| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | fedgmullem1.g | . . . . 5
⊢ (𝜑 → 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) | 
| 2 |  | simpllr 776 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) | 
| 3 |  | simplr 769 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ 𝑌) | 
| 4 | 2, 3 | ffvelcdmd 7105 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝐺‘𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋)) | 
| 5 |  | elmapi 8889 | . . . . . . . . . . . 12
⊢ ((𝐺‘𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶))) | 
| 6 | 4, 5 | syl 17 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶))) | 
| 7 | 6 | anasss 466 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶))) | 
| 8 |  | simprr 773 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑖 ∈ 𝑋) | 
| 9 | 7, 8 | ffvelcdmd 7105 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶))) | 
| 10 |  | fedgmul.k | . . . . . . . . . . . . 13
⊢ 𝐾 = (𝐸 ↾s 𝑉) | 
| 11 |  | fedgmul.a | . . . . . . . . . . . . . . 15
⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑉) | 
| 12 | 11 | a1i 11 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝐸)‘𝑉)) | 
| 13 |  | fedgmul.4 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸)) | 
| 14 |  | fedgmul.5 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐹)) | 
| 15 |  | fedgmul.f | . . . . . . . . . . . . . . . . . . 19
⊢ 𝐹 = (𝐸 ↾s 𝑈) | 
| 16 | 15 | subsubrg 20598 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑈 ∈ (SubRing‘𝐸) → (𝑉 ∈ (SubRing‘𝐹) ↔ (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈))) | 
| 17 | 16 | biimpa 476 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ∈ (SubRing‘𝐹)) → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈)) | 
| 18 | 13, 14, 17 | syl2anc 584 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈)) | 
| 19 | 18 | simpld 494 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐸)) | 
| 20 |  | eqid 2737 | . . . . . . . . . . . . . . . 16
⊢
(Base‘𝐸) =
(Base‘𝐸) | 
| 21 | 20 | subrgss 20572 | . . . . . . . . . . . . . . 15
⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸)) | 
| 22 | 19, 21 | syl 17 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐸)) | 
| 23 | 12, 22 | srasca 21183 | . . . . . . . . . . . . 13
⊢ (𝜑 → (𝐸 ↾s 𝑉) = (Scalar‘𝐴)) | 
| 24 | 10, 23 | eqtrid 2789 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝐾 = (Scalar‘𝐴)) | 
| 25 | 18 | simprd 495 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑉 ⊆ 𝑈) | 
| 26 |  | ressabs 17294 | . . . . . . . . . . . . . . 15
⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈) → ((𝐸 ↾s 𝑈) ↾s 𝑉) = (𝐸 ↾s 𝑉)) | 
| 27 | 13, 25, 26 | syl2anc 584 | . . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝐸 ↾s 𝑈) ↾s 𝑉) = (𝐸 ↾s 𝑉)) | 
| 28 | 15 | oveq1i 7441 | . . . . . . . . . . . . . 14
⊢ (𝐹 ↾s 𝑉) = ((𝐸 ↾s 𝑈) ↾s 𝑉) | 
| 29 | 27, 28, 10 | 3eqtr4g 2802 | . . . . . . . . . . . . 13
⊢ (𝜑 → (𝐹 ↾s 𝑉) = 𝐾) | 
| 30 |  | fedgmul.c | . . . . . . . . . . . . . . 15
⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉) | 
| 31 | 30 | a1i 11 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐶 = ((subringAlg ‘𝐹)‘𝑉)) | 
| 32 |  | eqid 2737 | . . . . . . . . . . . . . . . 16
⊢
(Base‘𝐹) =
(Base‘𝐹) | 
| 33 | 32 | subrgss 20572 | . . . . . . . . . . . . . . 15
⊢ (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹)) | 
| 34 | 14, 33 | syl 17 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐹)) | 
| 35 | 31, 34 | srasca 21183 | . . . . . . . . . . . . 13
⊢ (𝜑 → (𝐹 ↾s 𝑉) = (Scalar‘𝐶)) | 
| 36 | 29, 35 | eqtr3d 2779 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝐾 = (Scalar‘𝐶)) | 
| 37 | 24, 36 | eqtr3d 2779 | . . . . . . . . . . 11
⊢ (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶)) | 
| 38 | 37 | fveq2d 6910 | . . . . . . . . . 10
⊢ (𝜑 →
(Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶))) | 
| 39 | 38 | ad2antrr 726 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (Base‘(Scalar‘𝐴)) =
(Base‘(Scalar‘𝐶))) | 
| 40 | 9, 39 | eleqtrrd 2844 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴))) | 
| 41 | 40 | ralrimivva 3202 | . . . . . . 7
⊢ ((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → ∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴))) | 
| 42 |  | fedgmullem.h | . . . . . . . 8
⊢ 𝐻 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖)) | 
| 43 | 42 | fmpo 8093 | . . . . . . 7
⊢
(∀𝑗 ∈
𝑌 ∀𝑖 ∈ 𝑋 ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)) ↔ 𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴))) | 
| 44 | 41, 43 | sylib 218 | . . . . . 6
⊢ ((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → 𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴))) | 
| 45 |  | fvexd 6921 | . . . . . . 7
⊢ ((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) →
(Base‘(Scalar‘𝐴)) ∈ V) | 
| 46 |  | fedgmullem.y | . . . . . . . . 9
⊢ (𝜑 → 𝑌 ∈ (LBasis‘𝐵)) | 
| 47 |  | fedgmullem.x | . . . . . . . . 9
⊢ (𝜑 → 𝑋 ∈ (LBasis‘𝐶)) | 
| 48 | 46, 47 | xpexd 7771 | . . . . . . . 8
⊢ (𝜑 → (𝑌 × 𝑋) ∈ V) | 
| 49 | 48 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → (𝑌 × 𝑋) ∈ V) | 
| 50 | 45, 49 | elmapd 8880 | . . . . . 6
⊢ ((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → (𝐻 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)) ↔ 𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))) | 
| 51 | 44, 50 | mpbird 257 | . . . . 5
⊢ ((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → 𝐻 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋))) | 
| 52 | 1, 51 | mpdan 687 | . . . 4
⊢ (𝜑 → 𝐻 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋))) | 
| 53 |  | simpl 482 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝜑) | 
| 54 | 53 | adantr 480 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝜑) | 
| 55 | 1 | ffvelcdmda 7104 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋)) | 
| 56 | 55, 5 | syl 17 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶))) | 
| 57 | 56 | adantr 480 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶))) | 
| 58 | 38 | feq3d 6723 | . . . . . . . . . . 11
⊢ (𝜑 → ((𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐴)) ↔ (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))) | 
| 59 | 58 | biimpar 477 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶))) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐴))) | 
| 60 | 54, 57, 59 | syl2anc 584 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐴))) | 
| 61 |  | simpr 484 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋) | 
| 62 | 60, 61 | ffvelcdmd 7105 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴))) | 
| 63 | 62 | ralrimiva 3146 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ∀𝑖 ∈ 𝑋 ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴))) | 
| 64 | 63 | ralrimiva 3146 | . . . . . 6
⊢ (𝜑 → ∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴))) | 
| 65 | 64, 43 | sylib 218 | . . . . 5
⊢ (𝜑 → 𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴))) | 
| 66 | 65 | ffund 6740 | . . . 4
⊢ (𝜑 → Fun 𝐻) | 
| 67 |  | fedgmul.1 | . . . . . 6
⊢ (𝜑 → 𝐸 ∈ DivRing) | 
| 68 |  | drngring 20736 | . . . . . 6
⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring) | 
| 69 | 67, 68 | syl 17 | . . . . 5
⊢ (𝜑 → 𝐸 ∈ Ring) | 
| 70 |  | ringgrp 20235 | . . . . 5
⊢ (𝐸 ∈ Ring → 𝐸 ∈ Grp) | 
| 71 |  | eqid 2737 | . . . . . 6
⊢
(0g‘𝐸) = (0g‘𝐸) | 
| 72 | 20, 71 | grpidcl 18983 | . . . . 5
⊢ (𝐸 ∈ Grp →
(0g‘𝐸)
∈ (Base‘𝐸)) | 
| 73 | 69, 70, 72 | 3syl 18 | . . . 4
⊢ (𝜑 → (0g‘𝐸) ∈ (Base‘𝐸)) | 
| 74 |  | fedgmullem1.1 | . . . . . . 7
⊢ (𝜑 → 𝐿 finSupp
(0g‘(Scalar‘𝐵))) | 
| 75 | 74 | fsuppimpd 9409 | . . . . . 6
⊢ (𝜑 → (𝐿 supp
(0g‘(Scalar‘𝐵))) ∈ Fin) | 
| 76 |  | simpl 482 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp
(0g‘(Scalar‘𝐵))))) → 𝜑) | 
| 77 |  | simpr 484 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp
(0g‘(Scalar‘𝐵))))) → 𝑗 ∈ (𝑌 ∖ (𝐿 supp
(0g‘(Scalar‘𝐵))))) | 
| 78 | 77 | eldifad 3963 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp
(0g‘(Scalar‘𝐵))))) → 𝑗 ∈ 𝑌) | 
| 79 |  | fedgmullem1.l | . . . . . . . . . 10
⊢ (𝜑 → 𝐿:𝑌⟶(Base‘(Scalar‘𝐵))) | 
| 80 |  | ssidd 4007 | . . . . . . . . . 10
⊢ (𝜑 → (𝐿 supp
(0g‘(Scalar‘𝐵))) ⊆ (𝐿 supp
(0g‘(Scalar‘𝐵)))) | 
| 81 |  | fvexd 6921 | . . . . . . . . . 10
⊢ (𝜑 →
(0g‘(Scalar‘𝐵)) ∈ V) | 
| 82 | 79, 80, 46, 81 | suppssr 8220 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp
(0g‘(Scalar‘𝐵))))) → (𝐿‘𝑗) = (0g‘(Scalar‘𝐵))) | 
| 83 |  | fedgmullem1.3 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐿‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖)))) | 
| 84 | 78, 83 | syldan 591 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp
(0g‘(Scalar‘𝐵))))) → (𝐿‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖)))) | 
| 85 |  | fedgmul.b | . . . . . . . . . . . . . . 15
⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) | 
| 86 | 85 | a1i 11 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐵 = ((subringAlg ‘𝐸)‘𝑈)) | 
| 87 | 20 | subrgss 20572 | . . . . . . . . . . . . . . 15
⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸)) | 
| 88 | 13, 87 | syl 17 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐸)) | 
| 89 | 86, 88 | srasca 21183 | . . . . . . . . . . . . 13
⊢ (𝜑 → (𝐸 ↾s 𝑈) = (Scalar‘𝐵)) | 
| 90 | 15, 89 | eqtrid 2789 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝐹 = (Scalar‘𝐵)) | 
| 91 | 90 | fveq2d 6910 | . . . . . . . . . . 11
⊢ (𝜑 → (0g‘𝐹) =
(0g‘(Scalar‘𝐵))) | 
| 92 |  | fedgmul.2 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝐹 ∈ DivRing) | 
| 93 | 30, 92, 14 | drgext0g 33640 | . . . . . . . . . . 11
⊢ (𝜑 → (0g‘𝐹) = (0g‘𝐶)) | 
| 94 | 91, 93 | eqtr3d 2779 | . . . . . . . . . 10
⊢ (𝜑 →
(0g‘(Scalar‘𝐵)) = (0g‘𝐶)) | 
| 95 | 94 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp
(0g‘(Scalar‘𝐵))))) →
(0g‘(Scalar‘𝐵)) = (0g‘𝐶)) | 
| 96 | 82, 84, 95 | 3eqtr3d 2785 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp
(0g‘(Scalar‘𝐵))))) → (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (0g‘𝐶)) | 
| 97 |  | fedgmullem1.2 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) finSupp
(0g‘(Scalar‘𝐶))) | 
| 98 |  | breq1 5146 | . . . . . . . . . . . . 13
⊢ (𝑔 = (𝐺‘𝑗) → (𝑔 finSupp
(0g‘(Scalar‘𝐶)) ↔ (𝐺‘𝑗) finSupp
(0g‘(Scalar‘𝐶)))) | 
| 99 |  | fveq1 6905 | . . . . . . . . . . . . . . . . 17
⊢ (𝑔 = (𝐺‘𝑗) → (𝑔‘𝑖) = ((𝐺‘𝑗)‘𝑖)) | 
| 100 | 99 | oveq1d 7446 | . . . . . . . . . . . . . . . 16
⊢ (𝑔 = (𝐺‘𝑗) → ((𝑔‘𝑖)( ·𝑠
‘𝐶)𝑖) = (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖)) | 
| 101 | 100 | mpteq2dv 5244 | . . . . . . . . . . . . . . 15
⊢ (𝑔 = (𝐺‘𝑗) → (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠
‘𝐶)𝑖)) = (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖))) | 
| 102 | 101 | oveq2d 7447 | . . . . . . . . . . . . . 14
⊢ (𝑔 = (𝐺‘𝑗) → (𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖)))) | 
| 103 | 102 | eqeq1d 2739 | . . . . . . . . . . . . 13
⊢ (𝑔 = (𝐺‘𝑗) → ((𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (0g‘𝐶) ↔ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (0g‘𝐶))) | 
| 104 | 98, 103 | anbi12d 632 | . . . . . . . . . . . 12
⊢ (𝑔 = (𝐺‘𝑗) → ((𝑔 finSupp
(0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (0g‘𝐶)) ↔ ((𝐺‘𝑗) finSupp
(0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (0g‘𝐶)))) | 
| 105 |  | eqeq1 2741 | . . . . . . . . . . . 12
⊢ (𝑔 = (𝐺‘𝑗) → (𝑔 = (𝑋 ×
{(0g‘(Scalar‘𝐶))}) ↔ (𝐺‘𝑗) = (𝑋 ×
{(0g‘(Scalar‘𝐶))}))) | 
| 106 | 104, 105 | imbi12d 344 | . . . . . . . . . . 11
⊢ (𝑔 = (𝐺‘𝑗) → (((𝑔 finSupp
(0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (0g‘𝐶)) → 𝑔 = (𝑋 ×
{(0g‘(Scalar‘𝐶))})) ↔ (((𝐺‘𝑗) finSupp
(0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (0g‘𝐶)) → (𝐺‘𝑗) = (𝑋 ×
{(0g‘(Scalar‘𝐶))})))) | 
| 107 |  | fedgmul.3 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐾 ∈ DivRing) | 
| 108 | 29, 107 | eqeltrd 2841 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐹 ↾s 𝑉) ∈ DivRing) | 
| 109 |  | eqid 2737 | . . . . . . . . . . . . . . . 16
⊢ (𝐹 ↾s 𝑉) = (𝐹 ↾s 𝑉) | 
| 110 | 30, 109 | sralvec 33636 | . . . . . . . . . . . . . . 15
⊢ ((𝐹 ∈ DivRing ∧ (𝐹 ↾s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec) | 
| 111 | 92, 108, 14, 110 | syl3anc 1373 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐶 ∈ LVec) | 
| 112 |  | lveclmod 21105 | . . . . . . . . . . . . . 14
⊢ (𝐶 ∈ LVec → 𝐶 ∈ LMod) | 
| 113 | 111, 112 | syl 17 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝐶 ∈ LMod) | 
| 114 | 113 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐶 ∈ LMod) | 
| 115 |  | eqid 2737 | . . . . . . . . . . . . . . 15
⊢
(Base‘𝐶) =
(Base‘𝐶) | 
| 116 |  | eqid 2737 | . . . . . . . . . . . . . . 15
⊢
(LBasis‘𝐶) =
(LBasis‘𝐶) | 
| 117 | 115, 116 | lbsss 21076 | . . . . . . . . . . . . . 14
⊢ (𝑋 ∈ (LBasis‘𝐶) → 𝑋 ⊆ (Base‘𝐶)) | 
| 118 | 47, 117 | syl 17 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐶)) | 
| 119 | 118 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑋 ⊆ (Base‘𝐶)) | 
| 120 |  | eqid 2737 | . . . . . . . . . . . . . . . 16
⊢
(LSpan‘𝐶) =
(LSpan‘𝐶) | 
| 121 | 115, 116,
120 | islbs4 21852 | . . . . . . . . . . . . . . 15
⊢ (𝑋 ∈ (LBasis‘𝐶) ↔ (𝑋 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑋) = (Base‘𝐶))) | 
| 122 | 47, 121 | sylib 218 | . . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑋 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑋) = (Base‘𝐶))) | 
| 123 | 122 | simpld 494 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝑋 ∈ (LIndS‘𝐶)) | 
| 124 | 123 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑋 ∈ (LIndS‘𝐶)) | 
| 125 |  | eqid 2737 | . . . . . . . . . . . . . 14
⊢
(Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶)) | 
| 126 |  | eqid 2737 | . . . . . . . . . . . . . 14
⊢
(Scalar‘𝐶) =
(Scalar‘𝐶) | 
| 127 |  | eqid 2737 | . . . . . . . . . . . . . 14
⊢ (
·𝑠 ‘𝐶) = ( ·𝑠
‘𝐶) | 
| 128 |  | eqid 2737 | . . . . . . . . . . . . . 14
⊢
(0g‘𝐶) = (0g‘𝐶) | 
| 129 |  | eqid 2737 | . . . . . . . . . . . . . 14
⊢
(0g‘(Scalar‘𝐶)) =
(0g‘(Scalar‘𝐶)) | 
| 130 | 115, 125,
126, 127, 128, 129 | islinds5 33395 | . . . . . . . . . . . . 13
⊢ ((𝐶 ∈ LMod ∧ 𝑋 ⊆ (Base‘𝐶)) → (𝑋 ∈ (LIndS‘𝐶) ↔ ∀𝑔 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋)((𝑔 finSupp
(0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (0g‘𝐶)) → 𝑔 = (𝑋 ×
{(0g‘(Scalar‘𝐶))})))) | 
| 131 | 130 | biimpa 476 | . . . . . . . . . . . 12
⊢ (((𝐶 ∈ LMod ∧ 𝑋 ⊆ (Base‘𝐶)) ∧ 𝑋 ∈ (LIndS‘𝐶)) → ∀𝑔 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋)((𝑔 finSupp
(0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (0g‘𝐶)) → 𝑔 = (𝑋 ×
{(0g‘(Scalar‘𝐶))}))) | 
| 132 | 114, 119,
124, 131 | syl21anc 838 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ∀𝑔 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋)((𝑔 finSupp
(0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (0g‘𝐶)) → 𝑔 = (𝑋 ×
{(0g‘(Scalar‘𝐶))}))) | 
| 133 | 106, 132,
55 | rspcdva 3623 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (((𝐺‘𝑗) finSupp
(0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (0g‘𝐶)) → (𝐺‘𝑗) = (𝑋 ×
{(0g‘(Scalar‘𝐶))}))) | 
| 134 | 97, 133 | mpand 695 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (0g‘𝐶) → (𝐺‘𝑗) = (𝑋 ×
{(0g‘(Scalar‘𝐶))}))) | 
| 135 | 134 | imp 406 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (0g‘𝐶)) → (𝐺‘𝑗) = (𝑋 ×
{(0g‘(Scalar‘𝐶))})) | 
| 136 | 76, 78, 96, 135 | syl21anc 838 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp
(0g‘(Scalar‘𝐵))))) → (𝐺‘𝑗) = (𝑋 ×
{(0g‘(Scalar‘𝐶))})) | 
| 137 | 1, 136 | suppss 8219 | . . . . . 6
⊢ (𝜑 → (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))})) ⊆ (𝐿 supp
(0g‘(Scalar‘𝐵)))) | 
| 138 | 75, 137 | ssfid 9301 | . . . . 5
⊢ (𝜑 → (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))})) ∈ Fin) | 
| 139 |  | suppssdm 8202 | . . . . . . . . . 10
⊢ (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))})) ⊆ dom 𝐺 | 
| 140 | 139, 1 | fssdm 6755 | . . . . . . . . 9
⊢ (𝜑 → (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))})) ⊆ 𝑌) | 
| 141 | 140 | sselda 3983 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))) → 𝑤 ∈ 𝑌) | 
| 142 |  | eleq1w 2824 | . . . . . . . . . . . 12
⊢ (𝑗 = 𝑤 → (𝑗 ∈ 𝑌 ↔ 𝑤 ∈ 𝑌)) | 
| 143 | 142 | anbi2d 630 | . . . . . . . . . . 11
⊢ (𝑗 = 𝑤 → ((𝜑 ∧ 𝑗 ∈ 𝑌) ↔ (𝜑 ∧ 𝑤 ∈ 𝑌))) | 
| 144 |  | fveq2 6906 | . . . . . . . . . . . 12
⊢ (𝑗 = 𝑤 → (𝐺‘𝑗) = (𝐺‘𝑤)) | 
| 145 | 144 | breq1d 5153 | . . . . . . . . . . 11
⊢ (𝑗 = 𝑤 → ((𝐺‘𝑗) finSupp
(0g‘(Scalar‘𝐶)) ↔ (𝐺‘𝑤) finSupp
(0g‘(Scalar‘𝐶)))) | 
| 146 | 143, 145 | imbi12d 344 | . . . . . . . . . 10
⊢ (𝑗 = 𝑤 → (((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) finSupp
(0g‘(Scalar‘𝐶))) ↔ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝐺‘𝑤) finSupp
(0g‘(Scalar‘𝐶))))) | 
| 147 | 146, 97 | chvarvv 1998 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝐺‘𝑤) finSupp
(0g‘(Scalar‘𝐶))) | 
| 148 | 147 | fsuppimpd 9409 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝐺‘𝑤) supp
(0g‘(Scalar‘𝐶))) ∈ Fin) | 
| 149 | 141, 148 | syldan 591 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))) → ((𝐺‘𝑤) supp
(0g‘(Scalar‘𝐶))) ∈ Fin) | 
| 150 | 149 | ralrimiva 3146 | . . . . . 6
⊢ (𝜑 → ∀𝑤 ∈ (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp
(0g‘(Scalar‘𝐶))) ∈ Fin) | 
| 151 |  | iunfi 9383 | . . . . . 6
⊢ (((𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))})) ∈ Fin ∧ ∀𝑤 ∈ (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp
(0g‘(Scalar‘𝐶))) ∈ Fin) → ∪ 𝑤 ∈ (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp
(0g‘(Scalar‘𝐶))) ∈ Fin) | 
| 152 | 138, 150,
151 | syl2anc 584 | . . . . 5
⊢ (𝜑 → ∪ 𝑤 ∈ (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp
(0g‘(Scalar‘𝐶))) ∈ Fin) | 
| 153 |  | xpfi 9358 | . . . . 5
⊢ (((𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))})) ∈ Fin ∧ ∪ 𝑤 ∈ (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp
(0g‘(Scalar‘𝐶))) ∈ Fin) → ((𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))})) × ∪ 𝑤 ∈ (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp
(0g‘(Scalar‘𝐶)))) ∈ Fin) | 
| 154 | 138, 152,
153 | syl2anc 584 | . . . 4
⊢ (𝜑 → ((𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))})) × ∪ 𝑤 ∈ (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp
(0g‘(Scalar‘𝐶)))) ∈ Fin) | 
| 155 |  | fveq2 6906 | . . . . . . . . . 10
⊢ (𝑣 = 𝑗 → (𝐺‘𝑣) = (𝐺‘𝑗)) | 
| 156 | 155 | fveq1d 6908 | . . . . . . . . 9
⊢ (𝑣 = 𝑗 → ((𝐺‘𝑣)‘𝑢) = ((𝐺‘𝑗)‘𝑢)) | 
| 157 | 156 | mpteq2dv 5244 | . . . . . . . 8
⊢ (𝑣 = 𝑗 → (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)) = (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑢))) | 
| 158 |  | fveq2 6906 | . . . . . . . . 9
⊢ (𝑢 = 𝑖 → ((𝐺‘𝑗)‘𝑢) = ((𝐺‘𝑗)‘𝑖)) | 
| 159 | 158 | cbvmptv 5255 | . . . . . . . 8
⊢ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑢)) = (𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖)) | 
| 160 | 157, 159 | eqtrdi 2793 | . . . . . . 7
⊢ (𝑣 = 𝑗 → (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)) = (𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖))) | 
| 161 | 160 | cbvmptv 5255 | . . . . . 6
⊢ (𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) = (𝑗 ∈ 𝑌 ↦ (𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖))) | 
| 162 |  | fvexd 6921 | . . . . . 6
⊢ (𝜑 →
(0g‘(Scalar‘𝐶)) ∈ V) | 
| 163 |  | fvexd 6921 | . . . . . 6
⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → ((𝐺‘𝑗)‘𝑖) ∈ V) | 
| 164 | 42, 161, 46, 47, 162, 163 | suppovss 32690 | . . . . 5
⊢ (𝜑 → (𝐻 supp
(0g‘(Scalar‘𝐶))) ⊆ (((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) supp (𝑋 ×
{(0g‘(Scalar‘𝐶))})) × ∪ 𝑤 ∈ ((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))(((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))‘𝑤) supp
(0g‘(Scalar‘𝐶))))) | 
| 165 | 10, 71 | subrg0 20579 | . . . . . . . 8
⊢ (𝑉 ∈ (SubRing‘𝐸) →
(0g‘𝐸) =
(0g‘𝐾)) | 
| 166 | 19, 165 | syl 17 | . . . . . . 7
⊢ (𝜑 → (0g‘𝐸) = (0g‘𝐾)) | 
| 167 | 36 | fveq2d 6910 | . . . . . . 7
⊢ (𝜑 → (0g‘𝐾) =
(0g‘(Scalar‘𝐶))) | 
| 168 | 166, 167 | eqtr2d 2778 | . . . . . 6
⊢ (𝜑 →
(0g‘(Scalar‘𝐶)) = (0g‘𝐸)) | 
| 169 | 168 | oveq2d 7447 | . . . . 5
⊢ (𝜑 → (𝐻 supp
(0g‘(Scalar‘𝐶))) = (𝐻 supp (0g‘𝐸))) | 
| 170 | 1 | feqmptd 6977 | . . . . . . . 8
⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝑌 ↦ (𝐺‘𝑣))) | 
| 171 |  | eleq1w 2824 | . . . . . . . . . . . . 13
⊢ (𝑗 = 𝑣 → (𝑗 ∈ 𝑌 ↔ 𝑣 ∈ 𝑌)) | 
| 172 | 171 | anbi2d 630 | . . . . . . . . . . . 12
⊢ (𝑗 = 𝑣 → ((𝜑 ∧ 𝑗 ∈ 𝑌) ↔ (𝜑 ∧ 𝑣 ∈ 𝑌))) | 
| 173 |  | fveq2 6906 | . . . . . . . . . . . . 13
⊢ (𝑗 = 𝑣 → (𝐺‘𝑗) = (𝐺‘𝑣)) | 
| 174 | 173 | feq1d 6720 | . . . . . . . . . . . 12
⊢ (𝑗 = 𝑣 → ((𝐺‘𝑗):𝑋⟶(Base‘𝐸) ↔ (𝐺‘𝑣):𝑋⟶(Base‘𝐸))) | 
| 175 | 172, 174 | imbi12d 344 | . . . . . . . . . . 11
⊢ (𝑗 = 𝑣 → (((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗):𝑋⟶(Base‘𝐸)) ↔ ((𝜑 ∧ 𝑣 ∈ 𝑌) → (𝐺‘𝑣):𝑋⟶(Base‘𝐸)))) | 
| 176 | 10, 20 | ressbas2 17283 | . . . . . . . . . . . . . . . 16
⊢ (𝑉 ⊆ (Base‘𝐸) → 𝑉 = (Base‘𝐾)) | 
| 177 | 22, 176 | syl 17 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑉 = (Base‘𝐾)) | 
| 178 | 36 | fveq2d 6910 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → (Base‘𝐾) =
(Base‘(Scalar‘𝐶))) | 
| 179 | 177, 178 | eqtrd 2777 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑉 = (Base‘(Scalar‘𝐶))) | 
| 180 | 179, 22 | eqsstrrd 4019 | . . . . . . . . . . . . 13
⊢ (𝜑 →
(Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸)) | 
| 181 | 180 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸)) | 
| 182 | 56, 181 | fssd 6753 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗):𝑋⟶(Base‘𝐸)) | 
| 183 | 175, 182 | chvarvv 1998 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑣 ∈ 𝑌) → (𝐺‘𝑣):𝑋⟶(Base‘𝐸)) | 
| 184 | 183 | feqmptd 6977 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑣 ∈ 𝑌) → (𝐺‘𝑣) = (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) | 
| 185 | 184 | mpteq2dva 5242 | . . . . . . . 8
⊢ (𝜑 → (𝑣 ∈ 𝑌 ↦ (𝐺‘𝑣)) = (𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))) | 
| 186 | 170, 185 | eqtr2d 2778 | . . . . . . 7
⊢ (𝜑 → (𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) = 𝐺) | 
| 187 | 186 | oveq1d 7446 | . . . . . 6
⊢ (𝜑 → ((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) supp (𝑋 ×
{(0g‘(Scalar‘𝐶))})) = (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))) | 
| 188 | 186 | fveq1d 6908 | . . . . . . . 8
⊢ (𝜑 → ((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))‘𝑤) = (𝐺‘𝑤)) | 
| 189 | 188 | oveq1d 7446 | . . . . . . 7
⊢ (𝜑 → (((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))‘𝑤) supp
(0g‘(Scalar‘𝐶))) = ((𝐺‘𝑤) supp
(0g‘(Scalar‘𝐶)))) | 
| 190 | 187, 189 | iuneq12d 5021 | . . . . . 6
⊢ (𝜑 → ∪ 𝑤 ∈ ((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))(((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))‘𝑤) supp
(0g‘(Scalar‘𝐶))) = ∪
𝑤 ∈ (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp
(0g‘(Scalar‘𝐶)))) | 
| 191 | 187, 190 | xpeq12d 5716 | . . . . 5
⊢ (𝜑 → (((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) supp (𝑋 ×
{(0g‘(Scalar‘𝐶))})) × ∪ 𝑤 ∈ ((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))(((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))‘𝑤) supp
(0g‘(Scalar‘𝐶)))) = ((𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))})) × ∪ 𝑤 ∈ (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp
(0g‘(Scalar‘𝐶))))) | 
| 192 | 164, 169,
191 | 3sstr3d 4038 | . . . 4
⊢ (𝜑 → (𝐻 supp (0g‘𝐸)) ⊆ ((𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))})) × ∪ 𝑤 ∈ (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp
(0g‘(Scalar‘𝐶))))) | 
| 193 |  | suppssfifsupp 9420 | . . . 4
⊢ (((𝐻 ∈
((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)) ∧ Fun 𝐻 ∧ (0g‘𝐸) ∈ (Base‘𝐸)) ∧ (((𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))})) × ∪ 𝑤 ∈ (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp
(0g‘(Scalar‘𝐶)))) ∈ Fin ∧ (𝐻 supp (0g‘𝐸)) ⊆ ((𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))})) × ∪ 𝑤 ∈ (𝐺 supp (𝑋 ×
{(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp
(0g‘(Scalar‘𝐶)))))) → 𝐻 finSupp (0g‘𝐸)) | 
| 194 | 52, 66, 73, 154, 192, 193 | syl32anc 1380 | . . 3
⊢ (𝜑 → 𝐻 finSupp (0g‘𝐸)) | 
| 195 | 37 | fveq2d 6910 | . . . 4
⊢ (𝜑 →
(0g‘(Scalar‘𝐴)) =
(0g‘(Scalar‘𝐶))) | 
| 196 | 195, 168 | eqtr2d 2778 | . . 3
⊢ (𝜑 → (0g‘𝐸) =
(0g‘(Scalar‘𝐴))) | 
| 197 | 194, 196 | breqtrd 5169 | . 2
⊢ (𝜑 → 𝐻 finSupp
(0g‘(Scalar‘𝐴))) | 
| 198 |  | fedgmullem1.z | . . 3
⊢ (𝜑 → 𝑍 = (𝐵 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠
‘𝐵)𝑗)))) | 
| 199 | 85, 67, 13, 15, 92, 46 | drgextgsum 33645 | . . 3
⊢ (𝜑 → (𝐸 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠
‘𝐵)𝑗))) = (𝐵 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠
‘𝐵)𝑗)))) | 
| 200 | 47 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑋 ∈ (LBasis‘𝐶)) | 
| 201 | 13 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑈 ∈ (SubRing‘𝐸)) | 
| 202 |  | subrgsubg 20577 | . . . . . . . . . . . . 13
⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ∈ (SubGrp‘𝐸)) | 
| 203 |  | subgsubm 19166 | . . . . . . . . . . . . 13
⊢ (𝑈 ∈ (SubGrp‘𝐸) → 𝑈 ∈ (SubMnd‘𝐸)) | 
| 204 | 201, 202,
203 | 3syl 18 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑈 ∈ (SubMnd‘𝐸)) | 
| 205 | 113 | ad2antrr 726 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐶 ∈ LMod) | 
| 206 | 56 | ffvelcdmda 7104 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶))) | 
| 207 | 118 | ad2antrr 726 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑋 ⊆ (Base‘𝐶)) | 
| 208 | 207, 61 | sseldd 3984 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐶)) | 
| 209 | 115, 126,
127, 125 | lmodvscl 20876 | . . . . . . . . . . . . . . 15
⊢ ((𝐶 ∈ LMod ∧ ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑖 ∈ (Base‘𝐶)) → (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖) ∈ (Base‘𝐶)) | 
| 210 | 205, 206,
208, 209 | syl3anc 1373 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖) ∈ (Base‘𝐶)) | 
| 211 | 15, 20 | ressbas2 17283 | . . . . . . . . . . . . . . . . 17
⊢ (𝑈 ⊆ (Base‘𝐸) → 𝑈 = (Base‘𝐹)) | 
| 212 | 88, 211 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑈 = (Base‘𝐹)) | 
| 213 | 31, 34 | srabase 21177 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (Base‘𝐹) = (Base‘𝐶)) | 
| 214 | 212, 213 | eqtrd 2777 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑈 = (Base‘𝐶)) | 
| 215 | 214 | ad2antrr 726 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑈 = (Base‘𝐶)) | 
| 216 | 210, 215 | eleqtrrd 2844 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖) ∈ 𝑈) | 
| 217 | 216 | fmpttd 7135 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖)):𝑋⟶𝑈) | 
| 218 | 200, 204,
217, 15 | gsumsubm 18848 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (𝐹 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖)))) | 
| 219 |  | eqid 2737 | . . . . . . . . . . . . . . . . . 18
⊢
(.r‘𝐸) = (.r‘𝐸) | 
| 220 | 15, 219 | ressmulr 17351 | . . . . . . . . . . . . . . . . 17
⊢ (𝑈 ∈ (SubRing‘𝐸) →
(.r‘𝐸) =
(.r‘𝐹)) | 
| 221 | 13, 220 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (.r‘𝐸) = (.r‘𝐹)) | 
| 222 | 31, 34 | sravsca 21185 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (.r‘𝐹) = (
·𝑠 ‘𝐶)) | 
| 223 | 221, 222 | eqtr2d 2778 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → (
·𝑠 ‘𝐶) = (.r‘𝐸)) | 
| 224 | 223 | ad2antrr 726 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (
·𝑠 ‘𝐶) = (.r‘𝐸)) | 
| 225 | 224 | oveqd 7448 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖) = (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)) | 
| 226 | 225 | mpteq2dva 5242 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖)) = (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖))) | 
| 227 | 226 | oveq2d 7447 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)))) | 
| 228 | 30, 92, 14, 109, 108, 47 | drgextgsum 33645 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝐹 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖)))) | 
| 229 | 228 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐹 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖))) = (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖)))) | 
| 230 | 218, 227,
229 | 3eqtr3d 2785 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖))) = (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖)))) | 
| 231 | 230 | oveq1d 7446 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐸 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)))(.r‘𝐸)𝑗) = ((𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖)))(.r‘𝐸)𝑗)) | 
| 232 | 69 | ad2antrr 726 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ Ring) | 
| 233 | 180 | ad2antrr 726 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸)) | 
| 234 | 233, 206 | sseldd 3984 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘𝐸)) | 
| 235 | 214, 88 | eqsstrrd 4019 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸)) | 
| 236 | 118, 235 | sstrd 3994 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐸)) | 
| 237 | 236 | ad2antrr 726 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑋 ⊆ (Base‘𝐸)) | 
| 238 | 237, 61 | sseldd 3984 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐸)) | 
| 239 |  | eqid 2737 | . . . . . . . . . . . . . . . . . 18
⊢
(Base‘𝐵) =
(Base‘𝐵) | 
| 240 |  | eqid 2737 | . . . . . . . . . . . . . . . . . 18
⊢
(LBasis‘𝐵) =
(LBasis‘𝐵) | 
| 241 | 239, 240 | lbsss 21076 | . . . . . . . . . . . . . . . . 17
⊢ (𝑌 ∈ (LBasis‘𝐵) → 𝑌 ⊆ (Base‘𝐵)) | 
| 242 | 46, 241 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑌 ⊆ (Base‘𝐵)) | 
| 243 | 86, 88 | srabase 21177 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐵)) | 
| 244 | 242, 243 | sseqtrrd 4021 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑌 ⊆ (Base‘𝐸)) | 
| 245 | 244 | ad2antrr 726 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑌 ⊆ (Base‘𝐸)) | 
| 246 |  | simplr 769 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ 𝑌) | 
| 247 | 245, 246 | sseldd 3984 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ (Base‘𝐸)) | 
| 248 | 20, 219 | ringass 20250 | . . . . . . . . . . . . 13
⊢ ((𝐸 ∈ Ring ∧ (((𝐺‘𝑗)‘𝑖) ∈ (Base‘𝐸) ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸))) → ((((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)(.r‘𝐸)𝑗) = (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗))) | 
| 249 | 232, 234,
238, 247, 248 | syl13anc 1374 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)(.r‘𝐸)𝑗) = (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗))) | 
| 250 | 249 | mpteq2dva 5242 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ ((((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)(.r‘𝐸)𝑗)) = (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))) | 
| 251 | 250 | oveq2d 7447 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐸 Σg (𝑖 ∈ 𝑋 ↦ ((((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)(.r‘𝐸)𝑗))) = (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗))))) | 
| 252 | 69 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐸 ∈ Ring) | 
| 253 | 242 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑌 ⊆ (Base‘𝐵)) | 
| 254 | 243 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (Base‘𝐸) = (Base‘𝐵)) | 
| 255 | 253, 254 | sseqtrrd 4021 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑌 ⊆ (Base‘𝐸)) | 
| 256 |  | simpr 484 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ 𝑌) | 
| 257 | 255, 256 | sseldd 3984 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ (Base‘𝐸)) | 
| 258 | 20, 219 | ringcl 20247 | . . . . . . . . . . . 12
⊢ ((𝐸 ∈ Ring ∧ ((𝐺‘𝑗)‘𝑖) ∈ (Base‘𝐸) ∧ 𝑖 ∈ (Base‘𝐸)) → (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖) ∈ (Base‘𝐸)) | 
| 259 | 232, 234,
238, 258 | syl3anc 1373 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖) ∈ (Base‘𝐸)) | 
| 260 | 168 | breq2d 5155 | . . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝐺‘𝑗) finSupp
(0g‘(Scalar‘𝐶)) ↔ (𝐺‘𝑗) finSupp (0g‘𝐸))) | 
| 261 | 260 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐺‘𝑗) finSupp
(0g‘(Scalar‘𝐶)) ↔ (𝐺‘𝑗) finSupp (0g‘𝐸))) | 
| 262 | 97, 261 | mpbid 232 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) finSupp (0g‘𝐸)) | 
| 263 | 20, 252, 200, 238, 182, 262 | rmfsupp2 33242 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)) finSupp (0g‘𝐸)) | 
| 264 | 20, 71, 219, 252, 200, 257, 259, 263 | gsummulc1 20313 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐸 Σg (𝑖 ∈ 𝑋 ↦ ((((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)(.r‘𝐸)𝑗))) = ((𝐸 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)))(.r‘𝐸)𝑗)) | 
| 265 | 251, 264 | eqtr3d 2779 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))) = ((𝐸 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)))(.r‘𝐸)𝑗)) | 
| 266 | 83 | oveq1d 7446 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐿‘𝑗)(.r‘𝐸)𝑗) = ((𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠
‘𝐶)𝑖)))(.r‘𝐸)𝑗)) | 
| 267 | 231, 265,
266 | 3eqtr4rd 2788 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐿‘𝑗)(.r‘𝐸)𝑗) = (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗))))) | 
| 268 | 86, 88 | sravsca 21185 | . . . . . . . . . 10
⊢ (𝜑 → (.r‘𝐸) = (
·𝑠 ‘𝐵)) | 
| 269 | 268 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (.r‘𝐸) = (
·𝑠 ‘𝐵)) | 
| 270 | 269 | oveqd 7448 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐿‘𝑗)(.r‘𝐸)𝑗) = ((𝐿‘𝑗)( ·𝑠
‘𝐵)𝑗)) | 
| 271 |  | fvexd 6921 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ V) | 
| 272 |  | ovexd 7466 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (𝑖(.r‘𝐸)𝑗) ∈ V) | 
| 273 | 42 | a1i 11 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐻 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖))) | 
| 274 |  | fedgmullem.d | . . . . . . . . . . . . . . 15
⊢ 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(.r‘𝐸)𝑗)) | 
| 275 | 274 | a1i 11 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(.r‘𝐸)𝑗))) | 
| 276 | 46, 47, 271, 272, 273, 275 | offval22 8113 | . . . . . . . . . . . . 13
⊢ (𝜑 → (𝐻 ∘f
(.r‘𝐸)𝐷) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))) | 
| 277 | 276 | oveqd 7448 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝑗(𝐻 ∘f
(.r‘𝐸)𝐷)𝑖) = (𝑗(𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))𝑖)) | 
| 278 | 277 | ad2antrr 726 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗(𝐻 ∘f
(.r‘𝐸)𝐷)𝑖) = (𝑗(𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))𝑖)) | 
| 279 |  | ovexd 7466 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)) ∈ V) | 
| 280 |  | eqid 2737 | . . . . . . . . . . . . 13
⊢ (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗))) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗))) | 
| 281 | 280 | ovmpt4g 7580 | . . . . . . . . . . . 12
⊢ ((𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ∧ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)) ∈ V) → (𝑗(𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))𝑖) = (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗))) | 
| 282 | 246, 61, 279, 281 | syl3anc 1373 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗(𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))𝑖) = (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗))) | 
| 283 | 278, 282 | eqtr2d 2778 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)) = (𝑗(𝐻 ∘f
(.r‘𝐸)𝐷)𝑖)) | 
| 284 | 283 | mpteq2dva 5242 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗))) = (𝑖 ∈ 𝑋 ↦ (𝑗(𝐻 ∘f
(.r‘𝐸)𝐷)𝑖))) | 
| 285 | 284 | oveq2d 7447 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))) = (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (𝑗(𝐻 ∘f
(.r‘𝐸)𝐷)𝑖)))) | 
| 286 | 267, 270,
285 | 3eqtr3d 2785 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐿‘𝑗)( ·𝑠
‘𝐵)𝑗) = (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (𝑗(𝐻 ∘f
(.r‘𝐸)𝐷)𝑖)))) | 
| 287 | 286 | mpteq2dva 5242 | . . . . . 6
⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠
‘𝐵)𝑗)) = (𝑗 ∈ 𝑌 ↦ (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (𝑗(𝐻 ∘f
(.r‘𝐸)𝐷)𝑖))))) | 
| 288 | 287 | oveq2d 7447 | . . . . 5
⊢ (𝜑 → (𝐸 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠
‘𝐵)𝑗))) = (𝐸 Σg (𝑗 ∈ 𝑌 ↦ (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (𝑗(𝐻 ∘f
(.r‘𝐸)𝐷)𝑖)))))) | 
| 289 |  | ringcmn 20279 | . . . . . . 7
⊢ (𝐸 ∈ Ring → 𝐸 ∈ CMnd) | 
| 290 | 69, 289 | syl 17 | . . . . . 6
⊢ (𝜑 → 𝐸 ∈ CMnd) | 
| 291 | 69 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝐸 ∈ Ring) | 
| 292 | 38, 180 | eqsstrd 4018 | . . . . . . . . . 10
⊢ (𝜑 →
(Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸)) | 
| 293 | 292 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸)) | 
| 294 |  | simprl 771 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑙 ∈ (Base‘(Scalar‘𝐴))) | 
| 295 | 293, 294 | sseldd 3984 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑙 ∈ (Base‘𝐸)) | 
| 296 |  | simprr 773 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑘 ∈ (Base‘𝐴)) | 
| 297 | 12, 22 | srabase 21177 | . . . . . . . . . 10
⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐴)) | 
| 298 | 297 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (Base‘𝐸) = (Base‘𝐴)) | 
| 299 | 296, 298 | eleqtrrd 2844 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑘 ∈ (Base‘𝐸)) | 
| 300 | 20, 219 | ringcl 20247 | . . . . . . . 8
⊢ ((𝐸 ∈ Ring ∧ 𝑙 ∈ (Base‘𝐸) ∧ 𝑘 ∈ (Base‘𝐸)) → (𝑙(.r‘𝐸)𝑘) ∈ (Base‘𝐸)) | 
| 301 | 291, 295,
299, 300 | syl3anc 1373 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (𝑙(.r‘𝐸)𝑘) ∈ (Base‘𝐸)) | 
| 302 | 20, 219 | ringcl 20247 | . . . . . . . . . . . 12
⊢ ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸)) | 
| 303 | 232, 238,
247, 302 | syl3anc 1373 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸)) | 
| 304 | 297 | ad2antrr 726 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (Base‘𝐸) = (Base‘𝐴)) | 
| 305 | 303, 304 | eleqtrd 2843 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴)) | 
| 306 | 305 | anasss 466 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴)) | 
| 307 | 306 | ralrimivva 3202 | . . . . . . . 8
⊢ (𝜑 → ∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴)) | 
| 308 | 274 | fmpo 8093 | . . . . . . . 8
⊢
(∀𝑗 ∈
𝑌 ∀𝑖 ∈ 𝑋 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴) ↔ 𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴)) | 
| 309 | 307, 308 | sylib 218 | . . . . . . 7
⊢ (𝜑 → 𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴)) | 
| 310 |  | inidm 4227 | . . . . . . 7
⊢ ((𝑌 × 𝑋) ∩ (𝑌 × 𝑋)) = (𝑌 × 𝑋) | 
| 311 | 301, 65, 309, 48, 48, 310 | off 7715 | . . . . . 6
⊢ (𝜑 → (𝐻 ∘f
(.r‘𝐸)𝐷):(𝑌 × 𝑋)⟶(Base‘𝐸)) | 
| 312 | 69 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → 𝐸 ∈ Ring) | 
| 313 |  | simpr 484 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → 𝑢 ∈ (Base‘𝐴)) | 
| 314 | 297 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → (Base‘𝐸) = (Base‘𝐴)) | 
| 315 | 313, 314 | eleqtrrd 2844 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → 𝑢 ∈ (Base‘𝐸)) | 
| 316 | 20, 219, 71 | ringlz 20290 | . . . . . . . 8
⊢ ((𝐸 ∈ Ring ∧ 𝑢 ∈ (Base‘𝐸)) →
((0g‘𝐸)(.r‘𝐸)𝑢) = (0g‘𝐸)) | 
| 317 | 312, 315,
316 | syl2anc 584 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → ((0g‘𝐸)(.r‘𝐸)𝑢) = (0g‘𝐸)) | 
| 318 | 48, 73, 73, 65, 309, 194, 317 | offinsupp1 32738 | . . . . . 6
⊢ (𝜑 → (𝐻 ∘f
(.r‘𝐸)𝐷) finSupp (0g‘𝐸)) | 
| 319 | 20, 71, 290, 46, 47, 311, 318 | gsumxp 19994 | . . . . 5
⊢ (𝜑 → (𝐸 Σg (𝐻 ∘f
(.r‘𝐸)𝐷)) = (𝐸 Σg (𝑗 ∈ 𝑌 ↦ (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (𝑗(𝐻 ∘f
(.r‘𝐸)𝐷)𝑖)))))) | 
| 320 | 12, 22 | sravsca 21185 | . . . . . . . 8
⊢ (𝜑 → (.r‘𝐸) = (
·𝑠 ‘𝐴)) | 
| 321 | 320 | ofeqd 7699 | . . . . . . 7
⊢ (𝜑 → ∘f
(.r‘𝐸) =
∘f ( ·𝑠 ‘𝐴)) | 
| 322 | 321 | oveqd 7448 | . . . . . 6
⊢ (𝜑 → (𝐻 ∘f
(.r‘𝐸)𝐷) = (𝐻 ∘f (
·𝑠 ‘𝐴)𝐷)) | 
| 323 | 322 | oveq2d 7447 | . . . . 5
⊢ (𝜑 → (𝐸 Σg (𝐻 ∘f
(.r‘𝐸)𝐷)) = (𝐸 Σg (𝐻 ∘f (
·𝑠 ‘𝐴)𝐷))) | 
| 324 | 288, 319,
323 | 3eqtr2rd 2784 | . . . 4
⊢ (𝜑 → (𝐸 Σg (𝐻 ∘f (
·𝑠 ‘𝐴)𝐷)) = (𝐸 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠
‘𝐵)𝑗)))) | 
| 325 |  | ovexd 7466 | . . . . 5
⊢ (𝜑 → (𝐻 ∘f (
·𝑠 ‘𝐴)𝐷) ∈ V) | 
| 326 |  | fedgmullem1.a | . . . . . 6
⊢ (𝜑 → 𝑍 ∈ (Base‘𝐴)) | 
| 327 | 326 | elfvexd 6945 | . . . . 5
⊢ (𝜑 → 𝐴 ∈ V) | 
| 328 | 11, 325, 67, 327, 22 | gsumsra 33050 | . . . 4
⊢ (𝜑 → (𝐸 Σg (𝐻 ∘f (
·𝑠 ‘𝐴)𝐷)) = (𝐴 Σg (𝐻 ∘f (
·𝑠 ‘𝐴)𝐷))) | 
| 329 | 324, 328 | eqtr3d 2779 | . . 3
⊢ (𝜑 → (𝐸 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠
‘𝐵)𝑗))) = (𝐴 Σg (𝐻 ∘f (
·𝑠 ‘𝐴)𝐷))) | 
| 330 | 198, 199,
329 | 3eqtr2d 2783 | . 2
⊢ (𝜑 → 𝑍 = (𝐴 Σg (𝐻 ∘f (
·𝑠 ‘𝐴)𝐷))) | 
| 331 | 197, 330 | jca 511 | 1
⊢ (𝜑 → (𝐻 finSupp
(0g‘(Scalar‘𝐴)) ∧ 𝑍 = (𝐴 Σg (𝐻 ∘f (
·𝑠 ‘𝐴)𝐷)))) |