Step | Hyp | Ref
| Expression |
1 | | cpmatsrngpmat.s |
. . . . . 6
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) |
2 | | cpmatsrngpmat.p |
. . . . . 6
⊢ 𝑃 = (Poly1‘𝑅) |
3 | | cpmatsrngpmat.c |
. . . . . 6
⊢ 𝐶 = (𝑁 Mat 𝑃) |
4 | | eqid 2610 |
. . . . . 6
⊢
(Base‘𝐶) =
(Base‘𝐶) |
5 | | eqid 2610 |
. . . . . 6
⊢
(Base‘𝑅) =
(Base‘𝑅) |
6 | | eqid 2610 |
. . . . . 6
⊢
(algSc‘𝑃) =
(algSc‘𝑃) |
7 | 1, 2, 3, 4, 5, 6 | cpmatelimp2 20338 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ 𝑆 → (𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)))) |
8 | 2 | ply1sca 19444 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃)) |
9 | 8 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝑃)) |
10 | 9 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑎 ∈ (Base‘𝑅)) → 𝑅 = (Scalar‘𝑃)) |
11 | 10 | eqcomd 2616 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑎 ∈ (Base‘𝑅)) → (Scalar‘𝑃) = 𝑅) |
12 | 11 | fveq2d 6107 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑎 ∈ (Base‘𝑅)) →
(invg‘(Scalar‘𝑃)) = (invg‘𝑅)) |
13 | 12 | fveq1d 6105 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑎 ∈ (Base‘𝑅)) →
((invg‘(Scalar‘𝑃))‘𝑎) = ((invg‘𝑅)‘𝑎)) |
14 | | ringgrp 18375 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) |
15 | 14 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Grp) |
16 | | eqid 2610 |
. . . . . . . . . . . . . . . . 17
⊢
(invg‘𝑅) = (invg‘𝑅) |
17 | 5, 16 | grpinvcl 17290 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑅 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑅)) →
((invg‘𝑅)‘𝑎) ∈ (Base‘𝑅)) |
18 | 15, 17 | sylan 487 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑎 ∈ (Base‘𝑅)) →
((invg‘𝑅)‘𝑎) ∈ (Base‘𝑅)) |
19 | 13, 18 | eqeltrd 2688 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑎 ∈ (Base‘𝑅)) →
((invg‘(Scalar‘𝑃))‘𝑎) ∈ (Base‘𝑅)) |
20 | 19 | ex 449 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑎 ∈ (Base‘𝑅) →
((invg‘(Scalar‘𝑃))‘𝑎) ∈ (Base‘𝑅))) |
21 | 20 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑎 ∈ (Base‘𝑅) →
((invg‘(Scalar‘𝑃))‘𝑎) ∈ (Base‘𝑅))) |
22 | 21 | imp 444 |
. . . . . . . . . . 11
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) →
((invg‘(Scalar‘𝑃))‘𝑎) ∈ (Base‘𝑅)) |
23 | 22 | adantr 480 |
. . . . . . . . . 10
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) →
((invg‘(Scalar‘𝑃))‘𝑎) ∈ (Base‘𝑅)) |
24 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑐 =
((invg‘(Scalar‘𝑃))‘𝑎) → ((algSc‘𝑃)‘𝑐) = ((algSc‘𝑃)‘((invg‘(Scalar‘𝑃))‘𝑎))) |
25 | 24 | eqeq2d 2620 |
. . . . . . . . . . 11
⊢ (𝑐 =
((invg‘(Scalar‘𝑃))‘𝑎) → ((𝑖((invg‘𝐶)‘𝑥)𝑗) = ((algSc‘𝑃)‘𝑐) ↔ (𝑖((invg‘𝐶)‘𝑥)𝑗) = ((algSc‘𝑃)‘((invg‘(Scalar‘𝑃))‘𝑎)))) |
26 | 25 | adantl 481 |
. . . . . . . . . 10
⊢
(((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) ∧ 𝑐 =
((invg‘(Scalar‘𝑃))‘𝑎)) → ((𝑖((invg‘𝐶)‘𝑥)𝑗) = ((algSc‘𝑃)‘𝑐) ↔ (𝑖((invg‘𝐶)‘𝑥)𝑗) = ((algSc‘𝑃)‘((invg‘(Scalar‘𝑃))‘𝑎)))) |
27 | 2 | ply1ring 19439 |
. . . . . . . . . . . . . . 15
⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
28 | 27 | ad3antlr 763 |
. . . . . . . . . . . . . 14
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑃 ∈ Ring) |
29 | | simplr 788 |
. . . . . . . . . . . . . 14
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑥 ∈ (Base‘𝐶)) |
30 | | simpr 476 |
. . . . . . . . . . . . . 14
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) |
31 | 28, 29, 30 | 3jca 1235 |
. . . . . . . . . . . . 13
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑃 ∈ Ring ∧ 𝑥 ∈ (Base‘𝐶) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) |
32 | 31 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → (𝑃 ∈ Ring ∧ 𝑥 ∈ (Base‘𝐶) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) |
33 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢
(invg‘𝑃) = (invg‘𝑃) |
34 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢
(invg‘𝐶) = (invg‘𝐶) |
35 | 3, 4, 33, 34 | matinvgcell 20060 |
. . . . . . . . . . . 12
⊢ ((𝑃 ∈ Ring ∧ 𝑥 ∈ (Base‘𝐶) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖((invg‘𝐶)‘𝑥)𝑗) = ((invg‘𝑃)‘(𝑖𝑥𝑗))) |
36 | 32, 35 | syl 17 |
. . . . . . . . . . 11
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → (𝑖((invg‘𝐶)‘𝑥)𝑗) = ((invg‘𝑃)‘(𝑖𝑥𝑗))) |
37 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ ((𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → ((invg‘𝑃)‘(𝑖𝑥𝑗)) = ((invg‘𝑃)‘((algSc‘𝑃)‘𝑎))) |
38 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢
(Scalar‘𝑃) =
(Scalar‘𝑃) |
39 | 28 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) → 𝑃 ∈ Ring) |
40 | 2 | ply1lmod 19443 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
41 | 40 | ad3antlr 763 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑃 ∈ LMod) |
42 | 41 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) → 𝑃 ∈ LMod) |
43 | 6, 38, 39, 42 | asclghm 19159 |
. . . . . . . . . . . . . 14
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) → (algSc‘𝑃) ∈ ((Scalar‘𝑃) GrpHom 𝑃)) |
44 | 9 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) →
(Base‘𝑅) =
(Base‘(Scalar‘𝑃))) |
45 | 44 | eleq2d 2673 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑎 ∈ (Base‘𝑅) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑃)))) |
46 | 45 | biimpd 218 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑎 ∈ (Base‘𝑅) → 𝑎 ∈ (Base‘(Scalar‘𝑃)))) |
47 | 46 | ad2antrr 758 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑎 ∈ (Base‘𝑅) → 𝑎 ∈ (Base‘(Scalar‘𝑃)))) |
48 | 47 | imp 444 |
. . . . . . . . . . . . . 14
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) → 𝑎 ∈ (Base‘(Scalar‘𝑃))) |
49 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢
(Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) |
50 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢
(invg‘(Scalar‘𝑃)) =
(invg‘(Scalar‘𝑃)) |
51 | 49, 50, 33 | ghminv 17490 |
. . . . . . . . . . . . . 14
⊢
(((algSc‘𝑃)
∈ ((Scalar‘𝑃)
GrpHom 𝑃) ∧ 𝑎 ∈
(Base‘(Scalar‘𝑃))) → ((algSc‘𝑃)‘((invg‘(Scalar‘𝑃))‘𝑎)) = ((invg‘𝑃)‘((algSc‘𝑃)‘𝑎))) |
52 | 43, 48, 51 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) → ((algSc‘𝑃)‘((invg‘(Scalar‘𝑃))‘𝑎)) = ((invg‘𝑃)‘((algSc‘𝑃)‘𝑎))) |
53 | 52 | eqcomd 2616 |
. . . . . . . . . . . 12
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) → ((invg‘𝑃)‘((algSc‘𝑃)‘𝑎)) = ((algSc‘𝑃)‘((invg‘(Scalar‘𝑃))‘𝑎))) |
54 | 37, 53 | sylan9eqr 2666 |
. . . . . . . . . . 11
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → ((invg‘𝑃)‘(𝑖𝑥𝑗)) = ((algSc‘𝑃)‘((invg‘(Scalar‘𝑃))‘𝑎))) |
55 | 36, 54 | eqtrd 2644 |
. . . . . . . . . 10
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → (𝑖((invg‘𝐶)‘𝑥)𝑗) = ((algSc‘𝑃)‘((invg‘(Scalar‘𝑃))‘𝑎))) |
56 | 23, 26, 55 | rspcedvd 3289 |
. . . . . . . . 9
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → ∃𝑐 ∈ (Base‘𝑅)(𝑖((invg‘𝐶)‘𝑥)𝑗) = ((algSc‘𝑃)‘𝑐)) |
57 | 56 | ex 449 |
. . . . . . . 8
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) → ((𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → ∃𝑐 ∈ (Base‘𝑅)(𝑖((invg‘𝐶)‘𝑥)𝑗) = ((algSc‘𝑃)‘𝑐))) |
58 | 57 | rexlimdva 3013 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → ∃𝑐 ∈ (Base‘𝑅)(𝑖((invg‘𝐶)‘𝑥)𝑗) = ((algSc‘𝑃)‘𝑐))) |
59 | 58 | ralimdvva 2947 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖((invg‘𝐶)‘𝑥)𝑗) = ((algSc‘𝑃)‘𝑐))) |
60 | 59 | expimpd 627 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖((invg‘𝐶)‘𝑥)𝑗) = ((algSc‘𝑃)‘𝑐))) |
61 | 7, 60 | syld 46 |
. . . 4
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ 𝑆 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖((invg‘𝐶)‘𝑥)𝑗) = ((algSc‘𝑃)‘𝑐))) |
62 | 61 | imp 444 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝑆) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖((invg‘𝐶)‘𝑥)𝑗) = ((algSc‘𝑃)‘𝑐)) |
63 | | simpll 786 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝑆) → 𝑁 ∈ Fin) |
64 | | simplr 788 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝑆) → 𝑅 ∈ Ring) |
65 | 2, 3 | pmatring 20317 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring) |
66 | | ringgrp 18375 |
. . . . . . 7
⊢ (𝐶 ∈ Ring → 𝐶 ∈ Grp) |
67 | 65, 66 | syl 17 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Grp) |
68 | 67 | adantr 480 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝑆) → 𝐶 ∈ Grp) |
69 | 1, 2, 3, 4 | cpmatpmat 20334 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐶)) |
70 | 69 | 3expa 1257 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐶)) |
71 | 4, 34 | grpinvcl 17290 |
. . . . 5
⊢ ((𝐶 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐶)) →
((invg‘𝐶)‘𝑥) ∈ (Base‘𝐶)) |
72 | 68, 70, 71 | syl2anc 691 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝑆) → ((invg‘𝐶)‘𝑥) ∈ (Base‘𝐶)) |
73 | 1, 2, 3, 4, 5, 6 | cpmatel2 20337 |
. . . 4
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧
((invg‘𝐶)‘𝑥) ∈ (Base‘𝐶)) → (((invg‘𝐶)‘𝑥) ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖((invg‘𝐶)‘𝑥)𝑗) = ((algSc‘𝑃)‘𝑐))) |
74 | 63, 64, 72, 73 | syl3anc 1318 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝑆) → (((invg‘𝐶)‘𝑥) ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖((invg‘𝐶)‘𝑥)𝑗) = ((algSc‘𝑃)‘𝑐))) |
75 | 62, 74 | mpbird 246 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝑆) → ((invg‘𝐶)‘𝑥) ∈ 𝑆) |
76 | 75 | ralrimiva 2949 |
1
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) →
∀𝑥 ∈ 𝑆 ((invg‘𝐶)‘𝑥) ∈ 𝑆) |