Theorem fedgmullem1 33822

Description: Lemma for fedgmul 33824. (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 = (𝐸 ↾s 𝑈)
fedgmul.k	⊢ 𝐾 = (𝐸 ↾s 𝑉)
fedgmul.1	⊢ (𝜑 → 𝐸 ∈ DivRing)
fedgmul.2	⊢ (𝜑 → 𝐹 ∈ DivRing)
fedgmul.3	⊢ (𝜑 → 𝐾 ∈ DivRing)
fedgmul.4	⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))
fedgmul.5	⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐹))
fedgmullem.d	⊢ 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(.r‘𝐸)𝑗))
fedgmullem.h	⊢ 𝐻 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖))
fedgmullem.x	⊢ (𝜑 → 𝑋 ∈ (LBasis‘𝐶))
fedgmullem.y	⊢ (𝜑 → 𝑌 ∈ (LBasis‘𝐵))
fedgmullem1.a	⊢ (𝜑 → 𝑍 ∈ (Base‘𝐴))
fedgmullem1.l	⊢ (𝜑 → 𝐿:𝑌⟶(Base‘(Scalar‘𝐵)))
fedgmullem1.1	⊢ (𝜑 → 𝐿 finSupp (0g‘(Scalar‘𝐵)))
fedgmullem1.z	⊢ (𝜑 → 𝑍 = (𝐵 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))))
fedgmullem1.g	⊢ (𝜑 → 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋))
fedgmullem1.2	⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶)))
fedgmullem1.3	⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐿‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))))

Assertion

Ref	Expression
fedgmullem1	⊢ (𝜑 → (𝐻 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑍 = (𝐴 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)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.

Step	Hyp	Ref	Expression
1		fedgmullem1.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋))
2		simpllr 781	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋))
3		simplr 774	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ 𝑌)
4	2, 3	ffvelcdmd 7027	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝐺‘𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋))
5		elmapi 8787	. . . . . . . . . . . 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 467	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
8		simprr 778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑖 ∈ 𝑋)
9	7, 8	ffvelcdmd 7027	. . . . . . . . 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 20571	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (𝑉 ∈ (SubRing‘𝐹) ↔ (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈)))
17	16	biimpa 477	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ∈ (SubRing‘𝐹)) → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈))
18	13, 14, 17	syl2anc 590	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈))
19	18	simpld 495	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐸))
20		eqid 2739	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐸) = (Base‘𝐸)
21	20	subrgss 20545	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
22	19, 21	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐸))
23	12, 22	srasca 21171	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ↾s 𝑉) = (Scalar‘𝐴))
24	10, 23	eqtrid 2786	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐴))
25	18	simprd 496	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 ⊆ 𝑈)
26		ressabs 17210	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈) → ((𝐸 ↾s 𝑈) ↾s 𝑉) = (𝐸 ↾s 𝑉))
27	13, 25, 26	syl2anc 590	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐸 ↾s 𝑈) ↾s 𝑉) = (𝐸 ↾s 𝑉))
28	15	oveq1i 7367	. . . . . . . . . . . . . 14 ⊢ (𝐹 ↾s 𝑉) = ((𝐸 ↾s 𝑈) ↾s 𝑉)
29	27, 28, 10	3eqtr4g 2799	. . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐹) = (Base‘𝐹)
33	32	subrgss 20545	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
34	14, 33	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐹))
35	31, 34	srasca 21171	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ↾s 𝑉) = (Scalar‘𝐶))
36	29, 35	eqtr3d 2776	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐶))
37	24, 36	eqtr3d 2776	. . . . . . . . . . 11 ⊢ (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
38	37	fveq2d 6832	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
39	38	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
40	9, 39	eleqtrrd 2842	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
41	40	ralrimivva 3182	. . . . . . 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 8011	. . . . . . 7 ⊢ (∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)) ↔ 𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
44	41, 43	sylib 219	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → 𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
45		fvexd 6843	. . . . . . 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 7695	. . . . . . . 8 ⊢ (𝜑 → (𝑌 × 𝑋) ∈ V)
49	48	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → (𝑌 × 𝑋) ∈ V)
50	45, 49	elmapd 8778	. . . . . 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 258	. . . . 5 ⊢ ((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → 𝐻 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)))
52	1, 51	mpdan 693	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)))
53		simpl 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝜑)
54	53	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝜑)
55	1	ffvelcdmda 7026	. . . . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
58	38	feq3d 6641	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐴)) ↔ (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶))))
59	58	biimpar 478	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶))) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐴)))
60	54, 57, 59	syl2anc 590	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐴)))
61		simpr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
62	60, 61	ffvelcdmd 7027	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
63	62	ralrimiva 3131	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ∀𝑖 ∈ 𝑋 ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
64	63	ralrimiva 3131	. . . . . 6 ⊢ (𝜑 → ∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
65	64, 43	sylib 219	. . . . 5 ⊢ (𝜑 → 𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
66	65	ffund 6660	. . . 4 ⊢ (𝜑 → Fun 𝐻)
67		fedgmul.1	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ DivRing)
68		drngring 20709	. . . . . 6 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
69	67, 68	syl 17	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Ring)
70		ringgrp 20211	. . . . 5 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Grp)
71		eqid 2739	. . . . . 6 ⊢ (0g‘𝐸) = (0g‘𝐸)
72	20, 71	grpidcl 18933	. . . . 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 9273	. . . . . 6 ⊢ (𝜑 → (𝐿 supp (0g‘(Scalar‘𝐵))) ∈ Fin)
76		simpl 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → 𝜑)
77		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵)))))
78	77	eldifad 3895	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → 𝑗 ∈ 𝑌)
79		fedgmullem1.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿:𝑌⟶(Base‘(Scalar‘𝐵)))
80		ssidd 3938	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 supp (0g‘(Scalar‘𝐵))) ⊆ (𝐿 supp (0g‘(Scalar‘𝐵))))
81		fvexd 6843	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘(Scalar‘𝐵)) ∈ V)
82	79, 80, 46, 81	suppssr 8136	. . . . . . . . 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 597	. . . . . . . . 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 20545	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
88	13, 87	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐸))
89	86, 88	srasca 21171	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ↾s 𝑈) = (Scalar‘𝐵))
90	15, 89	eqtrid 2786	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐵))
91	90	fveq2d 6832	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝐹) = (0g‘(Scalar‘𝐵)))
92		fedgmul.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ DivRing)
93	30, 92, 14	drgext0g 33783	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝐹) = (0g‘𝐶))
94	91, 93	eqtr3d 2776	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘(Scalar‘𝐵)) = (0g‘𝐶))
95	94	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → (0g‘(Scalar‘𝐵)) = (0g‘𝐶))
96	82, 84, 95	3eqtr3d 2782	. . . . . . . 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 5076	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝐺‘𝑗) → (𝑔 finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶))))
99		fveq1 6827	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (𝐺‘𝑗) → (𝑔‘𝑖) = ((𝐺‘𝑗)‘𝑖))
100	99	oveq1d 7372	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝐺‘𝑗) → ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖) = (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
101	100	mpteq2dv 5167	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝐺‘𝑗) → (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖)) = (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))
102	101	oveq2d 7373	. . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . 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 638	. . . . . . . . . . . 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 2743	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐺‘𝑗) → (𝑔 = (𝑋 × {(0g‘(Scalar‘𝐶))}) ↔ (𝐺‘𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))})))
106	104, 105	imbi12d 345	. . . . . . . . . . 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 2839	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 ↾s 𝑉) ∈ DivRing)
109		eqid 2739	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ↾s 𝑉) = (𝐹 ↾s 𝑉)
110	30, 109	sralvec 33778	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ DivRing ∧ (𝐹 ↾s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
111	92, 108, 14, 110	syl3anc 1379	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ LVec)
112		lveclmod 21097	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ LVec → 𝐶 ∈ LMod)
113	111, 112	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ LMod)
114	113	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐶 ∈ LMod)
115		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐶) = (Base‘𝐶)
116		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (LBasis‘𝐶) = (LBasis‘𝐶)
117	115, 116	lbsss 21068	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ (LBasis‘𝐶) → 𝑋 ⊆ (Base‘𝐶))
118	47, 117	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐶))
119	118	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑋 ⊆ (Base‘𝐶))
120		eqid 2739	. . . . . . . . . . . . . . . 16 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
121	115, 116, 120	islbs4 21808	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (LBasis‘𝐶) ↔ (𝑋 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑋) = (Base‘𝐶)))
122	47, 121	sylib 219	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑋) = (Base‘𝐶)))
123	122	simpld 495	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ (LIndS‘𝐶))
124	123	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑋 ∈ (LIndS‘𝐶))
125		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
126		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
127		eqid 2739	. . . . . . . . . . . . . 14 ⊢ ( ·𝑠 ‘𝐶) = ( ·𝑠 ‘𝐶)
128		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (0g‘𝐶) = (0g‘𝐶)
129		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘𝐶))
130	115, 125, 126, 127, 128, 129	islinds5 33451	. . . . . . . . . . . . 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 477	. . . . . . . . . . . 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 843	. . . . . . . . . . 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 3561	. . . . . . . . . 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 701	. . . . . . . . 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 407	. . . . . . . 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 843	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → (𝐺‘𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))}))
137	1, 136	suppss 8135	. . . . . 6 ⊢ (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ (𝐿 supp (0g‘(Scalar‘𝐵))))
138	75, 137	ssfid 9170	. . . . 5 ⊢ (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ∈ Fin)
139		suppssdm 8118	. . . . . . . . . 10 ⊢ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ dom 𝐺
140	139, 1	fssdm 6675	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ 𝑌)
141	140	sselda 3915	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))) → 𝑤 ∈ 𝑌)
142		eleq1w 2822	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑤 → (𝑗 ∈ 𝑌 ↔ 𝑤 ∈ 𝑌))
143	142	anbi2d 636	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑤 → ((𝜑 ∧ 𝑗 ∈ 𝑌) ↔ (𝜑 ∧ 𝑤 ∈ 𝑌)))
144		fveq2 6828	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑤 → (𝐺‘𝑗) = (𝐺‘𝑤))
145	144	breq1d 5083	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑤 → ((𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺‘𝑤) finSupp (0g‘(Scalar‘𝐶))))
146	143, 145	imbi12d 345	. . . . . . . . . 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 1996	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝐺‘𝑤) finSupp (0g‘(Scalar‘𝐶)))
148	147	fsuppimpd 9273	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
149	141, 148	syldan 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))) → ((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
150	149	ralrimiva 3131	. . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
151		iunfi 9244	. . . . . 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 590	. . . . 5 ⊢ (𝜑 → ∪ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
153		xpfi 9221	. . . . 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 590	. . . 4 ⊢ (𝜑 → ((𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) × ∪ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶)))) ∈ Fin)
155		fveq2 6828	. . . . . . . . . 10 ⊢ (𝑣 = 𝑗 → (𝐺‘𝑣) = (𝐺‘𝑗))
156	155	fveq1d 6830	. . . . . . . . 9 ⊢ (𝑣 = 𝑗 → ((𝐺‘𝑣)‘𝑢) = ((𝐺‘𝑗)‘𝑢))
157	156	mpteq2dv 5167	. . . . . . . 8 ⊢ (𝑣 = 𝑗 → (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)) = (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑢)))
158		fveq2 6828	. . . . . . . . 9 ⊢ (𝑢 = 𝑖 → ((𝐺‘𝑗)‘𝑢) = ((𝐺‘𝑗)‘𝑖))
159	158	cbvmptv 5177	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑢)) = (𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖))
160	157, 159	eqtrdi 2790	. . . . . . 7 ⊢ (𝑣 = 𝑗 → (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)) = (𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖)))
161	160	cbvmptv 5177	. . . . . 6 ⊢ (𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) = (𝑗 ∈ 𝑌 ↦ (𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖)))
162		fvexd 6843	. . . . . 6 ⊢ (𝜑 → (0g‘(Scalar‘𝐶)) ∈ V)
163		fvexd 6843	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → ((𝐺‘𝑗)‘𝑖) ∈ V)
164	42, 161, 46, 47, 162, 163	suppovss 32774	. . . . 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 20552	. . . . . . . 8 ⊢ (𝑉 ∈ (SubRing‘𝐸) → (0g‘𝐸) = (0g‘𝐾))
166	19, 165	syl 17	. . . . . . 7 ⊢ (𝜑 → (0g‘𝐸) = (0g‘𝐾))
167	36	fveq2d 6832	. . . . . . 7 ⊢ (𝜑 → (0g‘𝐾) = (0g‘(Scalar‘𝐶)))
168	166, 167	eqtr2d 2775	. . . . . 6 ⊢ (𝜑 → (0g‘(Scalar‘𝐶)) = (0g‘𝐸))
169	168	oveq2d 7373	. . . . 5 ⊢ (𝜑 → (𝐻 supp (0g‘(Scalar‘𝐶))) = (𝐻 supp (0g‘𝐸)))
170	1	feqmptd 6896	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝑌 ↦ (𝐺‘𝑣)))
171		eleq1w 2822	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑣 → (𝑗 ∈ 𝑌 ↔ 𝑣 ∈ 𝑌))
172	171	anbi2d 636	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑣 → ((𝜑 ∧ 𝑗 ∈ 𝑌) ↔ (𝜑 ∧ 𝑣 ∈ 𝑌)))
173		fveq2 6828	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑣 → (𝐺‘𝑗) = (𝐺‘𝑣))
174	173	feq1d 6638	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑣 → ((𝐺‘𝑗):𝑋⟶(Base‘𝐸) ↔ (𝐺‘𝑣):𝑋⟶(Base‘𝐸)))
175	172, 174	imbi12d 345	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑣 → (((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗):𝑋⟶(Base‘𝐸)) ↔ ((𝜑 ∧ 𝑣 ∈ 𝑌) → (𝐺‘𝑣):𝑋⟶(Base‘𝐸))))
176	10, 20	ressbas2 17200	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ⊆ (Base‘𝐸) → 𝑉 = (Base‘𝐾))
177	22, 176	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 = (Base‘𝐾))
178	36	fveq2d 6832	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝐶)))
179	177, 178	eqtrd 2774	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 = (Base‘(Scalar‘𝐶)))
180	179, 22	eqsstrrd 3950	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
181	180	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
182	56, 181	fssd 6673	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗):𝑋⟶(Base‘𝐸))
183	175, 182	chvarvv 1996	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑌) → (𝐺‘𝑣):𝑋⟶(Base‘𝐸))
184	183	feqmptd 6896	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑌) → (𝐺‘𝑣) = (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))
185	184	mpteq2dva 5166	. . . . . . . 8 ⊢ (𝜑 → (𝑣 ∈ 𝑌 ↦ (𝐺‘𝑣)) = (𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))))
186	170, 185	eqtr2d 2775	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) = 𝐺)
187	186	oveq1d 7372	. . . . . 6 ⊢ (𝜑 → ((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) supp (𝑋 × {(0g‘(Scalar‘𝐶))})) = (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})))
188	186	fveq1d 6830	. . . . . . . 8 ⊢ (𝜑 → ((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))‘𝑤) = (𝐺‘𝑤))
189	188	oveq1d 7372	. . . . . . 7 ⊢ (𝜑 → (((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))‘𝑤) supp (0g‘(Scalar‘𝐶))) = ((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))))
190	187, 189	iuneq12d 4952	. . . . . 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 5650	. . . . 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 3969	. . . 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 9284	. . . 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 1386	. . 3 ⊢ (𝜑 → 𝐻 finSupp (0g‘𝐸))
195	37	fveq2d 6832	. . . 4 ⊢ (𝜑 → (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐶)))
196	195, 168	eqtr2d 2775	. . 3 ⊢ (𝜑 → (0g‘𝐸) = (0g‘(Scalar‘𝐴)))
197	194, 196	breqtrd 5099	. 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 33788	. . 3 ⊢ (𝜑 → (𝐸 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))) = (𝐵 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))))
200	47	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑋 ∈ (LBasis‘𝐶))
201	13	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑈 ∈ (SubRing‘𝐸))
202		subrgsubg 20550	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ∈ (SubGrp‘𝐸))
203		subgsubm 19116	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (SubGrp‘𝐸) → 𝑈 ∈ (SubMnd‘𝐸))
204	201, 202, 203	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑈 ∈ (SubMnd‘𝐸))
205	113	ad2antrr 732	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐶 ∈ LMod)
206	56	ffvelcdmda 7026	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶)))
207	118	ad2antrr 732	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑋 ⊆ (Base‘𝐶))
208	207, 61	sseldd 3916	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐶))
209	115, 126, 127, 125	lmodvscl 20869	. . . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖) ∈ (Base‘𝐶))
211	15, 20	ressbas2 17200	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ⊆ (Base‘𝐸) → 𝑈 = (Base‘𝐹))
212	88, 211	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 = (Base‘𝐹))
213	31, 34	srabase 21168	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐹) = (Base‘𝐶))
214	212, 213	eqtrd 2774	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
215	214	ad2antrr 732	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑈 = (Base‘𝐶))
216	210, 215	eleqtrrd 2842	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖) ∈ 𝑈)
217	216	fmpttd 7057	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)):𝑋⟶𝑈)
218	200, 204, 217, 15	gsumsubm 18795	. . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (.r‘𝐸) = (.r‘𝐸)
220	15, 219	ressmulr 17262	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (.r‘𝐸) = (.r‘𝐹))
221	13, 220	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (.r‘𝐸) = (.r‘𝐹))
222	31, 34	sravsca 21172	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (.r‘𝐹) = ( ·𝑠 ‘𝐶))
223	221, 222	eqtr2d 2775	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ( ·𝑠 ‘𝐶) = (.r‘𝐸))
224	223	ad2antrr 732	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ( ·𝑠 ‘𝐶) = (.r‘𝐸))
225	224	oveqd 7374	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖) = (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖))
226	225	mpteq2dva 5166	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)) = (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)))
227	226	oveq2d 7373	. . . . . . . . . . 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 33788	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 2782	. . . . . . . . . 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 7372	. . . . . . . . 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 732	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ Ring)
233	180	ad2antrr 732	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
234	233, 206	sseldd 3916	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘𝐸))
235	214, 88	eqsstrrd 3950	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
236	118, 235	sstrd 3925	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐸))
237	236	ad2antrr 732	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑋 ⊆ (Base‘𝐸))
238	237, 61	sseldd 3916	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐸))
239		eqid 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝐵) = (Base‘𝐵)
240		eqid 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (LBasis‘𝐵) = (LBasis‘𝐵)
241	239, 240	lbsss 21068	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 ∈ (LBasis‘𝐵) → 𝑌 ⊆ (Base‘𝐵))
242	46, 241	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ⊆ (Base‘𝐵))
243	86, 88	srabase 21168	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐵))
244	242, 243	sseqtrrd 3952	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ⊆ (Base‘𝐸))
245	244	ad2antrr 732	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑌 ⊆ (Base‘𝐸))
246		simplr 774	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ 𝑌)
247	245, 246	sseldd 3916	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ (Base‘𝐸))
248	20, 219	ringass 20226	. . . . . . . . . . . . 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 1380	. . . . . . . . . . . 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 5166	. . . . . . . . . . 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 7373	. . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐸 ∈ Ring)
253	242	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑌 ⊆ (Base‘𝐵))
254	243	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (Base‘𝐸) = (Base‘𝐵))
255	253, 254	sseqtrrd 3952	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑌 ⊆ (Base‘𝐸))
256		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ 𝑌)
257	255, 256	sseldd 3916	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ (Base‘𝐸))
258	20, 219	ringcl 20223	. . . . . . . . . . . 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 1379	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖) ∈ (Base‘𝐸))
260	168	breq2d 5085	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺‘𝑗) finSupp (0g‘𝐸)))
261	260	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺‘𝑗) finSupp (0g‘𝐸)))
262	97, 261	mpbid 233	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) finSupp (0g‘𝐸))
263	20, 252, 200, 238, 182, 262	rmfsupp2 33320	. . . . . . . . . . 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 20287	. . . . . . . . . 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 2776	. . . . . . . . 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 7372	. . . . . . . . 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 2785	. . . . . . . 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 21172	. . . . . . . . . 10 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘𝐵))
269	268	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (.r‘𝐸) = ( ·𝑠 ‘𝐵))
270	269	oveqd 7374	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐿‘𝑗)(.r‘𝐸)𝑗) = ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))
271		fvexd 6843	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ V)
272		ovexd 7392	. . . . . . . . . . . . . 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 8028	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐻 ∘f (.r‘𝐸)𝐷) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗))))
277	276	oveqd 7374	. . . . . . . . . . . 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 732	. . . . . . . . . . 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 7392	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)) ∈ V)
280		eqid 2739	. . . . . . . . . . . . 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 7504	. . . . . . . . . . . 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 1379	. . . . . . . . . . 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 2775	. . . . . . . . . 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 5166	. . . . . . . . 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 7373	. . . . . . . 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 2782	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗) = (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (𝑗(𝐻 ∘f (.r‘𝐸)𝐷)𝑖))))
287	286	mpteq2dva 5166	. . . . . 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 7373	. . . . 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 20255	. . . . . . 7 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ CMnd)
290	69, 289	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ CMnd)
291	69	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝐸 ∈ Ring)
292	38, 180	eqsstrd 3949	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸))
293	292	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸))
294		simprl 776	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑙 ∈ (Base‘(Scalar‘𝐴)))
295	293, 294	sseldd 3916	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑙 ∈ (Base‘𝐸))
296		simprr 778	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑘 ∈ (Base‘𝐴))
297	12, 22	srabase 21168	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐴))
298	297	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (Base‘𝐸) = (Base‘𝐴))
299	296, 298	eleqtrrd 2842	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑘 ∈ (Base‘𝐸))
300	20, 219	ringcl 20223	. . . . . . . 8 ⊢ ((𝐸 ∈ Ring ∧ 𝑙 ∈ (Base‘𝐸) ∧ 𝑘 ∈ (Base‘𝐸)) → (𝑙(.r‘𝐸)𝑘) ∈ (Base‘𝐸))
301	291, 295, 299, 300	syl3anc 1379	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (𝑙(.r‘𝐸)𝑘) ∈ (Base‘𝐸))
302	20, 219	ringcl 20223	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
303	232, 238, 247, 302	syl3anc 1379	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
304	297	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (Base‘𝐸) = (Base‘𝐴))
305	303, 304	eleqtrd 2841	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴))
306	305	anasss 467	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴))
307	306	ralrimivva 3182	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴))
308	274	fmpo 8011	. . . . . . . 8 ⊢ (∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴) ↔ 𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
309	307, 308	sylib 219	. . . . . . 7 ⊢ (𝜑 → 𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
310		inidm 4156	. . . . . . 7 ⊢ ((𝑌 × 𝑋) ∩ (𝑌 × 𝑋)) = (𝑌 × 𝑋)
311	301, 65, 309, 48, 48, 310	off 7639	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘f (.r‘𝐸)𝐷):(𝑌 × 𝑋)⟶(Base‘𝐸))
312	69	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → 𝐸 ∈ Ring)
313		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → 𝑢 ∈ (Base‘𝐴))
314	297	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → (Base‘𝐸) = (Base‘𝐴))
315	313, 314	eleqtrrd 2842	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → 𝑢 ∈ (Base‘𝐸))
316	20, 219, 71	ringlz 20266	. . . . . . . 8 ⊢ ((𝐸 ∈ Ring ∧ 𝑢 ∈ (Base‘𝐸)) → ((0g‘𝐸)(.r‘𝐸)𝑢) = (0g‘𝐸))
317	312, 315, 316	syl2anc 590	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → ((0g‘𝐸)(.r‘𝐸)𝑢) = (0g‘𝐸))
318	48, 73, 73, 65, 309, 194, 317	offinsupp1 32819	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘f (.r‘𝐸)𝐷) finSupp (0g‘𝐸))
319	20, 71, 290, 46, 47, 311, 318	gsumxp 19943	. . . . 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 21172	. . . . . . . 8 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘𝐴))
321	320	ofeqd 7623	. . . . . . 7 ⊢ (𝜑 → ∘f (.r‘𝐸) = ∘f ( ·𝑠 ‘𝐴))
322	321	oveqd 7374	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘f (.r‘𝐸)𝐷) = (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷))
323	322	oveq2d 7373	. . . . 5 ⊢ (𝜑 → (𝐸 Σg (𝐻 ∘f (.r‘𝐸)𝐷)) = (𝐸 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)))
324	288, 319, 323	3eqtr2rd 2781	. . . 4 ⊢ (𝜑 → (𝐸 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)) = (𝐸 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))))
325		ovexd 7392	. . . . 5 ⊢ (𝜑 → (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷) ∈ V)
326		fedgmullem1.a	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐴))
327	326	elfvexd 6864	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
328	11, 325, 67, 327, 22	gsumsra 33129	. . . 4 ⊢ (𝜑 → (𝐸 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)) = (𝐴 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)))
329	324, 328	eqtr3d 2776	. . 3 ⊢ (𝜑 → (𝐸 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))) = (𝐴 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)))
330	198, 199, 329	3eqtr2d 2780	. 2 ⊢ (𝜑 → 𝑍 = (𝐴 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)))
331	197, 330	jca 516	1 ⊢ (𝜑 → (𝐻 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑍 = (𝐴 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  Vcvv 3431   ∖ cdif 3880   ⊆ wss 3883  {csn 4556  ∪ ciun 4922   class class class wbr 5073   ↦ cmpt 5154   × cxp 5617  Fun wfun 6480  ⟶wf 6482  ‘cfv 6486  (class class class)co 7357   ∈ cmpo 7359   ∘_f cof 7619   supp csupp 8101   ↑_m cmap 8764  Fincfn 8884   finSupp cfsupp 9265  Basecbs 17171   ↾_s cress 17192  ._rcmulr 17213  Scalarcsca 17215   ·_𝑠 cvsca 17216  0_gc0g 17394   Σ_g cgsu 17395  SubMndcsubmnd 18742  Grpcgrp 18901  SubGrpcsubg 19088  CMndccmn 19747  Ringcrg 20206  SubRingcsubrg 20542  DivRingcdr 20702  LModclmod 20851  LSpanclspn 20962  LBasisclbs 21065  LVecclvec 21093  subringAlg csra 21162  LIndSclinds 21781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4840  df-int 4879  df-iun 4924  df-iin 4925  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7621  df-om 7808  df-1st 7932  df-2nd 7933  df-supp 8102  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-er 8634  df-map 8766  df-ixp 8837  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-fsupp 9266  df-sup 9346  df-oi 9416  df-card 9855  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12167  df-2 12236  df-3 12237  df-4 12238  df-5 12239  df-6 12240  df-7 12241  df-8 12242  df-9 12243  df-n0 12430  df-z 12517  df-dec 12637  df-uz 12781  df-fz 13454  df-fzo 13601  df-seq 13956  df-hash 14285  df-struct 17109  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17172  df-ress 17193  df-plusg 17225  df-mulr 17226  df-sca 17228  df-vsca 17229  df-ip 17230  df-tset 17231  df-ple 17232  df-ds 17234  df-hom 17236  df-cco 17237  df-0g 17396  df-gsum 17397  df-prds 17402  df-pws 17404  df-mre 17540  df-mrc 17541  df-acs 17543  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-mhm 18743  df-submnd 18744  df-grp 18904  df-minusg 18905  df-sbg 18906  df-mulg 19036  df-subg 19091  df-ghm 19180  df-cntz 19284  df-cmn 19749  df-abl 19750  df-mgp 20114  df-rng 20126  df-ur 20155  df-ring 20208  df-nzr 20486  df-subrg 20543  df-drng 20704  df-lmod 20853  df-lss 20923  df-lsp 20963  df-lmhm 21013  df-lbs 21066  df-lvec 21094  df-sra 21164  df-rgmod 21165  df-dsmm 21708  df-frlm 21723  df-uvc 21759  df-lindf 21782  df-linds 21783

This theorem is referenced by:  fedgmul  33824