⊢ (𝜑 → (𝐸 ∘ 𝑉) = 𝐺) |
396 | 301, 316,
395 | 3jca 1242 |
1
⊢ (𝜑 → (𝐸 ∈ (𝑃 RingHom 𝑆) ∧ (𝐸 ∘ 𝐴) = 𝐹 ∧ (𝐸 ∘ 𝑉) = 𝐺)) |