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Theorem fedgmullem1 33657
Description: Lemma for fedgmul 33659. (Contributed by Thierry Arnoux, 20-Jul-2023.)
Hypotheses
Ref Expression
fedgmul.a 𝐴 = ((subringAlg ‘𝐸)‘𝑉)
fedgmul.b 𝐵 = ((subringAlg ‘𝐸)‘𝑈)
fedgmul.c 𝐶 = ((subringAlg ‘𝐹)‘𝑉)
fedgmul.f 𝐹 = (𝐸s 𝑈)
fedgmul.k 𝐾 = (𝐸s 𝑉)
fedgmul.1 (𝜑𝐸 ∈ DivRing)
fedgmul.2 (𝜑𝐹 ∈ DivRing)
fedgmul.3 (𝜑𝐾 ∈ DivRing)
fedgmul.4 (𝜑𝑈 ∈ (SubRing‘𝐸))
fedgmul.5 (𝜑𝑉 ∈ (SubRing‘𝐹))
fedgmullem.d 𝐷 = (𝑗𝑌, 𝑖𝑋 ↦ (𝑖(.r𝐸)𝑗))
fedgmullem.h 𝐻 = (𝑗𝑌, 𝑖𝑋 ↦ ((𝐺𝑗)‘𝑖))
fedgmullem.x (𝜑𝑋 ∈ (LBasis‘𝐶))
fedgmullem.y (𝜑𝑌 ∈ (LBasis‘𝐵))
fedgmullem1.a (𝜑𝑍 ∈ (Base‘𝐴))
fedgmullem1.l (𝜑𝐿:𝑌⟶(Base‘(Scalar‘𝐵)))
fedgmullem1.1 (𝜑𝐿 finSupp (0g‘(Scalar‘𝐵)))
fedgmullem1.z (𝜑𝑍 = (𝐵 Σg (𝑗𝑌 ↦ ((𝐿𝑗)( ·𝑠𝐵)𝑗))))
fedgmullem1.g (𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋))
fedgmullem1.2 ((𝜑𝑗𝑌) → (𝐺𝑗) finSupp (0g‘(Scalar‘𝐶)))
fedgmullem1.3 ((𝜑𝑗𝑌) → (𝐿𝑗) = (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))
Assertion
Ref Expression
fedgmullem1 (𝜑 → (𝐻 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑍 = (𝐴 Σg (𝐻f ( ·𝑠𝐴)𝐷))))
Distinct variable groups:   𝐴,𝑖,𝑗   𝐵,𝑗   𝐶,𝑖,𝑗   𝐷,𝑖,𝑗   𝑖,𝐸,𝑗   𝑖,𝐺,𝑗   𝑖,𝐻,𝑗   𝑗,𝐿   𝑈,𝑖   𝑖,𝑋,𝑗   𝑖,𝑌,𝑗   𝜑,𝑖,𝑗
Allowed substitution hints:   𝐵(𝑖)   𝑈(𝑗)   𝐹(𝑖,𝑗)   𝐾(𝑖,𝑗)   𝐿(𝑖)   𝑉(𝑖,𝑗)   𝑍(𝑖,𝑗)

Proof of Theorem fedgmullem1
Dummy variables 𝑢 𝑘 𝑙 𝑔 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fedgmullem1.g . . . . 5 (𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋))
2 simpllr 776 . . . . . . . . . . . . 13 ((((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗𝑌) ∧ 𝑖𝑋) → 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋))
3 simplr 769 . . . . . . . . . . . . 13 ((((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗𝑌) ∧ 𝑖𝑋) → 𝑗𝑌)
42, 3ffvelcdmd 7105 . . . . . . . . . . . 12 ((((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗𝑌) ∧ 𝑖𝑋) → (𝐺𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋))
5 elmapi 8888 . . . . . . . . . . . 12 ((𝐺𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋) → (𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
64, 5syl 17 . . . . . . . . . . 11 ((((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗𝑌) ∧ 𝑖𝑋) → (𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
76anasss 466 . . . . . . . . . 10 (((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗𝑌𝑖𝑋)) → (𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
8 simprr 773 . . . . . . . . . 10 (((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗𝑌𝑖𝑋)) → 𝑖𝑋)
97, 8ffvelcdmd 7105 . . . . . . . . 9 (((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗𝑌𝑖𝑋)) → ((𝐺𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶)))
10 fedgmul.k . . . . . . . . . . . . 13 𝐾 = (𝐸s 𝑉)
11 fedgmul.a . . . . . . . . . . . . . . 15 𝐴 = ((subringAlg ‘𝐸)‘𝑉)
1211a1i 11 . . . . . . . . . . . . . 14 (𝜑𝐴 = ((subringAlg ‘𝐸)‘𝑉))
13 fedgmul.4 . . . . . . . . . . . . . . . . 17 (𝜑𝑈 ∈ (SubRing‘𝐸))
14 fedgmul.5 . . . . . . . . . . . . . . . . 17 (𝜑𝑉 ∈ (SubRing‘𝐹))
15 fedgmul.f . . . . . . . . . . . . . . . . . . 19 𝐹 = (𝐸s 𝑈)
1615subsubrg 20615 . . . . . . . . . . . . . . . . . 18 (𝑈 ∈ (SubRing‘𝐸) → (𝑉 ∈ (SubRing‘𝐹) ↔ (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈)))
1716biimpa 476 . . . . . . . . . . . . . . . . 17 ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ∈ (SubRing‘𝐹)) → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈))
1813, 14, 17syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈))
1918simpld 494 . . . . . . . . . . . . . . 15 (𝜑𝑉 ∈ (SubRing‘𝐸))
20 eqid 2735 . . . . . . . . . . . . . . . 16 (Base‘𝐸) = (Base‘𝐸)
2120subrgss 20589 . . . . . . . . . . . . . . 15 (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
2219, 21syl 17 . . . . . . . . . . . . . 14 (𝜑𝑉 ⊆ (Base‘𝐸))
2312, 22srasca 21201 . . . . . . . . . . . . 13 (𝜑 → (𝐸s 𝑉) = (Scalar‘𝐴))
2410, 23eqtrid 2787 . . . . . . . . . . . 12 (𝜑𝐾 = (Scalar‘𝐴))
2518simprd 495 . . . . . . . . . . . . . . 15 (𝜑𝑉𝑈)
26 ressabs 17295 . . . . . . . . . . . . . . 15 ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈) → ((𝐸s 𝑈) ↾s 𝑉) = (𝐸s 𝑉))
2713, 25, 26syl2anc 584 . . . . . . . . . . . . . 14 (𝜑 → ((𝐸s 𝑈) ↾s 𝑉) = (𝐸s 𝑉))
2815oveq1i 7441 . . . . . . . . . . . . . 14 (𝐹s 𝑉) = ((𝐸s 𝑈) ↾s 𝑉)
2927, 28, 103eqtr4g 2800 . . . . . . . . . . . . 13 (𝜑 → (𝐹s 𝑉) = 𝐾)
30 fedgmul.c . . . . . . . . . . . . . . 15 𝐶 = ((subringAlg ‘𝐹)‘𝑉)
3130a1i 11 . . . . . . . . . . . . . 14 (𝜑𝐶 = ((subringAlg ‘𝐹)‘𝑉))
32 eqid 2735 . . . . . . . . . . . . . . . 16 (Base‘𝐹) = (Base‘𝐹)
3332subrgss 20589 . . . . . . . . . . . . . . 15 (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
3414, 33syl 17 . . . . . . . . . . . . . 14 (𝜑𝑉 ⊆ (Base‘𝐹))
3531, 34srasca 21201 . . . . . . . . . . . . 13 (𝜑 → (𝐹s 𝑉) = (Scalar‘𝐶))
3629, 35eqtr3d 2777 . . . . . . . . . . . 12 (𝜑𝐾 = (Scalar‘𝐶))
3724, 36eqtr3d 2777 . . . . . . . . . . 11 (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
3837fveq2d 6911 . . . . . . . . . 10 (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
3938ad2antrr 726 . . . . . . . . 9 (((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗𝑌𝑖𝑋)) → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
409, 39eleqtrrd 2842 . . . . . . . 8 (((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗𝑌𝑖𝑋)) → ((𝐺𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
4140ralrimivva 3200 . . . . . . 7 ((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → ∀𝑗𝑌𝑖𝑋 ((𝐺𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
42 fedgmullem.h . . . . . . . 8 𝐻 = (𝑗𝑌, 𝑖𝑋 ↦ ((𝐺𝑗)‘𝑖))
4342fmpo 8092 . . . . . . 7 (∀𝑗𝑌𝑖𝑋 ((𝐺𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)) ↔ 𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
4441, 43sylib 218 . . . . . 6 ((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → 𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
45 fvexd 6922 . . . . . . 7 ((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → (Base‘(Scalar‘𝐴)) ∈ V)
46 fedgmullem.y . . . . . . . . 9 (𝜑𝑌 ∈ (LBasis‘𝐵))
47 fedgmullem.x . . . . . . . . 9 (𝜑𝑋 ∈ (LBasis‘𝐶))
4846, 47xpexd 7770 . . . . . . . 8 (𝜑 → (𝑌 × 𝑋) ∈ V)
4948adantr 480 . . . . . . 7 ((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → (𝑌 × 𝑋) ∈ V)
5045, 49elmapd 8879 . . . . . 6 ((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → (𝐻 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)) ↔ 𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴))))
5144, 50mpbird 257 . . . . 5 ((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → 𝐻 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)))
521, 51mpdan 687 . . . 4 (𝜑𝐻 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)))
53 simpl 482 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → 𝜑)
5453adantr 480 . . . . . . . . . 10 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝜑)
551ffvelcdmda 7104 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → (𝐺𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋))
5655, 5syl 17 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
5756adantr 480 . . . . . . . . . 10 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
5838feq3d 6724 . . . . . . . . . . 11 (𝜑 → ((𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐴)) ↔ (𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐶))))
5958biimpar 477 . . . . . . . . . 10 ((𝜑 ∧ (𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐶))) → (𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐴)))
6054, 57, 59syl2anc 584 . . . . . . . . 9 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐴)))
61 simpr 484 . . . . . . . . 9 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑖𝑋)
6260, 61ffvelcdmd 7105 . . . . . . . 8 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝐺𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
6362ralrimiva 3144 . . . . . . 7 ((𝜑𝑗𝑌) → ∀𝑖𝑋 ((𝐺𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
6463ralrimiva 3144 . . . . . 6 (𝜑 → ∀𝑗𝑌𝑖𝑋 ((𝐺𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
6564, 43sylib 218 . . . . 5 (𝜑𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
6665ffund 6741 . . . 4 (𝜑 → Fun 𝐻)
67 fedgmul.1 . . . . . 6 (𝜑𝐸 ∈ DivRing)
68 drngring 20753 . . . . . 6 (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
6967, 68syl 17 . . . . 5 (𝜑𝐸 ∈ Ring)
70 ringgrp 20256 . . . . 5 (𝐸 ∈ Ring → 𝐸 ∈ Grp)
71 eqid 2735 . . . . . 6 (0g𝐸) = (0g𝐸)
7220, 71grpidcl 18996 . . . . 5 (𝐸 ∈ Grp → (0g𝐸) ∈ (Base‘𝐸))
7369, 70, 723syl 18 . . . 4 (𝜑 → (0g𝐸) ∈ (Base‘𝐸))
74 fedgmullem1.1 . . . . . . 7 (𝜑𝐿 finSupp (0g‘(Scalar‘𝐵)))
7574fsuppimpd 9407 . . . . . 6 (𝜑 → (𝐿 supp (0g‘(Scalar‘𝐵))) ∈ Fin)
76 simpl 482 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → 𝜑)
77 simpr 484 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵)))))
7877eldifad 3975 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → 𝑗𝑌)
79 fedgmullem1.l . . . . . . . . . 10 (𝜑𝐿:𝑌⟶(Base‘(Scalar‘𝐵)))
80 ssidd 4019 . . . . . . . . . 10 (𝜑 → (𝐿 supp (0g‘(Scalar‘𝐵))) ⊆ (𝐿 supp (0g‘(Scalar‘𝐵))))
81 fvexd 6922 . . . . . . . . . 10 (𝜑 → (0g‘(Scalar‘𝐵)) ∈ V)
8279, 80, 46, 81suppssr 8219 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → (𝐿𝑗) = (0g‘(Scalar‘𝐵)))
83 fedgmullem1.3 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (𝐿𝑗) = (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))
8478, 83syldan 591 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → (𝐿𝑗) = (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))
85 fedgmul.b . . . . . . . . . . . . . . 15 𝐵 = ((subringAlg ‘𝐸)‘𝑈)
8685a1i 11 . . . . . . . . . . . . . 14 (𝜑𝐵 = ((subringAlg ‘𝐸)‘𝑈))
8720subrgss 20589 . . . . . . . . . . . . . . 15 (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
8813, 87syl 17 . . . . . . . . . . . . . 14 (𝜑𝑈 ⊆ (Base‘𝐸))
8986, 88srasca 21201 . . . . . . . . . . . . 13 (𝜑 → (𝐸s 𝑈) = (Scalar‘𝐵))
9015, 89eqtrid 2787 . . . . . . . . . . . 12 (𝜑𝐹 = (Scalar‘𝐵))
9190fveq2d 6911 . . . . . . . . . . 11 (𝜑 → (0g𝐹) = (0g‘(Scalar‘𝐵)))
92 fedgmul.2 . . . . . . . . . . . 12 (𝜑𝐹 ∈ DivRing)
9330, 92, 14drgext0g 33619 . . . . . . . . . . 11 (𝜑 → (0g𝐹) = (0g𝐶))
9491, 93eqtr3d 2777 . . . . . . . . . 10 (𝜑 → (0g‘(Scalar‘𝐵)) = (0g𝐶))
9594adantr 480 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → (0g‘(Scalar‘𝐵)) = (0g𝐶))
9682, 84, 953eqtr3d 2783 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶))
97 fedgmullem1.2 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (𝐺𝑗) finSupp (0g‘(Scalar‘𝐶)))
98 breq1 5151 . . . . . . . . . . . . 13 (𝑔 = (𝐺𝑗) → (𝑔 finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺𝑗) finSupp (0g‘(Scalar‘𝐶))))
99 fveq1 6906 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝐺𝑗) → (𝑔𝑖) = ((𝐺𝑗)‘𝑖))
10099oveq1d 7446 . . . . . . . . . . . . . . . 16 (𝑔 = (𝐺𝑗) → ((𝑔𝑖)( ·𝑠𝐶)𝑖) = (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))
101100mpteq2dv 5250 . . . . . . . . . . . . . . 15 (𝑔 = (𝐺𝑗) → (𝑖𝑋 ↦ ((𝑔𝑖)( ·𝑠𝐶)𝑖)) = (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖)))
102101oveq2d 7447 . . . . . . . . . . . . . 14 (𝑔 = (𝐺𝑗) → (𝐶 Σg (𝑖𝑋 ↦ ((𝑔𝑖)( ·𝑠𝐶)𝑖))) = (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))
103102eqeq1d 2737 . . . . . . . . . . . . 13 (𝑔 = (𝐺𝑗) → ((𝐶 Σg (𝑖𝑋 ↦ ((𝑔𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶) ↔ (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶)))
10498, 103anbi12d 632 . . . . . . . . . . . 12 (𝑔 = (𝐺𝑗) → ((𝑔 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖𝑋 ↦ ((𝑔𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶)) ↔ ((𝐺𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶))))
105 eqeq1 2739 . . . . . . . . . . . 12 (𝑔 = (𝐺𝑗) → (𝑔 = (𝑋 × {(0g‘(Scalar‘𝐶))}) ↔ (𝐺𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))})))
106104, 105imbi12d 344 . . . . . . . . . . 11 (𝑔 = (𝐺𝑗) → (((𝑔 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖𝑋 ↦ ((𝑔𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶)) → 𝑔 = (𝑋 × {(0g‘(Scalar‘𝐶))})) ↔ (((𝐺𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶)) → (𝐺𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))}))))
107 fedgmul.3 . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ DivRing)
10829, 107eqeltrd 2839 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹s 𝑉) ∈ DivRing)
109 eqid 2735 . . . . . . . . . . . . . . . 16 (𝐹s 𝑉) = (𝐹s 𝑉)
11030, 109sralvec 33615 . . . . . . . . . . . . . . 15 ((𝐹 ∈ DivRing ∧ (𝐹s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
11192, 108, 14, 110syl3anc 1370 . . . . . . . . . . . . . 14 (𝜑𝐶 ∈ LVec)
112 lveclmod 21123 . . . . . . . . . . . . . 14 (𝐶 ∈ LVec → 𝐶 ∈ LMod)
113111, 112syl 17 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ LMod)
114113adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝐶 ∈ LMod)
115 eqid 2735 . . . . . . . . . . . . . . 15 (Base‘𝐶) = (Base‘𝐶)
116 eqid 2735 . . . . . . . . . . . . . . 15 (LBasis‘𝐶) = (LBasis‘𝐶)
117115, 116lbsss 21094 . . . . . . . . . . . . . 14 (𝑋 ∈ (LBasis‘𝐶) → 𝑋 ⊆ (Base‘𝐶))
11847, 117syl 17 . . . . . . . . . . . . 13 (𝜑𝑋 ⊆ (Base‘𝐶))
119118adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝑋 ⊆ (Base‘𝐶))
120 eqid 2735 . . . . . . . . . . . . . . . 16 (LSpan‘𝐶) = (LSpan‘𝐶)
121115, 116, 120islbs4 21870 . . . . . . . . . . . . . . 15 (𝑋 ∈ (LBasis‘𝐶) ↔ (𝑋 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑋) = (Base‘𝐶)))
12247, 121sylib 218 . . . . . . . . . . . . . 14 (𝜑 → (𝑋 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑋) = (Base‘𝐶)))
123122simpld 494 . . . . . . . . . . . . 13 (𝜑𝑋 ∈ (LIndS‘𝐶))
124123adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝑋 ∈ (LIndS‘𝐶))
125 eqid 2735 . . . . . . . . . . . . . 14 (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
126 eqid 2735 . . . . . . . . . . . . . 14 (Scalar‘𝐶) = (Scalar‘𝐶)
127 eqid 2735 . . . . . . . . . . . . . 14 ( ·𝑠𝐶) = ( ·𝑠𝐶)
128 eqid 2735 . . . . . . . . . . . . . 14 (0g𝐶) = (0g𝐶)
129 eqid 2735 . . . . . . . . . . . . . 14 (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘𝐶))
130115, 125, 126, 127, 128, 129islinds5 33375 . . . . . . . . . . . . 13 ((𝐶 ∈ LMod ∧ 𝑋 ⊆ (Base‘𝐶)) → (𝑋 ∈ (LIndS‘𝐶) ↔ ∀𝑔 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋)((𝑔 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖𝑋 ↦ ((𝑔𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶)) → 𝑔 = (𝑋 × {(0g‘(Scalar‘𝐶))}))))
131130biimpa 476 . . . . . . . . . . . 12 (((𝐶 ∈ LMod ∧ 𝑋 ⊆ (Base‘𝐶)) ∧ 𝑋 ∈ (LIndS‘𝐶)) → ∀𝑔 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋)((𝑔 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖𝑋 ↦ ((𝑔𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶)) → 𝑔 = (𝑋 × {(0g‘(Scalar‘𝐶))})))
132114, 119, 124, 131syl21anc 838 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → ∀𝑔 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋)((𝑔 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖𝑋 ↦ ((𝑔𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶)) → 𝑔 = (𝑋 × {(0g‘(Scalar‘𝐶))})))
133106, 132, 55rspcdva 3623 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (((𝐺𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶)) → (𝐺𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))})))
13497, 133mpand 695 . . . . . . . . 9 ((𝜑𝑗𝑌) → ((𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶) → (𝐺𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))})))
135134imp 406 . . . . . . . 8 (((𝜑𝑗𝑌) ∧ (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶)) → (𝐺𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))}))
13676, 78, 96, 135syl21anc 838 . . . . . . 7 ((𝜑𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → (𝐺𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))}))
1371, 136suppss 8218 . . . . . 6 (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ (𝐿 supp (0g‘(Scalar‘𝐵))))
13875, 137ssfid 9299 . . . . 5 (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ∈ Fin)
139 suppssdm 8201 . . . . . . . . . 10 (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ dom 𝐺
140139, 1fssdm 6756 . . . . . . . . 9 (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ 𝑌)
141140sselda 3995 . . . . . . . 8 ((𝜑𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))) → 𝑤𝑌)
142 eleq1w 2822 . . . . . . . . . . . 12 (𝑗 = 𝑤 → (𝑗𝑌𝑤𝑌))
143142anbi2d 630 . . . . . . . . . . 11 (𝑗 = 𝑤 → ((𝜑𝑗𝑌) ↔ (𝜑𝑤𝑌)))
144 fveq2 6907 . . . . . . . . . . . 12 (𝑗 = 𝑤 → (𝐺𝑗) = (𝐺𝑤))
145144breq1d 5158 . . . . . . . . . . 11 (𝑗 = 𝑤 → ((𝐺𝑗) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺𝑤) finSupp (0g‘(Scalar‘𝐶))))
146143, 145imbi12d 344 . . . . . . . . . 10 (𝑗 = 𝑤 → (((𝜑𝑗𝑌) → (𝐺𝑗) finSupp (0g‘(Scalar‘𝐶))) ↔ ((𝜑𝑤𝑌) → (𝐺𝑤) finSupp (0g‘(Scalar‘𝐶)))))
147146, 97chvarvv 1996 . . . . . . . . 9 ((𝜑𝑤𝑌) → (𝐺𝑤) finSupp (0g‘(Scalar‘𝐶)))
148147fsuppimpd 9407 . . . . . . . 8 ((𝜑𝑤𝑌) → ((𝐺𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
149141, 148syldan 591 . . . . . . 7 ((𝜑𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))) → ((𝐺𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
150149ralrimiva 3144 . . . . . 6 (𝜑 → ∀𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
151 iunfi 9381 . . . . . 6 (((𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ∈ Fin ∧ ∀𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin) → 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
152138, 150, 151syl2anc 584 . . . . 5 (𝜑 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
153 xpfi 9356 . . . . 5 (((𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ∈ Fin ∧ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin) → ((𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) × 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺𝑤) supp (0g‘(Scalar‘𝐶)))) ∈ Fin)
154138, 152, 153syl2anc 584 . . . 4 (𝜑 → ((𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) × 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺𝑤) supp (0g‘(Scalar‘𝐶)))) ∈ Fin)
155 fveq2 6907 . . . . . . . . . 10 (𝑣 = 𝑗 → (𝐺𝑣) = (𝐺𝑗))
156155fveq1d 6909 . . . . . . . . 9 (𝑣 = 𝑗 → ((𝐺𝑣)‘𝑢) = ((𝐺𝑗)‘𝑢))
157156mpteq2dv 5250 . . . . . . . 8 (𝑣 = 𝑗 → (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢)) = (𝑢𝑋 ↦ ((𝐺𝑗)‘𝑢)))
158 fveq2 6907 . . . . . . . . 9 (𝑢 = 𝑖 → ((𝐺𝑗)‘𝑢) = ((𝐺𝑗)‘𝑖))
159158cbvmptv 5261 . . . . . . . 8 (𝑢𝑋 ↦ ((𝐺𝑗)‘𝑢)) = (𝑖𝑋 ↦ ((𝐺𝑗)‘𝑖))
160157, 159eqtrdi 2791 . . . . . . 7 (𝑣 = 𝑗 → (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢)) = (𝑖𝑋 ↦ ((𝐺𝑗)‘𝑖)))
161160cbvmptv 5261 . . . . . 6 (𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢))) = (𝑗𝑌 ↦ (𝑖𝑋 ↦ ((𝐺𝑗)‘𝑖)))
162 fvexd 6922 . . . . . 6 (𝜑 → (0g‘(Scalar‘𝐶)) ∈ V)
163 fvexd 6922 . . . . . 6 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → ((𝐺𝑗)‘𝑖) ∈ V)
16442, 161, 46, 47, 162, 163suppovss 32696 . . . . 5 (𝜑 → (𝐻 supp (0g‘(Scalar‘𝐶))) ⊆ (((𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢))) supp (𝑋 × {(0g‘(Scalar‘𝐶))})) × 𝑤 ∈ ((𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢))) supp (𝑋 × {(0g‘(Scalar‘𝐶))}))(((𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢)))‘𝑤) supp (0g‘(Scalar‘𝐶)))))
16510, 71subrg0 20596 . . . . . . . 8 (𝑉 ∈ (SubRing‘𝐸) → (0g𝐸) = (0g𝐾))
16619, 165syl 17 . . . . . . 7 (𝜑 → (0g𝐸) = (0g𝐾))
16736fveq2d 6911 . . . . . . 7 (𝜑 → (0g𝐾) = (0g‘(Scalar‘𝐶)))
168166, 167eqtr2d 2776 . . . . . 6 (𝜑 → (0g‘(Scalar‘𝐶)) = (0g𝐸))
169168oveq2d 7447 . . . . 5 (𝜑 → (𝐻 supp (0g‘(Scalar‘𝐶))) = (𝐻 supp (0g𝐸)))
1701feqmptd 6977 . . . . . . . 8 (𝜑𝐺 = (𝑣𝑌 ↦ (𝐺𝑣)))
171 eleq1w 2822 . . . . . . . . . . . . 13 (𝑗 = 𝑣 → (𝑗𝑌𝑣𝑌))
172171anbi2d 630 . . . . . . . . . . . 12 (𝑗 = 𝑣 → ((𝜑𝑗𝑌) ↔ (𝜑𝑣𝑌)))
173 fveq2 6907 . . . . . . . . . . . . 13 (𝑗 = 𝑣 → (𝐺𝑗) = (𝐺𝑣))
174173feq1d 6721 . . . . . . . . . . . 12 (𝑗 = 𝑣 → ((𝐺𝑗):𝑋⟶(Base‘𝐸) ↔ (𝐺𝑣):𝑋⟶(Base‘𝐸)))
175172, 174imbi12d 344 . . . . . . . . . . 11 (𝑗 = 𝑣 → (((𝜑𝑗𝑌) → (𝐺𝑗):𝑋⟶(Base‘𝐸)) ↔ ((𝜑𝑣𝑌) → (𝐺𝑣):𝑋⟶(Base‘𝐸))))
17610, 20ressbas2 17283 . . . . . . . . . . . . . . . 16 (𝑉 ⊆ (Base‘𝐸) → 𝑉 = (Base‘𝐾))
17722, 176syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑉 = (Base‘𝐾))
17836fveq2d 6911 . . . . . . . . . . . . . . 15 (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝐶)))
179177, 178eqtrd 2775 . . . . . . . . . . . . . 14 (𝜑𝑉 = (Base‘(Scalar‘𝐶)))
180179, 22eqsstrrd 4035 . . . . . . . . . . . . 13 (𝜑 → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
181180adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
18256, 181fssd 6754 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝐺𝑗):𝑋⟶(Base‘𝐸))
183175, 182chvarvv 1996 . . . . . . . . . 10 ((𝜑𝑣𝑌) → (𝐺𝑣):𝑋⟶(Base‘𝐸))
184183feqmptd 6977 . . . . . . . . 9 ((𝜑𝑣𝑌) → (𝐺𝑣) = (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢)))
185184mpteq2dva 5248 . . . . . . . 8 (𝜑 → (𝑣𝑌 ↦ (𝐺𝑣)) = (𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢))))
186170, 185eqtr2d 2776 . . . . . . 7 (𝜑 → (𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢))) = 𝐺)
187186oveq1d 7446 . . . . . 6 (𝜑 → ((𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢))) supp (𝑋 × {(0g‘(Scalar‘𝐶))})) = (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})))
188186fveq1d 6909 . . . . . . . 8 (𝜑 → ((𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢)))‘𝑤) = (𝐺𝑤))
189188oveq1d 7446 . . . . . . 7 (𝜑 → (((𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢)))‘𝑤) supp (0g‘(Scalar‘𝐶))) = ((𝐺𝑤) supp (0g‘(Scalar‘𝐶))))
190187, 189iuneq12d 5026 . . . . . 6 (𝜑 𝑤 ∈ ((𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢))) supp (𝑋 × {(0g‘(Scalar‘𝐶))}))(((𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢)))‘𝑤) supp (0g‘(Scalar‘𝐶))) = 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺𝑤) supp (0g‘(Scalar‘𝐶))))
191187, 190xpeq12d 5720 . . . . 5 (𝜑 → (((𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢))) supp (𝑋 × {(0g‘(Scalar‘𝐶))})) × 𝑤 ∈ ((𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢))) supp (𝑋 × {(0g‘(Scalar‘𝐶))}))(((𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢)))‘𝑤) supp (0g‘(Scalar‘𝐶)))) = ((𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) × 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺𝑤) supp (0g‘(Scalar‘𝐶)))))
192164, 169, 1913sstr3d 4042 . . . 4 (𝜑 → (𝐻 supp (0g𝐸)) ⊆ ((𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) × 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺𝑤) supp (0g‘(Scalar‘𝐶)))))
193 suppssfifsupp 9418 . . . 4 (((𝐻 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)) ∧ Fun 𝐻 ∧ (0g𝐸) ∈ (Base‘𝐸)) ∧ (((𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) × 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺𝑤) supp (0g‘(Scalar‘𝐶)))) ∈ Fin ∧ (𝐻 supp (0g𝐸)) ⊆ ((𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) × 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺𝑤) supp (0g‘(Scalar‘𝐶)))))) → 𝐻 finSupp (0g𝐸))
19452, 66, 73, 154, 192, 193syl32anc 1377 . . 3 (𝜑𝐻 finSupp (0g𝐸))
19537fveq2d 6911 . . . 4 (𝜑 → (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐶)))
196195, 168eqtr2d 2776 . . 3 (𝜑 → (0g𝐸) = (0g‘(Scalar‘𝐴)))
197194, 196breqtrd 5174 . 2 (𝜑𝐻 finSupp (0g‘(Scalar‘𝐴)))
198 fedgmullem1.z . . 3 (𝜑𝑍 = (𝐵 Σg (𝑗𝑌 ↦ ((𝐿𝑗)( ·𝑠𝐵)𝑗))))
19985, 67, 13, 15, 92, 46drgextgsum 33624 . . 3 (𝜑 → (𝐸 Σg (𝑗𝑌 ↦ ((𝐿𝑗)( ·𝑠𝐵)𝑗))) = (𝐵 Σg (𝑗𝑌 ↦ ((𝐿𝑗)( ·𝑠𝐵)𝑗))))
20047adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝑋 ∈ (LBasis‘𝐶))
20113adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑗𝑌) → 𝑈 ∈ (SubRing‘𝐸))
202 subrgsubg 20594 . . . . . . . . . . . . 13 (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ∈ (SubGrp‘𝐸))
203 subgsubm 19179 . . . . . . . . . . . . 13 (𝑈 ∈ (SubGrp‘𝐸) → 𝑈 ∈ (SubMnd‘𝐸))
204201, 202, 2033syl 18 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝑈 ∈ (SubMnd‘𝐸))
205113ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝐶 ∈ LMod)
20656ffvelcdmda 7104 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝐺𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶)))
207118ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑋 ⊆ (Base‘𝐶))
208207, 61sseldd 3996 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑖 ∈ (Base‘𝐶))
209115, 126, 127, 125lmodvscl 20893 . . . . . . . . . . . . . . 15 ((𝐶 ∈ LMod ∧ ((𝐺𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑖 ∈ (Base‘𝐶)) → (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖) ∈ (Base‘𝐶))
210205, 206, 208, 209syl3anc 1370 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖) ∈ (Base‘𝐶))
21115, 20ressbas2 17283 . . . . . . . . . . . . . . . . 17 (𝑈 ⊆ (Base‘𝐸) → 𝑈 = (Base‘𝐹))
21288, 211syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑈 = (Base‘𝐹))
21331, 34srabase 21195 . . . . . . . . . . . . . . . 16 (𝜑 → (Base‘𝐹) = (Base‘𝐶))
214212, 213eqtrd 2775 . . . . . . . . . . . . . . 15 (𝜑𝑈 = (Base‘𝐶))
215214ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑈 = (Base‘𝐶))
216210, 215eleqtrrd 2842 . . . . . . . . . . . . 13 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖) ∈ 𝑈)
217216fmpttd 7135 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖)):𝑋𝑈)
218200, 204, 217, 15gsumsubm 18861 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (𝐹 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))
219 eqid 2735 . . . . . . . . . . . . . . . . . 18 (.r𝐸) = (.r𝐸)
22015, 219ressmulr 17353 . . . . . . . . . . . . . . . . 17 (𝑈 ∈ (SubRing‘𝐸) → (.r𝐸) = (.r𝐹))
22113, 220syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (.r𝐸) = (.r𝐹))
22231, 34sravsca 21203 . . . . . . . . . . . . . . . 16 (𝜑 → (.r𝐹) = ( ·𝑠𝐶))
223221, 222eqtr2d 2776 . . . . . . . . . . . . . . 15 (𝜑 → ( ·𝑠𝐶) = (.r𝐸))
224223ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ( ·𝑠𝐶) = (.r𝐸))
225224oveqd 7448 . . . . . . . . . . . . 13 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖) = (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖))
226225mpteq2dva 5248 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖)) = (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)))
227226oveq2d 7447 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖))))
22830, 92, 14, 109, 108, 47drgextgsum 33624 . . . . . . . . . . . 12 (𝜑 → (𝐹 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))
229228adantr 480 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝐹 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))
230218, 227, 2293eqtr3d 2783 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖))) = (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))
231230oveq1d 7446 . . . . . . . . 9 ((𝜑𝑗𝑌) → ((𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗) = ((𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖)))(.r𝐸)𝑗))
23269ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝐸 ∈ Ring)
233180ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
234233, 206sseldd 3996 . . . . . . . . . . . . 13 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝐺𝑗)‘𝑖) ∈ (Base‘𝐸))
235214, 88eqsstrrd 4035 . . . . . . . . . . . . . . . 16 (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
236118, 235sstrd 4006 . . . . . . . . . . . . . . 15 (𝜑𝑋 ⊆ (Base‘𝐸))
237236ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑋 ⊆ (Base‘𝐸))
238237, 61sseldd 3996 . . . . . . . . . . . . 13 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑖 ∈ (Base‘𝐸))
239 eqid 2735 . . . . . . . . . . . . . . . . . 18 (Base‘𝐵) = (Base‘𝐵)
240 eqid 2735 . . . . . . . . . . . . . . . . . 18 (LBasis‘𝐵) = (LBasis‘𝐵)
241239, 240lbsss 21094 . . . . . . . . . . . . . . . . 17 (𝑌 ∈ (LBasis‘𝐵) → 𝑌 ⊆ (Base‘𝐵))
24246, 241syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑌 ⊆ (Base‘𝐵))
24386, 88srabase 21195 . . . . . . . . . . . . . . . 16 (𝜑 → (Base‘𝐸) = (Base‘𝐵))
244242, 243sseqtrrd 4037 . . . . . . . . . . . . . . 15 (𝜑𝑌 ⊆ (Base‘𝐸))
245244ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑌 ⊆ (Base‘𝐸))
246 simplr 769 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑗𝑌)
247245, 246sseldd 3996 . . . . . . . . . . . . 13 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑗 ∈ (Base‘𝐸))
24820, 219ringass 20271 . . . . . . . . . . . . 13 ((𝐸 ∈ Ring ∧ (((𝐺𝑗)‘𝑖) ∈ (Base‘𝐸) ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸))) → ((((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗) = (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))
249232, 234, 238, 247, 248syl13anc 1371 . . . . . . . . . . . 12 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗) = (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))
250249mpteq2dva 5248 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ ((((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗)) = (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗))))
251250oveq2d 7447 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ ((((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))) = (𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))))
25269adantr 480 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → 𝐸 ∈ Ring)
253242adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑗𝑌) → 𝑌 ⊆ (Base‘𝐵))
254243adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑗𝑌) → (Base‘𝐸) = (Base‘𝐵))
255253, 254sseqtrrd 4037 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝑌 ⊆ (Base‘𝐸))
256 simpr 484 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝑗𝑌)
257255, 256sseldd 3996 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → 𝑗 ∈ (Base‘𝐸))
25820, 219ringcl 20268 . . . . . . . . . . . 12 ((𝐸 ∈ Ring ∧ ((𝐺𝑗)‘𝑖) ∈ (Base‘𝐸) ∧ 𝑖 ∈ (Base‘𝐸)) → (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖) ∈ (Base‘𝐸))
259232, 234, 238, 258syl3anc 1370 . . . . . . . . . . 11 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖) ∈ (Base‘𝐸))
260168breq2d 5160 . . . . . . . . . . . . . 14 (𝜑 → ((𝐺𝑗) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺𝑗) finSupp (0g𝐸)))
261260adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑗𝑌) → ((𝐺𝑗) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺𝑗) finSupp (0g𝐸)))
26297, 261mpbid 232 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → (𝐺𝑗) finSupp (0g𝐸))
26320, 252, 200, 238, 182, 262rmfsupp2 33228 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)) finSupp (0g𝐸))
26420, 71, 219, 252, 200, 257, 259, 263gsummulc1 20330 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ ((((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))) = ((𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗))
265251, 264eqtr3d 2777 . . . . . . . . 9 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))) = ((𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗))
26683oveq1d 7446 . . . . . . . . 9 ((𝜑𝑗𝑌) → ((𝐿𝑗)(.r𝐸)𝑗) = ((𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖)))(.r𝐸)𝑗))
267231, 265, 2663eqtr4rd 2786 . . . . . . . 8 ((𝜑𝑗𝑌) → ((𝐿𝑗)(.r𝐸)𝑗) = (𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))))
26886, 88sravsca 21203 . . . . . . . . . 10 (𝜑 → (.r𝐸) = ( ·𝑠𝐵))
269268adantr 480 . . . . . . . . 9 ((𝜑𝑗𝑌) → (.r𝐸) = ( ·𝑠𝐵))
270269oveqd 7448 . . . . . . . 8 ((𝜑𝑗𝑌) → ((𝐿𝑗)(.r𝐸)𝑗) = ((𝐿𝑗)( ·𝑠𝐵)𝑗))
271 fvexd 6922 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑌𝑖𝑋) → ((𝐺𝑗)‘𝑖) ∈ V)
272 ovexd 7466 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑌𝑖𝑋) → (𝑖(.r𝐸)𝑗) ∈ V)
27342a1i 11 . . . . . . . . . . . . . 14 (𝜑𝐻 = (𝑗𝑌, 𝑖𝑋 ↦ ((𝐺𝑗)‘𝑖)))
274 fedgmullem.d . . . . . . . . . . . . . . 15 𝐷 = (𝑗𝑌, 𝑖𝑋 ↦ (𝑖(.r𝐸)𝑗))
275274a1i 11 . . . . . . . . . . . . . 14 (𝜑𝐷 = (𝑗𝑌, 𝑖𝑋 ↦ (𝑖(.r𝐸)𝑗)))
27646, 47, 271, 272, 273, 275offval22 8112 . . . . . . . . . . . . 13 (𝜑 → (𝐻f (.r𝐸)𝐷) = (𝑗𝑌, 𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗))))
277276oveqd 7448 . . . . . . . . . . . 12 (𝜑 → (𝑗(𝐻f (.r𝐸)𝐷)𝑖) = (𝑗(𝑗𝑌, 𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))𝑖))
278277ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗(𝐻f (.r𝐸)𝐷)𝑖) = (𝑗(𝑗𝑌, 𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))𝑖))
279 ovexd 7466 . . . . . . . . . . . 12 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)) ∈ V)
280 eqid 2735 . . . . . . . . . . . . 13 (𝑗𝑌, 𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗))) = (𝑗𝑌, 𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))
281280ovmpt4g 7580 . . . . . . . . . . . 12 ((𝑗𝑌𝑖𝑋 ∧ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)) ∈ V) → (𝑗(𝑗𝑌, 𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))𝑖) = (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))
282246, 61, 279, 281syl3anc 1370 . . . . . . . . . . 11 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗(𝑗𝑌, 𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))𝑖) = (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))
283278, 282eqtr2d 2776 . . . . . . . . . 10 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)) = (𝑗(𝐻f (.r𝐸)𝐷)𝑖))
284283mpteq2dva 5248 . . . . . . . . 9 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗))) = (𝑖𝑋 ↦ (𝑗(𝐻f (.r𝐸)𝐷)𝑖)))
285284oveq2d 7447 . . . . . . . 8 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))) = (𝐸 Σg (𝑖𝑋 ↦ (𝑗(𝐻f (.r𝐸)𝐷)𝑖))))
286267, 270, 2853eqtr3d 2783 . . . . . . 7 ((𝜑𝑗𝑌) → ((𝐿𝑗)( ·𝑠𝐵)𝑗) = (𝐸 Σg (𝑖𝑋 ↦ (𝑗(𝐻f (.r𝐸)𝐷)𝑖))))
287286mpteq2dva 5248 . . . . . 6 (𝜑 → (𝑗𝑌 ↦ ((𝐿𝑗)( ·𝑠𝐵)𝑗)) = (𝑗𝑌 ↦ (𝐸 Σg (𝑖𝑋 ↦ (𝑗(𝐻f (.r𝐸)𝐷)𝑖)))))
288287oveq2d 7447 . . . . 5 (𝜑 → (𝐸 Σg (𝑗𝑌 ↦ ((𝐿𝑗)( ·𝑠𝐵)𝑗))) = (𝐸 Σg (𝑗𝑌 ↦ (𝐸 Σg (𝑖𝑋 ↦ (𝑗(𝐻f (.r𝐸)𝐷)𝑖))))))
289 ringcmn 20296 . . . . . . 7 (𝐸 ∈ Ring → 𝐸 ∈ CMnd)
29069, 289syl 17 . . . . . 6 (𝜑𝐸 ∈ CMnd)
29169adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝐸 ∈ Ring)
29238, 180eqsstrd 4034 . . . . . . . . . 10 (𝜑 → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸))
293292adantr 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)))
295293, 294sseldd 3996 . . . . . . . 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))
29712, 22srabase 21195 . . . . . . . . . 10 (𝜑 → (Base‘𝐸) = (Base‘𝐴))
298297adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (Base‘𝐸) = (Base‘𝐴))
299296, 298eleqtrrd 2842 . . . . . . . 8 ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑘 ∈ (Base‘𝐸))
30020, 219ringcl 20268 . . . . . . . 8 ((𝐸 ∈ Ring ∧ 𝑙 ∈ (Base‘𝐸) ∧ 𝑘 ∈ (Base‘𝐸)) → (𝑙(.r𝐸)𝑘) ∈ (Base‘𝐸))
301291, 295, 299, 300syl3anc 1370 . . . . . . 7 ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (𝑙(.r𝐸)𝑘) ∈ (Base‘𝐸))
30220, 219ringcl 20268 . . . . . . . . . . . 12 ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
303232, 238, 247, 302syl3anc 1370 . . . . . . . . . . 11 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
304297ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (Base‘𝐸) = (Base‘𝐴))
305303, 304eleqtrd 2841 . . . . . . . . . 10 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
306305anasss 466 . . . . . . . . 9 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
307306ralrimivva 3200 . . . . . . . 8 (𝜑 → ∀𝑗𝑌𝑖𝑋 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
308274fmpo 8092 . . . . . . . 8 (∀𝑗𝑌𝑖𝑋 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴) ↔ 𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
309307, 308sylib 218 . . . . . . 7 (𝜑𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
310 inidm 4235 . . . . . . 7 ((𝑌 × 𝑋) ∩ (𝑌 × 𝑋)) = (𝑌 × 𝑋)
311301, 65, 309, 48, 48, 310off 7715 . . . . . 6 (𝜑 → (𝐻f (.r𝐸)𝐷):(𝑌 × 𝑋)⟶(Base‘𝐸))
31269adantr 480 . . . . . . . 8 ((𝜑𝑢 ∈ (Base‘𝐴)) → 𝐸 ∈ Ring)
313 simpr 484 . . . . . . . . 9 ((𝜑𝑢 ∈ (Base‘𝐴)) → 𝑢 ∈ (Base‘𝐴))
314297adantr 480 . . . . . . . . 9 ((𝜑𝑢 ∈ (Base‘𝐴)) → (Base‘𝐸) = (Base‘𝐴))
315313, 314eleqtrrd 2842 . . . . . . . 8 ((𝜑𝑢 ∈ (Base‘𝐴)) → 𝑢 ∈ (Base‘𝐸))
31620, 219, 71ringlz 20307 . . . . . . . 8 ((𝐸 ∈ Ring ∧ 𝑢 ∈ (Base‘𝐸)) → ((0g𝐸)(.r𝐸)𝑢) = (0g𝐸))
317312, 315, 316syl2anc 584 . . . . . . 7 ((𝜑𝑢 ∈ (Base‘𝐴)) → ((0g𝐸)(.r𝐸)𝑢) = (0g𝐸))
31848, 73, 73, 65, 309, 194, 317offinsupp1 32745 . . . . . 6 (𝜑 → (𝐻f (.r𝐸)𝐷) finSupp (0g𝐸))
31920, 71, 290, 46, 47, 311, 318gsumxp 20009 . . . . 5 (𝜑 → (𝐸 Σg (𝐻f (.r𝐸)𝐷)) = (𝐸 Σg (𝑗𝑌 ↦ (𝐸 Σg (𝑖𝑋 ↦ (𝑗(𝐻f (.r𝐸)𝐷)𝑖))))))
32012, 22sravsca 21203 . . . . . . . 8 (𝜑 → (.r𝐸) = ( ·𝑠𝐴))
321320ofeqd 7699 . . . . . . 7 (𝜑 → ∘f (.r𝐸) = ∘f ( ·𝑠𝐴))
322321oveqd 7448 . . . . . 6 (𝜑 → (𝐻f (.r𝐸)𝐷) = (𝐻f ( ·𝑠𝐴)𝐷))
323322oveq2d 7447 . . . . 5 (𝜑 → (𝐸 Σg (𝐻f (.r𝐸)𝐷)) = (𝐸 Σg (𝐻f ( ·𝑠𝐴)𝐷)))
324288, 319, 3233eqtr2rd 2782 . . . 4 (𝜑 → (𝐸 Σg (𝐻f ( ·𝑠𝐴)𝐷)) = (𝐸 Σg (𝑗𝑌 ↦ ((𝐿𝑗)( ·𝑠𝐵)𝑗))))
325 ovexd 7466 . . . . 5 (𝜑 → (𝐻f ( ·𝑠𝐴)𝐷) ∈ V)
326 fedgmullem1.a . . . . . 6 (𝜑𝑍 ∈ (Base‘𝐴))
327326elfvexd 6946 . . . . 5 (𝜑𝐴 ∈ V)
32811, 325, 67, 327, 22gsumsra 33033 . . . 4 (𝜑 → (𝐸 Σg (𝐻f ( ·𝑠𝐴)𝐷)) = (𝐴 Σg (𝐻f ( ·𝑠𝐴)𝐷)))
329324, 328eqtr3d 2777 . . 3 (𝜑 → (𝐸 Σg (𝑗𝑌 ↦ ((𝐿𝑗)( ·𝑠𝐵)𝑗))) = (𝐴 Σg (𝐻f ( ·𝑠𝐴)𝐷)))
330198, 199, 3293eqtr2d 2781 . 2 (𝜑𝑍 = (𝐴 Σg (𝐻f ( ·𝑠𝐴)𝐷)))
331197, 330jca 511 1 (𝜑 → (𝐻 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑍 = (𝐴 Σg (𝐻f ( ·𝑠𝐴)𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cdif 3960  wss 3963  {csn 4631   ciun 4996   class class class wbr 5148  cmpt 5231   × cxp 5687  Fun wfun 6557  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  f cof 7695   supp csupp 8184  m cmap 8865  Fincfn 8984   finSupp cfsupp 9399  Basecbs 17245  s cress 17274  .rcmulr 17299  Scalarcsca 17301   ·𝑠 cvsca 17302  0gc0g 17486   Σg cgsu 17487  SubMndcsubmnd 18808  Grpcgrp 18964  SubGrpcsubg 19151  CMndccmn 19813  Ringcrg 20251  SubRingcsubrg 20586  DivRingcdr 20746  LModclmod 20875  LSpanclspn 20987  LBasisclbs 21091  LVecclvec 21119  subringAlg csra 21188  LIndSclinds 21843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-nzr 20530  df-subrg 20587  df-drng 20748  df-lmod 20877  df-lss 20948  df-lsp 20988  df-lmhm 21039  df-lbs 21092  df-lvec 21120  df-sra 21190  df-rgmod 21191  df-dsmm 21770  df-frlm 21785  df-uvc 21821  df-lindf 21844  df-linds 21845
This theorem is referenced by:  fedgmul  33659
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