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Theorem fedgmullem1 31113
 Description: Lemma for fedgmul 31115. (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 775 . . . . . . . . . . . . 13 ((((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗𝑌) ∧ 𝑖𝑋) → 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋))
3 simplr 768 . . . . . . . . . . . . 13 ((((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗𝑌) ∧ 𝑖𝑋) → 𝑗𝑌)
42, 3ffvelrnd 6833 . . . . . . . . . . . 12 ((((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗𝑌) ∧ 𝑖𝑋) → (𝐺𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋))
5 elmapi 8415 . . . . . . . . . . . 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 470 . . . . . . . . . 10 (((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗𝑌𝑖𝑋)) → (𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
8 simprr 772 . . . . . . . . . 10 (((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗𝑌𝑖𝑋)) → 𝑖𝑋)
97, 8ffvelrnd 6833 . . . . . . . . 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 19557 . . . . . . . . . . . . . . . . . 18 (𝑈 ∈ (SubRing‘𝐸) → (𝑉 ∈ (SubRing‘𝐹) ↔ (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈)))
1716biimpa 480 . . . . . . . . . . . . . . . . 17 ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ∈ (SubRing‘𝐹)) → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈))
1813, 14, 17syl2anc 587 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈))
1918simpld 498 . . . . . . . . . . . . . . 15 (𝜑𝑉 ∈ (SubRing‘𝐸))
20 eqid 2801 . . . . . . . . . . . . . . . 16 (Base‘𝐸) = (Base‘𝐸)
2120subrgss 19532 . . . . . . . . . . . . . . 15 (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
2219, 21syl 17 . . . . . . . . . . . . . 14 (𝜑𝑉 ⊆ (Base‘𝐸))
2312, 22srasca 19949 . . . . . . . . . . . . 13 (𝜑 → (𝐸s 𝑉) = (Scalar‘𝐴))
2410, 23syl5eq 2848 . . . . . . . . . . . 12 (𝜑𝐾 = (Scalar‘𝐴))
2518simprd 499 . . . . . . . . . . . . . . 15 (𝜑𝑉𝑈)
26 ressabs 16558 . . . . . . . . . . . . . . 15 ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈) → ((𝐸s 𝑈) ↾s 𝑉) = (𝐸s 𝑉))
2713, 25, 26syl2anc 587 . . . . . . . . . . . . . 14 (𝜑 → ((𝐸s 𝑈) ↾s 𝑉) = (𝐸s 𝑉))
2815oveq1i 7149 . . . . . . . . . . . . . 14 (𝐹s 𝑉) = ((𝐸s 𝑈) ↾s 𝑉)
2927, 28, 103eqtr4g 2861 . . . . . . . . . . . . 13 (𝜑 → (𝐹s 𝑉) = 𝐾)
30 fedgmul.c . . . . . . . . . . . . . . 15 𝐶 = ((subringAlg ‘𝐹)‘𝑉)
3130a1i 11 . . . . . . . . . . . . . 14 (𝜑𝐶 = ((subringAlg ‘𝐹)‘𝑉))
32 eqid 2801 . . . . . . . . . . . . . . . 16 (Base‘𝐹) = (Base‘𝐹)
3332subrgss 19532 . . . . . . . . . . . . . . 15 (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
3414, 33syl 17 . . . . . . . . . . . . . 14 (𝜑𝑉 ⊆ (Base‘𝐹))
3531, 34srasca 19949 . . . . . . . . . . . . 13 (𝜑 → (𝐹s 𝑉) = (Scalar‘𝐶))
3629, 35eqtr3d 2838 . . . . . . . . . . . 12 (𝜑𝐾 = (Scalar‘𝐶))
3724, 36eqtr3d 2838 . . . . . . . . . . 11 (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
3837fveq2d 6653 . . . . . . . . . 10 (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
3938ad2antrr 725 . . . . . . . . 9 (((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗𝑌𝑖𝑋)) → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
409, 39eleqtrrd 2896 . . . . . . . 8 (((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗𝑌𝑖𝑋)) → ((𝐺𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
4140ralrimivva 3159 . . . . . . 7 ((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → ∀𝑗𝑌𝑖𝑋 ((𝐺𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
42 fedgmullem.h . . . . . . . 8 𝐻 = (𝑗𝑌, 𝑖𝑋 ↦ ((𝐺𝑗)‘𝑖))
4342fmpo 7752 . . . . . . 7 (∀𝑗𝑌𝑖𝑋 ((𝐺𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)) ↔ 𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
4441, 43sylib 221 . . . . . 6 ((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → 𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
45 fvexd 6664 . . . . . . 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 7458 . . . . . . . 8 (𝜑 → (𝑌 × 𝑋) ∈ V)
4948adantr 484 . . . . . . 7 ((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → (𝑌 × 𝑋) ∈ V)
5045, 49elmapd 8407 . . . . . 6 ((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → (𝐻 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)) ↔ 𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴))))
5144, 50mpbird 260 . . . . 5 ((𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → 𝐻 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)))
521, 51mpdan 686 . . . 4 (𝜑𝐻 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)))
53 simpl 486 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → 𝜑)
5453adantr 484 . . . . . . . . . 10 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝜑)
551ffvelrnda 6832 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → (𝐺𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋))
5655, 5syl 17 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
5756adantr 484 . . . . . . . . . 10 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
5838feq3d 6478 . . . . . . . . . . 11 (𝜑 → ((𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐴)) ↔ (𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐶))))
5958biimpar 481 . . . . . . . . . 10 ((𝜑 ∧ (𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐶))) → (𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐴)))
6054, 57, 59syl2anc 587 . . . . . . . . 9 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝐺𝑗):𝑋⟶(Base‘(Scalar‘𝐴)))
61 simpr 488 . . . . . . . . 9 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑖𝑋)
6260, 61ffvelrnd 6833 . . . . . . . 8 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝐺𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
6362ralrimiva 3152 . . . . . . 7 ((𝜑𝑗𝑌) → ∀𝑖𝑋 ((𝐺𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
6463ralrimiva 3152 . . . . . 6 (𝜑 → ∀𝑗𝑌𝑖𝑋 ((𝐺𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
6564, 43sylib 221 . . . . 5 (𝜑𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
6665ffund 6495 . . . 4 (𝜑 → Fun 𝐻)
67 fedgmul.1 . . . . . 6 (𝜑𝐸 ∈ DivRing)
68 drngring 19505 . . . . . 6 (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
6967, 68syl 17 . . . . 5 (𝜑𝐸 ∈ Ring)
70 ringgrp 19298 . . . . 5 (𝐸 ∈ Ring → 𝐸 ∈ Grp)
71 eqid 2801 . . . . . 6 (0g𝐸) = (0g𝐸)
7220, 71grpidcl 18126 . . . . 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 8828 . . . . . 6 (𝜑 → (𝐿 supp (0g‘(Scalar‘𝐵))) ∈ Fin)
76 simpl 486 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → 𝜑)
77 simpr 488 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵)))))
7877eldifad 3896 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → 𝑗𝑌)
79 fedgmullem1.l . . . . . . . . . 10 (𝜑𝐿:𝑌⟶(Base‘(Scalar‘𝐵)))
80 ssidd 3941 . . . . . . . . . 10 (𝜑 → (𝐿 supp (0g‘(Scalar‘𝐵))) ⊆ (𝐿 supp (0g‘(Scalar‘𝐵))))
81 fvexd 6664 . . . . . . . . . 10 (𝜑 → (0g‘(Scalar‘𝐵)) ∈ V)
8279, 80, 46, 81suppssr 7848 . . . . . . . . 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 594 . . . . . . . . 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 19532 . . . . . . . . . . . . . . 15 (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
8813, 87syl 17 . . . . . . . . . . . . . 14 (𝜑𝑈 ⊆ (Base‘𝐸))
8986, 88srasca 19949 . . . . . . . . . . . . 13 (𝜑 → (𝐸s 𝑈) = (Scalar‘𝐵))
9015, 89syl5eq 2848 . . . . . . . . . . . 12 (𝜑𝐹 = (Scalar‘𝐵))
9190fveq2d 6653 . . . . . . . . . . 11 (𝜑 → (0g𝐹) = (0g‘(Scalar‘𝐵)))
92 fedgmul.2 . . . . . . . . . . . 12 (𝜑𝐹 ∈ DivRing)
9330, 92, 14drgext0g 31080 . . . . . . . . . . 11 (𝜑 → (0g𝐹) = (0g𝐶))
9491, 93eqtr3d 2838 . . . . . . . . . 10 (𝜑 → (0g‘(Scalar‘𝐵)) = (0g𝐶))
9594adantr 484 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → (0g‘(Scalar‘𝐵)) = (0g𝐶))
9682, 84, 953eqtr3d 2844 . . . . . . . 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 5036 . . . . . . . . . . . . 13 (𝑔 = (𝐺𝑗) → (𝑔 finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺𝑗) finSupp (0g‘(Scalar‘𝐶))))
99 fveq1 6648 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝐺𝑗) → (𝑔𝑖) = ((𝐺𝑗)‘𝑖))
10099oveq1d 7154 . . . . . . . . . . . . . . . 16 (𝑔 = (𝐺𝑗) → ((𝑔𝑖)( ·𝑠𝐶)𝑖) = (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))
101100mpteq2dv 5129 . . . . . . . . . . . . . . 15 (𝑔 = (𝐺𝑗) → (𝑖𝑋 ↦ ((𝑔𝑖)( ·𝑠𝐶)𝑖)) = (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖)))
102101oveq2d 7155 . . . . . . . . . . . . . 14 (𝑔 = (𝐺𝑗) → (𝐶 Σg (𝑖𝑋 ↦ ((𝑔𝑖)( ·𝑠𝐶)𝑖))) = (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))
103102eqeq1d 2803 . . . . . . . . . . . . 13 (𝑔 = (𝐺𝑗) → ((𝐶 Σg (𝑖𝑋 ↦ ((𝑔𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶) ↔ (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶)))
10498, 103anbi12d 633 . . . . . . . . . . . 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 2805 . . . . . . . . . . . 12 (𝑔 = (𝐺𝑗) → (𝑔 = (𝑋 × {(0g‘(Scalar‘𝐶))}) ↔ (𝐺𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))})))
106104, 105imbi12d 348 . . . . . . . . . . 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 2893 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹s 𝑉) ∈ DivRing)
109 eqid 2801 . . . . . . . . . . . . . . . 16 (𝐹s 𝑉) = (𝐹s 𝑉)
11030, 109sralvec 31078 . . . . . . . . . . . . . . 15 ((𝐹 ∈ DivRing ∧ (𝐹s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
11192, 108, 14, 110syl3anc 1368 . . . . . . . . . . . . . 14 (𝜑𝐶 ∈ LVec)
112 lveclmod 19874 . . . . . . . . . . . . . 14 (𝐶 ∈ LVec → 𝐶 ∈ LMod)
113111, 112syl 17 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ LMod)
114113adantr 484 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝐶 ∈ LMod)
115 eqid 2801 . . . . . . . . . . . . . . 15 (Base‘𝐶) = (Base‘𝐶)
116 eqid 2801 . . . . . . . . . . . . . . 15 (LBasis‘𝐶) = (LBasis‘𝐶)
117115, 116lbsss 19845 . . . . . . . . . . . . . 14 (𝑋 ∈ (LBasis‘𝐶) → 𝑋 ⊆ (Base‘𝐶))
11847, 117syl 17 . . . . . . . . . . . . 13 (𝜑𝑋 ⊆ (Base‘𝐶))
119118adantr 484 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝑋 ⊆ (Base‘𝐶))
120 eqid 2801 . . . . . . . . . . . . . . . 16 (LSpan‘𝐶) = (LSpan‘𝐶)
121115, 116, 120islbs4 20524 . . . . . . . . . . . . . . 15 (𝑋 ∈ (LBasis‘𝐶) ↔ (𝑋 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑋) = (Base‘𝐶)))
12247, 121sylib 221 . . . . . . . . . . . . . 14 (𝜑 → (𝑋 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑋) = (Base‘𝐶)))
123122simpld 498 . . . . . . . . . . . . 13 (𝜑𝑋 ∈ (LIndS‘𝐶))
124123adantr 484 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝑋 ∈ (LIndS‘𝐶))
125 eqid 2801 . . . . . . . . . . . . . 14 (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
126 eqid 2801 . . . . . . . . . . . . . 14 (Scalar‘𝐶) = (Scalar‘𝐶)
127 eqid 2801 . . . . . . . . . . . . . 14 ( ·𝑠𝐶) = ( ·𝑠𝐶)
128 eqid 2801 . . . . . . . . . . . . . 14 (0g𝐶) = (0g𝐶)
129 eqid 2801 . . . . . . . . . . . . . 14 (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘𝐶))
130115, 125, 126, 127, 128, 129islinds5 30986 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 836 . . . . . . . . . . 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 3576 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (((𝐺𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶)) → (𝐺𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))})))
13497, 133mpand 694 . . . . . . . . 9 ((𝜑𝑗𝑌) → ((𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶) → (𝐺𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))})))
135134imp 410 . . . . . . . 8 (((𝜑𝑗𝑌) ∧ (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (0g𝐶)) → (𝐺𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))}))
13676, 78, 96, 135syl21anc 836 . . . . . . 7 ((𝜑𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → (𝐺𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))}))
1371, 136suppss 7847 . . . . . 6 (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ (𝐿 supp (0g‘(Scalar‘𝐵))))
13875, 137ssfid 8729 . . . . 5 (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ∈ Fin)
139 suppssdm 7830 . . . . . . . . . 10 (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ dom 𝐺
140139, 1fssdm 6508 . . . . . . . . 9 (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ 𝑌)
141140sselda 3918 . . . . . . . 8 ((𝜑𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))) → 𝑤𝑌)
142 eleq1w 2875 . . . . . . . . . . . 12 (𝑗 = 𝑤 → (𝑗𝑌𝑤𝑌))
143142anbi2d 631 . . . . . . . . . . 11 (𝑗 = 𝑤 → ((𝜑𝑗𝑌) ↔ (𝜑𝑤𝑌)))
144 fveq2 6649 . . . . . . . . . . . 12 (𝑗 = 𝑤 → (𝐺𝑗) = (𝐺𝑤))
145144breq1d 5043 . . . . . . . . . . 11 (𝑗 = 𝑤 → ((𝐺𝑗) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺𝑤) finSupp (0g‘(Scalar‘𝐶))))
146143, 145imbi12d 348 . . . . . . . . . 10 (𝑗 = 𝑤 → (((𝜑𝑗𝑌) → (𝐺𝑗) finSupp (0g‘(Scalar‘𝐶))) ↔ ((𝜑𝑤𝑌) → (𝐺𝑤) finSupp (0g‘(Scalar‘𝐶)))))
147146, 97chvarvv 2005 . . . . . . . . 9 ((𝜑𝑤𝑌) → (𝐺𝑤) finSupp (0g‘(Scalar‘𝐶)))
148147fsuppimpd 8828 . . . . . . . 8 ((𝜑𝑤𝑌) → ((𝐺𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
149141, 148syldan 594 . . . . . . 7 ((𝜑𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))) → ((𝐺𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
150149ralrimiva 3152 . . . . . 6 (𝜑 → ∀𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
151 iunfi 8800 . . . . . 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 587 . . . . 5 (𝜑 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
153 xpfi 8777 . . . . 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 587 . . . 4 (𝜑 → ((𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) × 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺𝑤) supp (0g‘(Scalar‘𝐶)))) ∈ Fin)
155 fveq2 6649 . . . . . . . . . 10 (𝑣 = 𝑗 → (𝐺𝑣) = (𝐺𝑗))
156155fveq1d 6651 . . . . . . . . 9 (𝑣 = 𝑗 → ((𝐺𝑣)‘𝑢) = ((𝐺𝑗)‘𝑢))
157156mpteq2dv 5129 . . . . . . . 8 (𝑣 = 𝑗 → (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢)) = (𝑢𝑋 ↦ ((𝐺𝑗)‘𝑢)))
158 fveq2 6649 . . . . . . . . 9 (𝑢 = 𝑖 → ((𝐺𝑗)‘𝑢) = ((𝐺𝑗)‘𝑖))
159158cbvmptv 5136 . . . . . . . 8 (𝑢𝑋 ↦ ((𝐺𝑗)‘𝑢)) = (𝑖𝑋 ↦ ((𝐺𝑗)‘𝑖))
160157, 159eqtrdi 2852 . . . . . . 7 (𝑣 = 𝑗 → (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢)) = (𝑖𝑋 ↦ ((𝐺𝑗)‘𝑖)))
161160cbvmptv 5136 . . . . . 6 (𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢))) = (𝑗𝑌 ↦ (𝑖𝑋 ↦ ((𝐺𝑗)‘𝑖)))
162 fvexd 6664 . . . . . 6 (𝜑 → (0g‘(Scalar‘𝐶)) ∈ V)
163 fvexd 6664 . . . . . 6 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → ((𝐺𝑗)‘𝑖) ∈ V)
16442, 161, 46, 47, 162, 163suppovss 30446 . . . . 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 19538 . . . . . . . 8 (𝑉 ∈ (SubRing‘𝐸) → (0g𝐸) = (0g𝐾))
16619, 165syl 17 . . . . . . 7 (𝜑 → (0g𝐸) = (0g𝐾))
16736fveq2d 6653 . . . . . . 7 (𝜑 → (0g𝐾) = (0g‘(Scalar‘𝐶)))
168166, 167eqtr2d 2837 . . . . . 6 (𝜑 → (0g‘(Scalar‘𝐶)) = (0g𝐸))
169168oveq2d 7155 . . . . 5 (𝜑 → (𝐻 supp (0g‘(Scalar‘𝐶))) = (𝐻 supp (0g𝐸)))
1701feqmptd 6712 . . . . . . . 8 (𝜑𝐺 = (𝑣𝑌 ↦ (𝐺𝑣)))
171 eleq1w 2875 . . . . . . . . . . . . 13 (𝑗 = 𝑣 → (𝑗𝑌𝑣𝑌))
172171anbi2d 631 . . . . . . . . . . . 12 (𝑗 = 𝑣 → ((𝜑𝑗𝑌) ↔ (𝜑𝑣𝑌)))
173 fveq2 6649 . . . . . . . . . . . . 13 (𝑗 = 𝑣 → (𝐺𝑗) = (𝐺𝑣))
174173feq1d 6476 . . . . . . . . . . . 12 (𝑗 = 𝑣 → ((𝐺𝑗):𝑋⟶(Base‘𝐸) ↔ (𝐺𝑣):𝑋⟶(Base‘𝐸)))
175172, 174imbi12d 348 . . . . . . . . . . 11 (𝑗 = 𝑣 → (((𝜑𝑗𝑌) → (𝐺𝑗):𝑋⟶(Base‘𝐸)) ↔ ((𝜑𝑣𝑌) → (𝐺𝑣):𝑋⟶(Base‘𝐸))))
17610, 20ressbas2 16550 . . . . . . . . . . . . . . . 16 (𝑉 ⊆ (Base‘𝐸) → 𝑉 = (Base‘𝐾))
17722, 176syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑉 = (Base‘𝐾))
17836fveq2d 6653 . . . . . . . . . . . . . . 15 (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝐶)))
179177, 178eqtrd 2836 . . . . . . . . . . . . . 14 (𝜑𝑉 = (Base‘(Scalar‘𝐶)))
180179, 22eqsstrrd 3957 . . . . . . . . . . . . 13 (𝜑 → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
181180adantr 484 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
18256, 181fssd 6506 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝐺𝑗):𝑋⟶(Base‘𝐸))
183175, 182chvarvv 2005 . . . . . . . . . 10 ((𝜑𝑣𝑌) → (𝐺𝑣):𝑋⟶(Base‘𝐸))
184183feqmptd 6712 . . . . . . . . 9 ((𝜑𝑣𝑌) → (𝐺𝑣) = (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢)))
185184mpteq2dva 5128 . . . . . . . 8 (𝜑 → (𝑣𝑌 ↦ (𝐺𝑣)) = (𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢))))
186170, 185eqtr2d 2837 . . . . . . 7 (𝜑 → (𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢))) = 𝐺)
187186oveq1d 7154 . . . . . 6 (𝜑 → ((𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢))) supp (𝑋 × {(0g‘(Scalar‘𝐶))})) = (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})))
188186fveq1d 6651 . . . . . . . 8 (𝜑 → ((𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢)))‘𝑤) = (𝐺𝑤))
189188oveq1d 7154 . . . . . . 7 (𝜑 → (((𝑣𝑌 ↦ (𝑢𝑋 ↦ ((𝐺𝑣)‘𝑢)))‘𝑤) supp (0g‘(Scalar‘𝐶))) = ((𝐺𝑤) supp (0g‘(Scalar‘𝐶))))
190187, 189iuneq12d 4912 . . . . . 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 5554 . . . . 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 3964 . . . 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 8836 . . . 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 1375 . . 3 (𝜑𝐻 finSupp (0g𝐸))
19537fveq2d 6653 . . . 4 (𝜑 → (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐶)))
196195, 168eqtr2d 2837 . . 3 (𝜑 → (0g𝐸) = (0g‘(Scalar‘𝐴)))
197194, 196breqtrd 5059 . 2 (𝜑𝐻 finSupp (0g‘(Scalar‘𝐴)))
198 fedgmullem1.z . . 3 (𝜑𝑍 = (𝐵 Σg (𝑗𝑌 ↦ ((𝐿𝑗)( ·𝑠𝐵)𝑗))))
19985, 67, 13, 15, 92, 46drgextgsum 31085 . . 3 (𝜑 → (𝐸 Σg (𝑗𝑌 ↦ ((𝐿𝑗)( ·𝑠𝐵)𝑗))) = (𝐵 Σg (𝑗𝑌 ↦ ((𝐿𝑗)( ·𝑠𝐵)𝑗))))
20047adantr 484 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝑋 ∈ (LBasis‘𝐶))
20113adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑗𝑌) → 𝑈 ∈ (SubRing‘𝐸))
202 subrgsubg 19537 . . . . . . . . . . . . 13 (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ∈ (SubGrp‘𝐸))
203 subgsubm 18296 . . . . . . . . . . . . 13 (𝑈 ∈ (SubGrp‘𝐸) → 𝑈 ∈ (SubMnd‘𝐸))
204201, 202, 2033syl 18 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝑈 ∈ (SubMnd‘𝐸))
205113ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝐶 ∈ LMod)
20656ffvelrnda 6832 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝐺𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶)))
207118ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑋 ⊆ (Base‘𝐶))
208207, 61sseldd 3919 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑖 ∈ (Base‘𝐶))
209115, 126, 127, 125lmodvscl 19647 . . . . . . . . . . . . . . 15 ((𝐶 ∈ LMod ∧ ((𝐺𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑖 ∈ (Base‘𝐶)) → (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖) ∈ (Base‘𝐶))
210205, 206, 208, 209syl3anc 1368 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖) ∈ (Base‘𝐶))
21115, 20ressbas2 16550 . . . . . . . . . . . . . . . . 17 (𝑈 ⊆ (Base‘𝐸) → 𝑈 = (Base‘𝐹))
21288, 211syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑈 = (Base‘𝐹))
21331, 34srabase 19946 . . . . . . . . . . . . . . . 16 (𝜑 → (Base‘𝐹) = (Base‘𝐶))
214212, 213eqtrd 2836 . . . . . . . . . . . . . . 15 (𝜑𝑈 = (Base‘𝐶))
215214ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑈 = (Base‘𝐶))
216210, 215eleqtrrd 2896 . . . . . . . . . . . . 13 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖) ∈ 𝑈)
217216fmpttd 6860 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖)):𝑋𝑈)
218200, 204, 217, 15gsumsubm 17994 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (𝐹 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))
219 eqid 2801 . . . . . . . . . . . . . . . . . 18 (.r𝐸) = (.r𝐸)
22015, 219ressmulr 16620 . . . . . . . . . . . . . . . . 17 (𝑈 ∈ (SubRing‘𝐸) → (.r𝐸) = (.r𝐹))
22113, 220syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (.r𝐸) = (.r𝐹))
22231, 34sravsca 19950 . . . . . . . . . . . . . . . 16 (𝜑 → (.r𝐹) = ( ·𝑠𝐶))
223221, 222eqtr2d 2837 . . . . . . . . . . . . . . 15 (𝜑 → ( ·𝑠𝐶) = (.r𝐸))
224223ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ( ·𝑠𝐶) = (.r𝐸))
225224oveqd 7156 . . . . . . . . . . . . 13 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖) = (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖))
226225mpteq2dva 5128 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖)) = (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)))
227226oveq2d 7155 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖))))
22830, 92, 14, 109, 108, 47drgextgsum 31085 . . . . . . . . . . . 12 (𝜑 → (𝐹 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))
229228adantr 484 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝐹 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))) = (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))
230218, 227, 2293eqtr3d 2844 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖))) = (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))
231230oveq1d 7154 . . . . . . . . 9 ((𝜑𝑗𝑌) → ((𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗) = ((𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖)))(.r𝐸)𝑗))
23269ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝐸 ∈ Ring)
233180ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
234233, 206sseldd 3919 . . . . . . . . . . . . 13 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝐺𝑗)‘𝑖) ∈ (Base‘𝐸))
235214, 88eqsstrrd 3957 . . . . . . . . . . . . . . . 16 (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
236118, 235sstrd 3928 . . . . . . . . . . . . . . 15 (𝜑𝑋 ⊆ (Base‘𝐸))
237236ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑋 ⊆ (Base‘𝐸))
238237, 61sseldd 3919 . . . . . . . . . . . . 13 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑖 ∈ (Base‘𝐸))
239 eqid 2801 . . . . . . . . . . . . . . . . . 18 (Base‘𝐵) = (Base‘𝐵)
240 eqid 2801 . . . . . . . . . . . . . . . . . 18 (LBasis‘𝐵) = (LBasis‘𝐵)
241239, 240lbsss 19845 . . . . . . . . . . . . . . . . 17 (𝑌 ∈ (LBasis‘𝐵) → 𝑌 ⊆ (Base‘𝐵))
24246, 241syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑌 ⊆ (Base‘𝐵))
24386, 88srabase 19946 . . . . . . . . . . . . . . . 16 (𝜑 → (Base‘𝐸) = (Base‘𝐵))
244242, 243sseqtrrd 3959 . . . . . . . . . . . . . . 15 (𝜑𝑌 ⊆ (Base‘𝐸))
245244ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑌 ⊆ (Base‘𝐸))
246 simplr 768 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑗𝑌)
247245, 246sseldd 3919 . . . . . . . . . . . . 13 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑗 ∈ (Base‘𝐸))
24820, 219ringass 19313 . . . . . . . . . . . . 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 1369 . . . . . . . . . . . 12 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗) = (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))
250249mpteq2dva 5128 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ ((((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗)) = (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗))))
251250oveq2d 7155 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ ((((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))) = (𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))))
252 eqid 2801 . . . . . . . . . . 11 (+g𝐸) = (+g𝐸)
25369adantr 484 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → 𝐸 ∈ Ring)
254242adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑗𝑌) → 𝑌 ⊆ (Base‘𝐵))
255243adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑗𝑌) → (Base‘𝐸) = (Base‘𝐵))
256254, 255sseqtrrd 3959 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝑌 ⊆ (Base‘𝐸))
257 simpr 488 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝑗𝑌)
258256, 257sseldd 3919 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → 𝑗 ∈ (Base‘𝐸))
25920, 219ringcl 19310 . . . . . . . . . . . 12 ((𝐸 ∈ Ring ∧ ((𝐺𝑗)‘𝑖) ∈ (Base‘𝐸) ∧ 𝑖 ∈ (Base‘𝐸)) → (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖) ∈ (Base‘𝐸))
260232, 234, 238, 259syl3anc 1368 . . . . . . . . . . 11 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖) ∈ (Base‘𝐸))
261168breq2d 5045 . . . . . . . . . . . . . 14 (𝜑 → ((𝐺𝑗) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺𝑗) finSupp (0g𝐸)))
262261adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑗𝑌) → ((𝐺𝑗) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺𝑗) finSupp (0g𝐸)))
26397, 262mpbid 235 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → (𝐺𝑗) finSupp (0g𝐸))
26420, 253, 200, 238, 182, 263rmfsupp2 30920 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)) finSupp (0g𝐸))
26520, 71, 252, 219, 253, 200, 258, 260, 264gsummulc1 19355 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ ((((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))) = ((𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗))
266251, 265eqtr3d 2838 . . . . . . . . 9 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))) = ((𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗))
26783oveq1d 7154 . . . . . . . . 9 ((𝜑𝑗𝑌) → ((𝐿𝑗)(.r𝐸)𝑗) = ((𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖)))(.r𝐸)𝑗))
268231, 266, 2673eqtr4rd 2847 . . . . . . . 8 ((𝜑𝑗𝑌) → ((𝐿𝑗)(.r𝐸)𝑗) = (𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))))
26986, 88sravsca 19950 . . . . . . . . . 10 (𝜑 → (.r𝐸) = ( ·𝑠𝐵))
270269adantr 484 . . . . . . . . 9 ((𝜑𝑗𝑌) → (.r𝐸) = ( ·𝑠𝐵))
271270oveqd 7156 . . . . . . . 8 ((𝜑𝑗𝑌) → ((𝐿𝑗)(.r𝐸)𝑗) = ((𝐿𝑗)( ·𝑠𝐵)𝑗))
272 fvexd 6664 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑌𝑖𝑋) → ((𝐺𝑗)‘𝑖) ∈ V)
273 ovexd 7174 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑌𝑖𝑋) → (𝑖(.r𝐸)𝑗) ∈ V)
27442a1i 11 . . . . . . . . . . . . . 14 (𝜑𝐻 = (𝑗𝑌, 𝑖𝑋 ↦ ((𝐺𝑗)‘𝑖)))
275 fedgmullem.d . . . . . . . . . . . . . . 15 𝐷 = (𝑗𝑌, 𝑖𝑋 ↦ (𝑖(.r𝐸)𝑗))
276275a1i 11 . . . . . . . . . . . . . 14 (𝜑𝐷 = (𝑗𝑌, 𝑖𝑋 ↦ (𝑖(.r𝐸)𝑗)))
27746, 47, 272, 273, 274, 276offval22 7770 . . . . . . . . . . . . 13 (𝜑 → (𝐻f (.r𝐸)𝐷) = (𝑗𝑌, 𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗))))
278277oveqd 7156 . . . . . . . . . . . 12 (𝜑 → (𝑗(𝐻f (.r𝐸)𝐷)𝑖) = (𝑗(𝑗𝑌, 𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))𝑖))
279278ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗(𝐻f (.r𝐸)𝐷)𝑖) = (𝑗(𝑗𝑌, 𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))𝑖))
280 ovexd 7174 . . . . . . . . . . . 12 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)) ∈ V)
281 eqid 2801 . . . . . . . . . . . . 13 (𝑗𝑌, 𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗))) = (𝑗𝑌, 𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))
282281ovmpt4g 7280 . . . . . . . . . . . 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)))
283246, 61, 280, 282syl3anc 1368 . . . . . . . . . . 11 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗(𝑗𝑌, 𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))𝑖) = (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))
284279, 283eqtr2d 2837 . . . . . . . . . 10 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)) = (𝑗(𝐻f (.r𝐸)𝐷)𝑖))
285284mpteq2dva 5128 . . . . . . . . 9 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗))) = (𝑖𝑋 ↦ (𝑗(𝐻f (.r𝐸)𝐷)𝑖)))
286285oveq2d 7155 . . . . . . . 8 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))) = (𝐸 Σg (𝑖𝑋 ↦ (𝑗(𝐻f (.r𝐸)𝐷)𝑖))))
287268, 271, 2863eqtr3d 2844 . . . . . . 7 ((𝜑𝑗𝑌) → ((𝐿𝑗)( ·𝑠𝐵)𝑗) = (𝐸 Σg (𝑖𝑋 ↦ (𝑗(𝐻f (.r𝐸)𝐷)𝑖))))
288287mpteq2dva 5128 . . . . . 6 (𝜑 → (𝑗𝑌 ↦ ((𝐿𝑗)( ·𝑠𝐵)𝑗)) = (𝑗𝑌 ↦ (𝐸 Σg (𝑖𝑋 ↦ (𝑗(𝐻f (.r𝐸)𝐷)𝑖)))))
289288oveq2d 7155 . . . . 5 (𝜑 → (𝐸 Σg (𝑗𝑌 ↦ ((𝐿𝑗)( ·𝑠𝐵)𝑗))) = (𝐸 Σg (𝑗𝑌 ↦ (𝐸 Σg (𝑖𝑋 ↦ (𝑗(𝐻f (.r𝐸)𝐷)𝑖))))))
290 ringcmn 19330 . . . . . . 7 (𝐸 ∈ Ring → 𝐸 ∈ CMnd)
29169, 290syl 17 . . . . . 6 (𝜑𝐸 ∈ CMnd)
29269adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝐸 ∈ Ring)
29338, 180eqsstrd 3956 . . . . . . . . . 10 (𝜑 → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸))
294293adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸))
295 simprl 770 . . . . . . . . 9 ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑙 ∈ (Base‘(Scalar‘𝐴)))
296294, 295sseldd 3919 . . . . . . . 8 ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑙 ∈ (Base‘𝐸))
297 simprr 772 . . . . . . . . 9 ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑘 ∈ (Base‘𝐴))
29812, 22srabase 19946 . . . . . . . . . 10 (𝜑 → (Base‘𝐸) = (Base‘𝐴))
299298adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (Base‘𝐸) = (Base‘𝐴))
300297, 299eleqtrrd 2896 . . . . . . . 8 ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑘 ∈ (Base‘𝐸))
30120, 219ringcl 19310 . . . . . . . 8 ((𝐸 ∈ Ring ∧ 𝑙 ∈ (Base‘𝐸) ∧ 𝑘 ∈ (Base‘𝐸)) → (𝑙(.r𝐸)𝑘) ∈ (Base‘𝐸))
302292, 296, 300, 301syl3anc 1368 . . . . . . 7 ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (𝑙(.r𝐸)𝑘) ∈ (Base‘𝐸))
30320, 219ringcl 19310 . . . . . . . . . . . 12 ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
304232, 238, 247, 303syl3anc 1368 . . . . . . . . . . 11 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
305298ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (Base‘𝐸) = (Base‘𝐴))
306304, 305eleqtrd 2895 . . . . . . . . . 10 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
307306anasss 470 . . . . . . . . 9 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
308307ralrimivva 3159 . . . . . . . 8 (𝜑 → ∀𝑗𝑌𝑖𝑋 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
309275fmpo 7752 . . . . . . . 8 (∀𝑗𝑌𝑖𝑋 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴) ↔ 𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
310308, 309sylib 221 . . . . . . 7 (𝜑𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
311 inidm 4148 . . . . . . 7 ((𝑌 × 𝑋) ∩ (𝑌 × 𝑋)) = (𝑌 × 𝑋)
312302, 65, 310, 48, 48, 311off 7408 . . . . . 6 (𝜑 → (𝐻f (.r𝐸)𝐷):(𝑌 × 𝑋)⟶(Base‘𝐸))
31369adantr 484 . . . . . . . 8 ((𝜑𝑢 ∈ (Base‘𝐴)) → 𝐸 ∈ Ring)
314 simpr 488 . . . . . . . . 9 ((𝜑𝑢 ∈ (Base‘𝐴)) → 𝑢 ∈ (Base‘𝐴))
315298adantr 484 . . . . . . . . 9 ((𝜑𝑢 ∈ (Base‘𝐴)) → (Base‘𝐸) = (Base‘𝐴))
316314, 315eleqtrrd 2896 . . . . . . . 8 ((𝜑𝑢 ∈ (Base‘𝐴)) → 𝑢 ∈ (Base‘𝐸))
31720, 219, 71ringlz 19336 . . . . . . . 8 ((𝐸 ∈ Ring ∧ 𝑢 ∈ (Base‘𝐸)) → ((0g𝐸)(.r𝐸)𝑢) = (0g𝐸))
318313, 316, 317syl2anc 587 . . . . . . 7 ((𝜑𝑢 ∈ (Base‘𝐴)) → ((0g𝐸)(.r𝐸)𝑢) = (0g𝐸))
31948, 73, 73, 65, 310, 194, 318offinsupp1 30492 . . . . . 6 (𝜑 → (𝐻f (.r𝐸)𝐷) finSupp (0g𝐸))
32020, 71, 291, 46, 47, 312, 319gsumxp 19092 . . . . 5 (𝜑 → (𝐸 Σg (𝐻f (.r𝐸)𝐷)) = (𝐸 Σg (𝑗𝑌 ↦ (𝐸 Σg (𝑖𝑋 ↦ (𝑗(𝐻f (.r𝐸)𝐷)𝑖))))))
32112, 22sravsca 19950 . . . . . . . 8 (𝜑 → (.r𝐸) = ( ·𝑠𝐴))
322 ofeq 7395 . . . . . . . 8 ((.r𝐸) = ( ·𝑠𝐴) → ∘f (.r𝐸) = ∘f ( ·𝑠𝐴))
323321, 322syl 17 . . . . . . 7 (𝜑 → ∘f (.r𝐸) = ∘f ( ·𝑠𝐴))
324323oveqd 7156 . . . . . 6 (𝜑 → (𝐻f (.r𝐸)𝐷) = (𝐻f ( ·𝑠𝐴)𝐷))
325324oveq2d 7155 . . . . 5 (𝜑 → (𝐸 Σg (𝐻f (.r𝐸)𝐷)) = (𝐸 Σg (𝐻f ( ·𝑠𝐴)𝐷)))
326289, 320, 3253eqtr2rd 2843 . . . 4 (𝜑 → (𝐸 Σg (𝐻f ( ·𝑠𝐴)𝐷)) = (𝐸 Σg (𝑗𝑌 ↦ ((𝐿𝑗)( ·𝑠𝐵)𝑗))))
327 ovexd 7174 . . . . 5 (𝜑 → (𝐻f ( ·𝑠𝐴)𝐷) ∈ V)
328 fedgmullem1.a . . . . . 6 (𝜑𝑍 ∈ (Base‘𝐴))
329328elfvexd 6683 . . . . 5 (𝜑𝐴 ∈ V)
33011, 327, 67, 329, 22gsumsra 30735 . . . 4 (𝜑 → (𝐸 Σg (𝐻f ( ·𝑠𝐴)𝐷)) = (𝐴 Σg (𝐻f ( ·𝑠𝐴)𝐷)))
331326, 330eqtr3d 2838 . . 3 (𝜑 → (𝐸 Σg (𝑗𝑌 ↦ ((𝐿𝑗)( ·𝑠𝐵)𝑗))) = (𝐴 Σg (𝐻f ( ·𝑠𝐴)𝐷)))
332198, 199, 3313eqtr2d 2842 . 2 (𝜑𝑍 = (𝐴 Σg (𝐻f ( ·𝑠𝐴)𝐷)))
333197, 332jca 515 1 (𝜑 → (𝐻 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑍 = (𝐴 Σg (𝐻f ( ·𝑠𝐴)𝐷))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  ∀wral 3109  Vcvv 3444   ∖ cdif 3881   ⊆ wss 3884  {csn 4528  ∪ ciun 4884   class class class wbr 5033   ↦ cmpt 5113   × cxp 5521  Fun wfun 6322  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139   ∈ cmpo 7141   ∘f cof 7391   supp csupp 7817   ↑m cmap 8393  Fincfn 8496   finSupp cfsupp 8821  Basecbs 16478   ↾s cress 16479  +gcplusg 16560  .rcmulr 16561  Scalarcsca 16563   ·𝑠 cvsca 16564  0gc0g 16708   Σg cgsu 16709  SubMndcsubmnd 17950  Grpcgrp 18098  SubGrpcsubg 18268  CMndccmn 18901  Ringcrg 19293  DivRingcdr 19498  SubRingcsubrg 19527  LModclmod 19630  LSpanclspn 19739  LBasisclbs 19842  LVecclvec 19870  subringAlg csra 19936  LIndSclinds 20497 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12890  df-fzo 13033  df-seq 13369  df-hash 13691  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-sca 16576  df-vsca 16577  df-ip 16578  df-tset 16579  df-ple 16580  df-ds 16582  df-hom 16584  df-cco 16585  df-0g 16710  df-gsum 16711  df-prds 16716  df-pws 16718  df-mre 16852  df-mrc 16853  df-acs 16855  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-mhm 17951  df-submnd 17952  df-grp 18101  df-minusg 18102  df-sbg 18103  df-mulg 18220  df-subg 18271  df-ghm 18351  df-cntz 18442  df-cmn 18903  df-abl 18904  df-mgp 19236  df-ur 19248  df-ring 19295  df-drng 19500  df-subrg 19529  df-lmod 19632  df-lss 19700  df-lsp 19740  df-lmhm 19790  df-lbs 19843  df-lvec 19871  df-sra 19940  df-rgmod 19941  df-nzr 20027  df-dsmm 20424  df-frlm 20439  df-uvc 20475  df-lindf 20498  df-linds 20499 This theorem is referenced by:  fedgmul  31115
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