Theorem fedgmullem1 33735

Description: Lemma for fedgmul 33737. (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 775	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋))
3		simplr 768	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ 𝑌)
4	2, 3	ffvelcdmd 7028	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝐺‘𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋))
5		elmapi 8784	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
6	4, 5	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
7	6	anasss 466	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
8		simprr 772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑖 ∈ 𝑋)
9	7, 8	ffvelcdmd 7028	. . . . . . . . 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 20529	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (𝑉 ∈ (SubRing‘𝐹) ↔ (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈)))
17	16	biimpa 476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ∈ (SubRing‘𝐹)) → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈))
18	13, 14, 17	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈))
19	18	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐸))
20		eqid 2734	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐸) = (Base‘𝐸)
21	20	subrgss 20503	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
22	19, 21	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐸))
23	12, 22	srasca 21130	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ↾s 𝑉) = (Scalar‘𝐴))
24	10, 23	eqtrid 2781	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐴))
25	18	simprd 495	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 ⊆ 𝑈)
26		ressabs 17173	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈) → ((𝐸 ↾s 𝑈) ↾s 𝑉) = (𝐸 ↾s 𝑉))
27	13, 25, 26	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐸 ↾s 𝑈) ↾s 𝑉) = (𝐸 ↾s 𝑉))
28	15	oveq1i 7366	. . . . . . . . . . . . . 14 ⊢ (𝐹 ↾s 𝑉) = ((𝐸 ↾s 𝑈) ↾s 𝑉)
29	27, 28, 10	3eqtr4g 2794	. . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐹) = (Base‘𝐹)
33	32	subrgss 20503	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
34	14, 33	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐹))
35	31, 34	srasca 21130	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ↾s 𝑉) = (Scalar‘𝐶))
36	29, 35	eqtr3d 2771	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐶))
37	24, 36	eqtr3d 2771	. . . . . . . . . . 11 ⊢ (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
38	37	fveq2d 6836	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
39	38	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
40	9, 39	eleqtrrd 2837	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
41	40	ralrimivva 3177	. . . . . . 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 8010	. . . . . . 7 ⊢ (∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)) ↔ 𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
44	41, 43	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → 𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
45		fvexd 6847	. . . . . . 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 7694	. . . . . . . 8 ⊢ (𝜑 → (𝑌 × 𝑋) ∈ V)
49	48	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → (𝑌 × 𝑋) ∈ V)
50	45, 49	elmapd 8775	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → (𝐻 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)) ↔ 𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴))))
51	44, 50	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) → 𝐻 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)))
52	1, 51	mpdan 687	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)))
53		simpl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝜑)
54	53	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝜑)
55	1	ffvelcdmda 7027	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋))
56	55, 5	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
57	56	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
58	38	feq3d 6645	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐴)) ↔ (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶))))
59	58	biimpar 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶))) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐴)))
60	54, 57, 59	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐴)))
61		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
62	60, 61	ffvelcdmd 7028	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
63	62	ralrimiva 3126	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ∀𝑖 ∈ 𝑋 ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
64	63	ralrimiva 3126	. . . . . 6 ⊢ (𝜑 → ∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
65	64, 43	sylib 218	. . . . 5 ⊢ (𝜑 → 𝐻:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
66	65	ffund 6664	. . . 4 ⊢ (𝜑 → Fun 𝐻)
67		fedgmul.1	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ DivRing)
68		drngring 20667	. . . . . 6 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
69	67, 68	syl 17	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Ring)
70		ringgrp 20171	. . . . 5 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Grp)
71		eqid 2734	. . . . . 6 ⊢ (0g‘𝐸) = (0g‘𝐸)
72	20, 71	grpidcl 18893	. . . . 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 9270	. . . . . 6 ⊢ (𝜑 → (𝐿 supp (0g‘(Scalar‘𝐵))) ∈ Fin)
76		simpl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → 𝜑)
77		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵)))))
78	77	eldifad 3911	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → 𝑗 ∈ 𝑌)
79		fedgmullem1.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿:𝑌⟶(Base‘(Scalar‘𝐵)))
80		ssidd 3955	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 supp (0g‘(Scalar‘𝐵))) ⊆ (𝐿 supp (0g‘(Scalar‘𝐵))))
81		fvexd 6847	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘(Scalar‘𝐵)) ∈ V)
82	79, 80, 46, 81	suppssr 8135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → (𝐿‘𝑗) = (0g‘(Scalar‘𝐵)))
83		fedgmullem1.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐿‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))))
84	78, 83	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → (𝐿‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))))
85		fedgmul.b	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)
86	85	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 = ((subringAlg ‘𝐸)‘𝑈))
87	20	subrgss 20503	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
88	13, 87	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐸))
89	86, 88	srasca 21130	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ↾s 𝑈) = (Scalar‘𝐵))
90	15, 89	eqtrid 2781	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐵))
91	90	fveq2d 6836	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝐹) = (0g‘(Scalar‘𝐵)))
92		fedgmul.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ DivRing)
93	30, 92, 14	drgext0g 33695	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝐹) = (0g‘𝐶))
94	91, 93	eqtr3d 2771	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘(Scalar‘𝐵)) = (0g‘𝐶))
95	94	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → (0g‘(Scalar‘𝐵)) = (0g‘𝐶))
96	82, 84, 95	3eqtr3d 2777	. . . . . . . 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 5099	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝐺‘𝑗) → (𝑔 finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶))))
99		fveq1 6831	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (𝐺‘𝑗) → (𝑔‘𝑖) = ((𝐺‘𝑗)‘𝑖))
100	99	oveq1d 7371	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝐺‘𝑗) → ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖) = (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
101	100	mpteq2dv 5190	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝐺‘𝑗) → (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖)) = (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))
102	101	oveq2d 7372	. . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝐺‘𝑗) → ((𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶) ↔ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶)))
104	98, 103	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐺‘𝑗) → ((𝑔 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶)) ↔ ((𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶))))
105		eqeq1 2738	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐺‘𝑗) → (𝑔 = (𝑋 × {(0g‘(Scalar‘𝐶))}) ↔ (𝐺‘𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))})))
106	104, 105	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐺‘𝑗) → (((𝑔 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶)) → 𝑔 = (𝑋 × {(0g‘(Scalar‘𝐶))})) ↔ (((𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶)) → (𝐺‘𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))}))))
107		fedgmul.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ DivRing)
108	29, 107	eqeltrd 2834	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 ↾s 𝑉) ∈ DivRing)
109		eqid 2734	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ↾s 𝑉) = (𝐹 ↾s 𝑉)
110	30, 109	sralvec 33690	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ DivRing ∧ (𝐹 ↾s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
111	92, 108, 14, 110	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ LVec)
112		lveclmod 21056	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ LVec → 𝐶 ∈ LMod)
113	111, 112	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ LMod)
114	113	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐶 ∈ LMod)
115		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐶) = (Base‘𝐶)
116		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (LBasis‘𝐶) = (LBasis‘𝐶)
117	115, 116	lbsss 21027	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ (LBasis‘𝐶) → 𝑋 ⊆ (Base‘𝐶))
118	47, 117	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐶))
119	118	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑋 ⊆ (Base‘𝐶))
120		eqid 2734	. . . . . . . . . . . . . . . 16 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
121	115, 116, 120	islbs4 21785	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (LBasis‘𝐶) ↔ (𝑋 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑋) = (Base‘𝐶)))
122	47, 121	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑋) = (Base‘𝐶)))
123	122	simpld 494	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ (LIndS‘𝐶))
124	123	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑋 ∈ (LIndS‘𝐶))
125		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
126		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
127		eqid 2734	. . . . . . . . . . . . . 14 ⊢ ( ·𝑠 ‘𝐶) = ( ·𝑠 ‘𝐶)
128		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (0g‘𝐶) = (0g‘𝐶)
129		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘𝐶))
130	115, 125, 126, 127, 128, 129	islinds5 33397	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ LMod ∧ 𝑋 ⊆ (Base‘𝐶)) → (𝑋 ∈ (LIndS‘𝐶) ↔ ∀𝑔 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋)((𝑔 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶)) → 𝑔 = (𝑋 × {(0g‘(Scalar‘𝐶))}))))
131	130	biimpa 476	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ LMod ∧ 𝑋 ⊆ (Base‘𝐶)) ∧ 𝑋 ∈ (LIndS‘𝐶)) → ∀𝑔 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋)((𝑔 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶)) → 𝑔 = (𝑋 × {(0g‘(Scalar‘𝐶))})))
132	114, 119, 124, 131	syl21anc 837	. . . . . . . . . . 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 3575	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (((𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶)) → (𝐺‘𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))})))
134	97, 133	mpand 695	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶) → (𝐺‘𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))})))
135	134	imp 406	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶)) → (𝐺‘𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))}))
136	76, 78, 96, 135	syl21anc 837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → (𝐺‘𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))}))
137	1, 136	suppss 8134	. . . . . 6 ⊢ (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ (𝐿 supp (0g‘(Scalar‘𝐵))))
138	75, 137	ssfid 9167	. . . . 5 ⊢ (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ∈ Fin)
139		suppssdm 8117	. . . . . . . . . 10 ⊢ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ dom 𝐺
140	139, 1	fssdm 6679	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ 𝑌)
141	140	sselda 3931	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))) → 𝑤 ∈ 𝑌)
142		eleq1w 2817	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑤 → (𝑗 ∈ 𝑌 ↔ 𝑤 ∈ 𝑌))
143	142	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑤 → ((𝜑 ∧ 𝑗 ∈ 𝑌) ↔ (𝜑 ∧ 𝑤 ∈ 𝑌)))
144		fveq2 6832	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑤 → (𝐺‘𝑗) = (𝐺‘𝑤))
145	144	breq1d 5106	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑤 → ((𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺‘𝑤) finSupp (0g‘(Scalar‘𝐶))))
146	143, 145	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑗 = 𝑤 → (((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶))) ↔ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝐺‘𝑤) finSupp (0g‘(Scalar‘𝐶)))))
147	146, 97	chvarvv 1990	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝐺‘𝑤) finSupp (0g‘(Scalar‘𝐶)))
148	147	fsuppimpd 9270	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
149	141, 148	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))) → ((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
150	149	ralrimiva 3126	. . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
151		iunfi 9241	. . . . . 6 ⊢ (((𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ∈ Fin ∧ ∀𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin) → ∪ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
152	138, 150, 151	syl2anc 584	. . . . 5 ⊢ (𝜑 → ∪ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
153		xpfi 9218	. . . . 5 ⊢ (((𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ∈ Fin ∧ ∪ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin) → ((𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) × ∪ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶)))) ∈ Fin)
154	138, 152, 153	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) × ∪ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶)))) ∈ Fin)
155		fveq2 6832	. . . . . . . . . 10 ⊢ (𝑣 = 𝑗 → (𝐺‘𝑣) = (𝐺‘𝑗))
156	155	fveq1d 6834	. . . . . . . . 9 ⊢ (𝑣 = 𝑗 → ((𝐺‘𝑣)‘𝑢) = ((𝐺‘𝑗)‘𝑢))
157	156	mpteq2dv 5190	. . . . . . . 8 ⊢ (𝑣 = 𝑗 → (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)) = (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑢)))
158		fveq2 6832	. . . . . . . . 9 ⊢ (𝑢 = 𝑖 → ((𝐺‘𝑗)‘𝑢) = ((𝐺‘𝑗)‘𝑖))
159	158	cbvmptv 5200	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑢)) = (𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖))
160	157, 159	eqtrdi 2785	. . . . . . 7 ⊢ (𝑣 = 𝑗 → (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)) = (𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖)))
161	160	cbvmptv 5200	. . . . . 6 ⊢ (𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) = (𝑗 ∈ 𝑌 ↦ (𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖)))
162		fvexd 6847	. . . . . 6 ⊢ (𝜑 → (0g‘(Scalar‘𝐶)) ∈ V)
163		fvexd 6847	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → ((𝐺‘𝑗)‘𝑖) ∈ V)
164	42, 161, 46, 47, 162, 163	suppovss 32709	. . . . 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 20510	. . . . . . . 8 ⊢ (𝑉 ∈ (SubRing‘𝐸) → (0g‘𝐸) = (0g‘𝐾))
166	19, 165	syl 17	. . . . . . 7 ⊢ (𝜑 → (0g‘𝐸) = (0g‘𝐾))
167	36	fveq2d 6836	. . . . . . 7 ⊢ (𝜑 → (0g‘𝐾) = (0g‘(Scalar‘𝐶)))
168	166, 167	eqtr2d 2770	. . . . . 6 ⊢ (𝜑 → (0g‘(Scalar‘𝐶)) = (0g‘𝐸))
169	168	oveq2d 7372	. . . . 5 ⊢ (𝜑 → (𝐻 supp (0g‘(Scalar‘𝐶))) = (𝐻 supp (0g‘𝐸)))
170	1	feqmptd 6900	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝑌 ↦ (𝐺‘𝑣)))
171		eleq1w 2817	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑣 → (𝑗 ∈ 𝑌 ↔ 𝑣 ∈ 𝑌))
172	171	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑣 → ((𝜑 ∧ 𝑗 ∈ 𝑌) ↔ (𝜑 ∧ 𝑣 ∈ 𝑌)))
173		fveq2 6832	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑣 → (𝐺‘𝑗) = (𝐺‘𝑣))
174	173	feq1d 6642	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑣 → ((𝐺‘𝑗):𝑋⟶(Base‘𝐸) ↔ (𝐺‘𝑣):𝑋⟶(Base‘𝐸)))
175	172, 174	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑣 → (((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗):𝑋⟶(Base‘𝐸)) ↔ ((𝜑 ∧ 𝑣 ∈ 𝑌) → (𝐺‘𝑣):𝑋⟶(Base‘𝐸))))
176	10, 20	ressbas2 17163	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ⊆ (Base‘𝐸) → 𝑉 = (Base‘𝐾))
177	22, 176	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 = (Base‘𝐾))
178	36	fveq2d 6836	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝐶)))
179	177, 178	eqtrd 2769	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 = (Base‘(Scalar‘𝐶)))
180	179, 22	eqsstrrd 3967	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
181	180	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
182	56, 181	fssd 6677	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗):𝑋⟶(Base‘𝐸))
183	175, 182	chvarvv 1990	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑌) → (𝐺‘𝑣):𝑋⟶(Base‘𝐸))
184	183	feqmptd 6900	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑌) → (𝐺‘𝑣) = (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))
185	184	mpteq2dva 5189	. . . . . . . 8 ⊢ (𝜑 → (𝑣 ∈ 𝑌 ↦ (𝐺‘𝑣)) = (𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))))
186	170, 185	eqtr2d 2770	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) = 𝐺)
187	186	oveq1d 7371	. . . . . 6 ⊢ (𝜑 → ((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) supp (𝑋 × {(0g‘(Scalar‘𝐶))})) = (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})))
188	186	fveq1d 6834	. . . . . . . 8 ⊢ (𝜑 → ((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))‘𝑤) = (𝐺‘𝑤))
189	188	oveq1d 7371	. . . . . . 7 ⊢ (𝜑 → (((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))‘𝑤) supp (0g‘(Scalar‘𝐶))) = ((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))))
190	187, 189	iuneq12d 4974	. . . . . 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 5653	. . . . 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 3986	. . . 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 9281	. . . 4 ⊢ (((𝐻 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)) ∧ Fun 𝐻 ∧ (0g‘𝐸) ∈ (Base‘𝐸)) ∧ (((𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) × ∪ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶)))) ∈ Fin ∧ (𝐻 supp (0g‘𝐸)) ⊆ ((𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) × ∪ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶)))))) → 𝐻 finSupp (0g‘𝐸))
194	52, 66, 73, 154, 192, 193	syl32anc 1380	. . 3 ⊢ (𝜑 → 𝐻 finSupp (0g‘𝐸))
195	37	fveq2d 6836	. . . 4 ⊢ (𝜑 → (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐶)))
196	195, 168	eqtr2d 2770	. . 3 ⊢ (𝜑 → (0g‘𝐸) = (0g‘(Scalar‘𝐴)))
197	194, 196	breqtrd 5122	. 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 33700	. . 3 ⊢ (𝜑 → (𝐸 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))) = (𝐵 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))))
200	47	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑋 ∈ (LBasis‘𝐶))
201	13	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑈 ∈ (SubRing‘𝐸))
202		subrgsubg 20508	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ∈ (SubGrp‘𝐸))
203		subgsubm 19076	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (SubGrp‘𝐸) → 𝑈 ∈ (SubMnd‘𝐸))
204	201, 202, 203	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑈 ∈ (SubMnd‘𝐸))
205	113	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐶 ∈ LMod)
206	56	ffvelcdmda 7027	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶)))
207	118	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑋 ⊆ (Base‘𝐶))
208	207, 61	sseldd 3932	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐶))
209	115, 126, 127, 125	lmodvscl 20827	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ LMod ∧ ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑖 ∈ (Base‘𝐶)) → (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖) ∈ (Base‘𝐶))
210	205, 206, 208, 209	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖) ∈ (Base‘𝐶))
211	15, 20	ressbas2 17163	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ⊆ (Base‘𝐸) → 𝑈 = (Base‘𝐹))
212	88, 211	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 = (Base‘𝐹))
213	31, 34	srabase 21127	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐹) = (Base‘𝐶))
214	212, 213	eqtrd 2769	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
215	214	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑈 = (Base‘𝐶))
216	210, 215	eleqtrrd 2837	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖) ∈ 𝑈)
217	216	fmpttd 7058	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)):𝑋⟶𝑈)
218	200, 204, 217, 15	gsumsubm 18758	. . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . . . 18 ⊢ (.r‘𝐸) = (.r‘𝐸)
220	15, 219	ressmulr 17225	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (.r‘𝐸) = (.r‘𝐹))
221	13, 220	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (.r‘𝐸) = (.r‘𝐹))
222	31, 34	sravsca 21131	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (.r‘𝐹) = ( ·𝑠 ‘𝐶))
223	221, 222	eqtr2d 2770	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ( ·𝑠 ‘𝐶) = (.r‘𝐸))
224	223	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ( ·𝑠 ‘𝐶) = (.r‘𝐸))
225	224	oveqd 7373	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖) = (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖))
226	225	mpteq2dva 5189	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)) = (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)))
227	226	oveq2d 7372	. . . . . . . . . . 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 33700	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))))
229	228	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐹 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))))
230	218, 227, 229	3eqtr3d 2777	. . . . . . . . . 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 7371	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐸 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)))(.r‘𝐸)𝑗) = ((𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))(.r‘𝐸)𝑗))
232	69	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ Ring)
233	180	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
234	233, 206	sseldd 3932	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘𝐸))
235	214, 88	eqsstrrd 3967	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
236	118, 235	sstrd 3942	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐸))
237	236	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑋 ⊆ (Base‘𝐸))
238	237, 61	sseldd 3932	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐸))
239		eqid 2734	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝐵) = (Base‘𝐵)
240		eqid 2734	. . . . . . . . . . . . . . . . . 18 ⊢ (LBasis‘𝐵) = (LBasis‘𝐵)
241	239, 240	lbsss 21027	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 ∈ (LBasis‘𝐵) → 𝑌 ⊆ (Base‘𝐵))
242	46, 241	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ⊆ (Base‘𝐵))
243	86, 88	srabase 21127	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐵))
244	242, 243	sseqtrrd 3969	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ⊆ (Base‘𝐸))
245	244	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑌 ⊆ (Base‘𝐸))
246		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ 𝑌)
247	245, 246	sseldd 3932	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ (Base‘𝐸))
248	20, 219	ringass 20186	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ Ring ∧ (((𝐺‘𝑗)‘𝑖) ∈ (Base‘𝐸) ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸))) → ((((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)(.r‘𝐸)𝑗) = (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))
249	232, 234, 238, 247, 248	syl13anc 1374	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)(.r‘𝐸)𝑗) = (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))
250	249	mpteq2dva 5189	. . . . . . . . . . 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 7372	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐸 Σg (𝑖 ∈ 𝑋 ↦ ((((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)(.r‘𝐸)𝑗))) = (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))))
252	69	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐸 ∈ Ring)
253	242	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑌 ⊆ (Base‘𝐵))
254	243	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (Base‘𝐸) = (Base‘𝐵))
255	253, 254	sseqtrrd 3969	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑌 ⊆ (Base‘𝐸))
256		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ 𝑌)
257	255, 256	sseldd 3932	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ (Base‘𝐸))
258	20, 219	ringcl 20183	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ Ring ∧ ((𝐺‘𝑗)‘𝑖) ∈ (Base‘𝐸) ∧ 𝑖 ∈ (Base‘𝐸)) → (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖) ∈ (Base‘𝐸))
259	232, 234, 238, 258	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖) ∈ (Base‘𝐸))
260	168	breq2d 5108	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺‘𝑗) finSupp (0g‘𝐸)))
261	260	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺‘𝑗) finSupp (0g‘𝐸)))
262	97, 261	mpbid 232	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) finSupp (0g‘𝐸))
263	20, 252, 200, 238, 182, 262	rmfsupp2 33269	. . . . . . . . . . 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 20249	. . . . . . . . . 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 2771	. . . . . . . . 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 7371	. . . . . . . . 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 2780	. . . . . . . 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 21131	. . . . . . . . . 10 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘𝐵))
269	268	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (.r‘𝐸) = ( ·𝑠 ‘𝐵))
270	269	oveqd 7373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐿‘𝑗)(.r‘𝐸)𝑗) = ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))
271		fvexd 6847	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ V)
272		ovexd 7391	. . . . . . . . . . . . . 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 7373	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗(𝐻 ∘f (.r‘𝐸)𝐷)𝑖) = (𝑗(𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))𝑖))
278	277	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗(𝐻 ∘f (.r‘𝐸)𝐷)𝑖) = (𝑗(𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))𝑖))
279		ovexd 7391	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)) ∈ V)
280		eqid 2734	. . . . . . . . . . . . 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 7503	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ∧ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)) ∈ V) → (𝑗(𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))𝑖) = (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))
282	246, 61, 279, 281	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑗(𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))𝑖) = (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))
283	278, 282	eqtr2d 2770	. . . . . . . . . 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 5189	. . . . . . . . 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 7372	. . . . . . . 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 2777	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗) = (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (𝑗(𝐻 ∘f (.r‘𝐸)𝐷)𝑖))))
287	286	mpteq2dva 5189	. . . . . 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 7372	. . . . 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 20215	. . . . . . 7 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ CMnd)
290	69, 289	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ CMnd)
291	69	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝐸 ∈ Ring)
292	38, 180	eqsstrd 3966	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸))
293	292	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸))
294		simprl 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑙 ∈ (Base‘(Scalar‘𝐴)))
295	293, 294	sseldd 3932	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑙 ∈ (Base‘𝐸))
296		simprr 772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑘 ∈ (Base‘𝐴))
297	12, 22	srabase 21127	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐴))
298	297	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (Base‘𝐸) = (Base‘𝐴))
299	296, 298	eleqtrrd 2837	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑘 ∈ (Base‘𝐸))
300	20, 219	ringcl 20183	. . . . . . . 8 ⊢ ((𝐸 ∈ Ring ∧ 𝑙 ∈ (Base‘𝐸) ∧ 𝑘 ∈ (Base‘𝐸)) → (𝑙(.r‘𝐸)𝑘) ∈ (Base‘𝐸))
301	291, 295, 299, 300	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (𝑙(.r‘𝐸)𝑘) ∈ (Base‘𝐸))
302	20, 219	ringcl 20183	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
303	232, 238, 247, 302	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
304	297	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (Base‘𝐸) = (Base‘𝐴))
305	303, 304	eleqtrd 2836	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴))
306	305	anasss 466	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴))
307	306	ralrimivva 3177	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴))
308	274	fmpo 8010	. . . . . . . 8 ⊢ (∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴) ↔ 𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
309	307, 308	sylib 218	. . . . . . 7 ⊢ (𝜑 → 𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
310		inidm 4177	. . . . . . 7 ⊢ ((𝑌 × 𝑋) ∩ (𝑌 × 𝑋)) = (𝑌 × 𝑋)
311	301, 65, 309, 48, 48, 310	off 7638	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘f (.r‘𝐸)𝐷):(𝑌 × 𝑋)⟶(Base‘𝐸))
312	69	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → 𝐸 ∈ Ring)
313		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → 𝑢 ∈ (Base‘𝐴))
314	297	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → (Base‘𝐸) = (Base‘𝐴))
315	313, 314	eleqtrrd 2837	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → 𝑢 ∈ (Base‘𝐸))
316	20, 219, 71	ringlz 20226	. . . . . . . 8 ⊢ ((𝐸 ∈ Ring ∧ 𝑢 ∈ (Base‘𝐸)) → ((0g‘𝐸)(.r‘𝐸)𝑢) = (0g‘𝐸))
317	312, 315, 316	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → ((0g‘𝐸)(.r‘𝐸)𝑢) = (0g‘𝐸))
318	48, 73, 73, 65, 309, 194, 317	offinsupp1 32754	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘f (.r‘𝐸)𝐷) finSupp (0g‘𝐸))
319	20, 71, 290, 46, 47, 311, 318	gsumxp 19903	. . . . 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 21131	. . . . . . . 8 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘𝐴))
321	320	ofeqd 7622	. . . . . . 7 ⊢ (𝜑 → ∘f (.r‘𝐸) = ∘f ( ·𝑠 ‘𝐴))
322	321	oveqd 7373	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘f (.r‘𝐸)𝐷) = (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷))
323	322	oveq2d 7372	. . . . 5 ⊢ (𝜑 → (𝐸 Σg (𝐻 ∘f (.r‘𝐸)𝐷)) = (𝐸 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)))
324	288, 319, 323	3eqtr2rd 2776	. . . 4 ⊢ (𝜑 → (𝐸 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)) = (𝐸 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))))
325		ovexd 7391	. . . . 5 ⊢ (𝜑 → (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷) ∈ V)
326		fedgmullem1.a	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐴))
327	326	elfvexd 6868	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
328	11, 325, 67, 327, 22	gsumsra 33079	. . . 4 ⊢ (𝜑 → (𝐸 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)) = (𝐴 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)))
329	324, 328	eqtr3d 2771	. . 3 ⊢ (𝜑 → (𝐸 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))) = (𝐴 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)))
330	198, 199, 329	3eqtr2d 2775	. 2 ⊢ (𝜑 → 𝑍 = (𝐴 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)))
331	197, 330	jca 511	1 ⊢ (𝜑 → (𝐻 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑍 = (𝐴 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049  Vcvv 3438   ∖ cdif 3896   ⊆ wss 3899  {csn 4578  ∪ ciun 4944   class class class wbr 5096   ↦ cmpt 5177   × cxp 5620  Fun wfun 6484  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358   ∘_f cof 7618   supp csupp 8100   ↑_m cmap 8761  Fincfn 8881   finSupp cfsupp 9262  Basecbs 17134   ↾_s cress 17155  ._rcmulr 17176  Scalarcsca 17178   ·_𝑠 cvsca 17179  0_gc0g 17357   Σ_g cgsu 17358  SubMndcsubmnd 18705  Grpcgrp 18861  SubGrpcsubg 19048  CMndccmn 19707  Ringcrg 20166  SubRingcsubrg 20500  DivRingcdr 20660  LModclmod 20809  LSpanclspn 20920  LBasisclbs 21024  LVecclvec 21052  subringAlg csra 21121  LIndSclinds 21758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-map 8763  df-ixp 8834  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-fsupp 9263  df-sup 9343  df-oi 9413  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-9 12213  df-n0 12400  df-z 12487  df-dec 12606  df-uz 12750  df-fz 13422  df-fzo 13569  df-seq 13923  df-hash 14252  df-struct 17072  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-plusg 17188  df-mulr 17189  df-sca 17191  df-vsca 17192  df-ip 17193  df-tset 17194  df-ple 17195  df-ds 17197  df-hom 17199  df-cco 17200  df-0g 17359  df-gsum 17360  df-prds 17365  df-pws 17367  df-mre 17503  df-mrc 17504  df-acs 17506  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-mhm 18706  df-submnd 18707  df-grp 18864  df-minusg 18865  df-sbg 18866  df-mulg 18996  df-subg 19051  df-ghm 19140  df-cntz 19244  df-cmn 19709  df-abl 19710  df-mgp 20074  df-rng 20086  df-ur 20115  df-ring 20168  df-nzr 20444  df-subrg 20501  df-drng 20662  df-lmod 20811  df-lss 20881  df-lsp 20921  df-lmhm 20972  df-lbs 21025  df-lvec 21053  df-sra 21123  df-rgmod 21124  df-dsmm 21685  df-frlm 21700  df-uvc 21736  df-lindf 21759  df-linds 21760

This theorem is referenced by:  fedgmul  33737