Theorem fedgmullem1 33813

Description: Lemma for fedgmul 33815. (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 776	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋))
3		simplr 769	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ 𝑌)
4	2, 3	ffvelcdmd 7041	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝐺‘𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋))
5		elmapi 8800	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
6	4, 5	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
7	6	anasss 466	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (𝐺‘𝑗):𝑋⟶(Base‘(Scalar‘𝐶)))
8		simprr 773	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑖 ∈ 𝑋)
9	7, 8	ffvelcdmd 7041	. . . . . . . . 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 20548	. . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈))
19	18	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐸))
20		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐸) = (Base‘𝐸)
21	20	subrgss 20522	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
22	19, 21	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐸))
23	12, 22	srasca 21149	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ↾s 𝑉) = (Scalar‘𝐴))
24	10, 23	eqtrid 2784	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐴))
25	18	simprd 495	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 ⊆ 𝑈)
26		ressabs 17189	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈) → ((𝐸 ↾s 𝑈) ↾s 𝑉) = (𝐸 ↾s 𝑉))
27	13, 25, 26	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐸 ↾s 𝑈) ↾s 𝑉) = (𝐸 ↾s 𝑉))
28	15	oveq1i 7380	. . . . . . . . . . . . . 14 ⊢ (𝐹 ↾s 𝑉) = ((𝐸 ↾s 𝑈) ↾s 𝑉)
29	27, 28, 10	3eqtr4g 2797	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ↾s 𝑉) = 𝐾)
30		fedgmul.c	. . . . . . . . . . . . . . 15 ⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉)
31	30	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 = ((subringAlg ‘𝐹)‘𝑉))
32		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐹) = (Base‘𝐹)
33	32	subrgss 20522	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
34	14, 33	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐹))
35	31, 34	srasca 21149	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ↾s 𝑉) = (Scalar‘𝐶))
36	29, 35	eqtr3d 2774	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐶))
37	24, 36	eqtr3d 2774	. . . . . . . . . . 11 ⊢ (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
38	37	fveq2d 6848	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
39	38	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
40	9, 39	eleqtrrd 2840	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
41	40	ralrimivva 3181	. . . . . . 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 8024	. . . . . . 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 6859	. . . . . . 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 7708	. . . . . . . 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 8791	. . . . . 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 688	. . . 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 7040	. . . . . . . . . . . 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 6657	. . . . . . . . . . 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 585	. . . . . . . . 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 7041	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
63	62	ralrimiva 3130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ∀𝑖 ∈ 𝑋 ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
64	63	ralrimiva 3130	. . . . . 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 6676	. . . 4 ⊢ (𝜑 → Fun 𝐻)
67		fedgmul.1	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ DivRing)
68		drngring 20686	. . . . . 6 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
69	67, 68	syl 17	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Ring)
70		ringgrp 20190	. . . . 5 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Grp)
71		eqid 2737	. . . . . 6 ⊢ (0g‘𝐸) = (0g‘𝐸)
72	20, 71	grpidcl 18912	. . . . 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 9286	. . . . . 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 3915	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → 𝑗 ∈ 𝑌)
79		fedgmullem1.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿:𝑌⟶(Base‘(Scalar‘𝐵)))
80		ssidd 3959	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 supp (0g‘(Scalar‘𝐵))) ⊆ (𝐿 supp (0g‘(Scalar‘𝐵))))
81		fvexd 6859	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘(Scalar‘𝐵)) ∈ V)
82	79, 80, 46, 81	suppssr 8149	. . . . . . . . 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 592	. . . . . . . . 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 20522	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
88	13, 87	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐸))
89	86, 88	srasca 21149	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ↾s 𝑈) = (Scalar‘𝐵))
90	15, 89	eqtrid 2784	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐵))
91	90	fveq2d 6848	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝐹) = (0g‘(Scalar‘𝐵)))
92		fedgmul.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ DivRing)
93	30, 92, 14	drgext0g 33773	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝐹) = (0g‘𝐶))
94	91, 93	eqtr3d 2774	. . . . . . . . . 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 2780	. . . . . . . 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 5103	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝐺‘𝑗) → (𝑔 finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶))))
99		fveq1 6843	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (𝐺‘𝑗) → (𝑔‘𝑖) = ((𝐺‘𝑗)‘𝑖))
100	99	oveq1d 7385	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝐺‘𝑗) → ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖) = (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
101	100	mpteq2dv 5194	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝐺‘𝑗) → (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖)) = (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))
102	101	oveq2d 7386	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝐺‘𝑗) → (𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))))
103	102	eqeq1d 2739	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝐺‘𝑗) → ((𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶) ↔ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶)))
104	98, 103	anbi12d 633	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐺‘𝑗) → ((𝑔 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶)) ↔ ((𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶))))
105		eqeq1 2741	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐺‘𝑗) → (𝑔 = (𝑋 × {(0g‘(Scalar‘𝐶))}) ↔ (𝐺‘𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))})))
106	104, 105	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐺‘𝑗) → (((𝑔 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶)) → 𝑔 = (𝑋 × {(0g‘(Scalar‘𝐶))})) ↔ (((𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶)) → (𝐺‘𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))}))))
107		fedgmul.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ DivRing)
108	29, 107	eqeltrd 2837	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 ↾s 𝑉) ∈ DivRing)
109		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ↾s 𝑉) = (𝐹 ↾s 𝑉)
110	30, 109	sralvec 33768	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ DivRing ∧ (𝐹 ↾s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
111	92, 108, 14, 110	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ LVec)
112		lveclmod 21075	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ LVec → 𝐶 ∈ LMod)
113	111, 112	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ LMod)
114	113	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐶 ∈ LMod)
115		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐶) = (Base‘𝐶)
116		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (LBasis‘𝐶) = (LBasis‘𝐶)
117	115, 116	lbsss 21046	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ (LBasis‘𝐶) → 𝑋 ⊆ (Base‘𝐶))
118	47, 117	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐶))
119	118	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑋 ⊆ (Base‘𝐶))
120		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
121	115, 116, 120	islbs4 21804	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (LBasis‘𝐶) ↔ (𝑋 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑋) = (Base‘𝐶)))
122	47, 121	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 ∈ (LIndS‘𝐶) ∧ ((LSpan‘𝐶)‘𝑋) = (Base‘𝐶)))
123	122	simpld 494	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ (LIndS‘𝐶))
124	123	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑋 ∈ (LIndS‘𝐶))
125		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
126		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
127		eqid 2737	. . . . . . . . . . . . . 14 ⊢ ( ·𝑠 ‘𝐶) = ( ·𝑠 ‘𝐶)
128		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (0g‘𝐶) = (0g‘𝐶)
129		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘𝐶))
130	115, 125, 126, 127, 128, 129	islinds5 33466	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ LMod ∧ 𝑋 ⊆ (Base‘𝐶)) → (𝑋 ∈ (LIndS‘𝐶) ↔ ∀𝑔 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋)((𝑔 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶)) → 𝑔 = (𝑋 × {(0g‘(Scalar‘𝐶))}))))
131	130	biimpa 476	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ LMod ∧ 𝑋 ⊆ (Base‘𝐶)) ∧ 𝑋 ∈ (LIndS‘𝐶)) → ∀𝑔 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋)((𝑔 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶)) → 𝑔 = (𝑋 × {(0g‘(Scalar‘𝐶))})))
132	114, 119, 124, 131	syl21anc 838	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ∀𝑔 ∈ ((Base‘(Scalar‘𝐶)) ↑m 𝑋)((𝑔 finSupp (0g‘(Scalar‘𝐶)) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶)) → 𝑔 = (𝑋 × {(0g‘(Scalar‘𝐶))})))
133	106, 132, 55	rspcdva 3579	. . . . . . . . . 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 696	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶) → (𝐺‘𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))})))
135	134	imp 406	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (0g‘𝐶)) → (𝐺‘𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))}))
136	76, 78, 96, 135	syl21anc 838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → (𝐺‘𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))}))
137	1, 136	suppss 8148	. . . . . 6 ⊢ (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ (𝐿 supp (0g‘(Scalar‘𝐵))))
138	75, 137	ssfid 9183	. . . . 5 ⊢ (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ∈ Fin)
139		suppssdm 8131	. . . . . . . . . 10 ⊢ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ dom 𝐺
140	139, 1	fssdm 6691	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ 𝑌)
141	140	sselda 3935	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))) → 𝑤 ∈ 𝑌)
142		eleq1w 2820	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑤 → (𝑗 ∈ 𝑌 ↔ 𝑤 ∈ 𝑌))
143	142	anbi2d 631	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑤 → ((𝜑 ∧ 𝑗 ∈ 𝑌) ↔ (𝜑 ∧ 𝑤 ∈ 𝑌)))
144		fveq2 6844	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑤 → (𝐺‘𝑗) = (𝐺‘𝑤))
145	144	breq1d 5110	. . . . . . . . . . 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 1991	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝐺‘𝑤) finSupp (0g‘(Scalar‘𝐶)))
148	147	fsuppimpd 9286	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
149	141, 148	syldan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))) → ((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
150	149	ralrimiva 3130	. . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
151		iunfi 9257	. . . . . 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 585	. . . . 5 ⊢ (𝜑 → ∪ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
153		xpfi 9234	. . . . 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 585	. . . 4 ⊢ (𝜑 → ((𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) × ∪ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶)))) ∈ Fin)
155		fveq2 6844	. . . . . . . . . 10 ⊢ (𝑣 = 𝑗 → (𝐺‘𝑣) = (𝐺‘𝑗))
156	155	fveq1d 6846	. . . . . . . . 9 ⊢ (𝑣 = 𝑗 → ((𝐺‘𝑣)‘𝑢) = ((𝐺‘𝑗)‘𝑢))
157	156	mpteq2dv 5194	. . . . . . . 8 ⊢ (𝑣 = 𝑗 → (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)) = (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑢)))
158		fveq2 6844	. . . . . . . . 9 ⊢ (𝑢 = 𝑖 → ((𝐺‘𝑗)‘𝑢) = ((𝐺‘𝑗)‘𝑖))
159	158	cbvmptv 5204	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑢)) = (𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖))
160	157, 159	eqtrdi 2788	. . . . . . 7 ⊢ (𝑣 = 𝑗 → (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)) = (𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖)))
161	160	cbvmptv 5204	. . . . . 6 ⊢ (𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) = (𝑗 ∈ 𝑌 ↦ (𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖)))
162		fvexd 6859	. . . . . 6 ⊢ (𝜑 → (0g‘(Scalar‘𝐶)) ∈ V)
163		fvexd 6859	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → ((𝐺‘𝑗)‘𝑖) ∈ V)
164	42, 161, 46, 47, 162, 163	suppovss 32777	. . . . 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 20529	. . . . . . . 8 ⊢ (𝑉 ∈ (SubRing‘𝐸) → (0g‘𝐸) = (0g‘𝐾))
166	19, 165	syl 17	. . . . . . 7 ⊢ (𝜑 → (0g‘𝐸) = (0g‘𝐾))
167	36	fveq2d 6848	. . . . . . 7 ⊢ (𝜑 → (0g‘𝐾) = (0g‘(Scalar‘𝐶)))
168	166, 167	eqtr2d 2773	. . . . . 6 ⊢ (𝜑 → (0g‘(Scalar‘𝐶)) = (0g‘𝐸))
169	168	oveq2d 7386	. . . . 5 ⊢ (𝜑 → (𝐻 supp (0g‘(Scalar‘𝐶))) = (𝐻 supp (0g‘𝐸)))
170	1	feqmptd 6912	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝑌 ↦ (𝐺‘𝑣)))
171		eleq1w 2820	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑣 → (𝑗 ∈ 𝑌 ↔ 𝑣 ∈ 𝑌))
172	171	anbi2d 631	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑣 → ((𝜑 ∧ 𝑗 ∈ 𝑌) ↔ (𝜑 ∧ 𝑣 ∈ 𝑌)))
173		fveq2 6844	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑣 → (𝐺‘𝑗) = (𝐺‘𝑣))
174	173	feq1d 6654	. . . . . . . . . . . 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 17179	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ⊆ (Base‘𝐸) → 𝑉 = (Base‘𝐾))
177	22, 176	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 = (Base‘𝐾))
178	36	fveq2d 6848	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝐶)))
179	177, 178	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 = (Base‘(Scalar‘𝐶)))
180	179, 22	eqsstrrd 3971	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
181	180	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
182	56, 181	fssd 6689	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗):𝑋⟶(Base‘𝐸))
183	175, 182	chvarvv 1991	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑌) → (𝐺‘𝑣):𝑋⟶(Base‘𝐸))
184	183	feqmptd 6912	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑌) → (𝐺‘𝑣) = (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))
185	184	mpteq2dva 5193	. . . . . . . 8 ⊢ (𝜑 → (𝑣 ∈ 𝑌 ↦ (𝐺‘𝑣)) = (𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))))
186	170, 185	eqtr2d 2773	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) = 𝐺)
187	186	oveq1d 7385	. . . . . 6 ⊢ (𝜑 → ((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) supp (𝑋 × {(0g‘(Scalar‘𝐶))})) = (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})))
188	186	fveq1d 6846	. . . . . . . 8 ⊢ (𝜑 → ((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))‘𝑤) = (𝐺‘𝑤))
189	188	oveq1d 7385	. . . . . . 7 ⊢ (𝜑 → (((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))‘𝑤) supp (0g‘(Scalar‘𝐶))) = ((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))))
190	187, 189	iuneq12d 4978	. . . . . 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 5665	. . . . 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 3990	. . . 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 9297	. . . 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 1381	. . 3 ⊢ (𝜑 → 𝐻 finSupp (0g‘𝐸))
195	37	fveq2d 6848	. . . 4 ⊢ (𝜑 → (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐶)))
196	195, 168	eqtr2d 2773	. . 3 ⊢ (𝜑 → (0g‘𝐸) = (0g‘(Scalar‘𝐴)))
197	194, 196	breqtrd 5126	. 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 33778	. . 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 20527	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ∈ (SubGrp‘𝐸))
203		subgsubm 19095	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (SubGrp‘𝐸) → 𝑈 ∈ (SubMnd‘𝐸))
204	201, 202, 203	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑈 ∈ (SubMnd‘𝐸))
205	113	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐶 ∈ LMod)
206	56	ffvelcdmda 7040	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶)))
207	118	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑋 ⊆ (Base‘𝐶))
208	207, 61	sseldd 3936	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐶))
209	115, 126, 127, 125	lmodvscl 20846	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖) ∈ (Base‘𝐶))
211	15, 20	ressbas2 17179	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ⊆ (Base‘𝐸) → 𝑈 = (Base‘𝐹))
212	88, 211	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 = (Base‘𝐹))
213	31, 34	srabase 21146	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐹) = (Base‘𝐶))
214	212, 213	eqtrd 2772	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
215	214	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑈 = (Base‘𝐶))
216	210, 215	eleqtrrd 2840	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖) ∈ 𝑈)
217	216	fmpttd 7071	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)):𝑋⟶𝑈)
218	200, 204, 217, 15	gsumsubm 18774	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))) = (𝐹 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))))
219		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (.r‘𝐸) = (.r‘𝐸)
220	15, 219	ressmulr 17241	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (.r‘𝐸) = (.r‘𝐹))
221	13, 220	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (.r‘𝐸) = (.r‘𝐹))
222	31, 34	sravsca 21150	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (.r‘𝐹) = ( ·𝑠 ‘𝐶))
223	221, 222	eqtr2d 2773	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ( ·𝑠 ‘𝐶) = (.r‘𝐸))
224	223	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ( ·𝑠 ‘𝐶) = (.r‘𝐸))
225	224	oveqd 7387	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖) = (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖))
226	225	mpteq2dva 5193	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)) = (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)))
227	226	oveq2d 7386	. . . . . . . . . . 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 33778	. . . . . . . . . . . 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 2780	. . . . . . . . . 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 7385	. . . . . . . . 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 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ Ring)
233	180	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
234	233, 206	sseldd 3936	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘𝐸))
235	214, 88	eqsstrrd 3971	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
236	118, 235	sstrd 3946	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐸))
237	236	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑋 ⊆ (Base‘𝐸))
238	237, 61	sseldd 3936	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐸))
239		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝐵) = (Base‘𝐵)
240		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (LBasis‘𝐵) = (LBasis‘𝐵)
241	239, 240	lbsss 21046	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 ∈ (LBasis‘𝐵) → 𝑌 ⊆ (Base‘𝐵))
242	46, 241	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ⊆ (Base‘𝐵))
243	86, 88	srabase 21146	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐵))
244	242, 243	sseqtrrd 3973	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ⊆ (Base‘𝐸))
245	244	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑌 ⊆ (Base‘𝐸))
246		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ 𝑌)
247	245, 246	sseldd 3936	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ (Base‘𝐸))
248	20, 219	ringass 20205	. . . . . . . . . . . . 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 1375	. . . . . . . . . . . 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 5193	. . . . . . . . . . 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 7386	. . . . . . . . . 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 3973	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑌 ⊆ (Base‘𝐸))
256		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ 𝑌)
257	255, 256	sseldd 3936	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ (Base‘𝐸))
258	20, 219	ringcl 20202	. . . . . . . . . . . 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 1374	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖) ∈ (Base‘𝐸))
260	168	breq2d 5112	. . . . . . . . . . . . . 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 33338	. . . . . . . . . . 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 20268	. . . . . . . . . 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 2774	. . . . . . . . 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 7385	. . . . . . . . 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 2783	. . . . . . . 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 21150	. . . . . . . . . 10 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘𝐵))
269	268	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (.r‘𝐸) = ( ·𝑠 ‘𝐵))
270	269	oveqd 7387	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐿‘𝑗)(.r‘𝐸)𝑗) = ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))
271		fvexd 6859	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ V)
272		ovexd 7405	. . . . . . . . . . . . . 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 8042	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐻 ∘f (.r‘𝐸)𝐷) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗))))
277	276	oveqd 7387	. . . . . . . . . . . 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 727	. . . . . . . . . . 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 7405	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)) ∈ V)
280		eqid 2737	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗))) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))
281	280	ovmpt4g 7517	. . . . . . . . . . . 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 1374	. . . . . . . . . . 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 2773	. . . . . . . . . 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 5193	. . . . . . . . 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 7386	. . . . . . . 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 2780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗) = (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (𝑗(𝐻 ∘f (.r‘𝐸)𝐷)𝑖))))
287	286	mpteq2dva 5193	. . . . . 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 7386	. . . . 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 20234	. . . . . . 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 3970	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸))
293	292	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸))
294		simprl 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑙 ∈ (Base‘(Scalar‘𝐴)))
295	293, 294	sseldd 3936	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑙 ∈ (Base‘𝐸))
296		simprr 773	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑘 ∈ (Base‘𝐴))
297	12, 22	srabase 21146	. . . . . . . . . 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 2840	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑘 ∈ (Base‘𝐸))
300	20, 219	ringcl 20202	. . . . . . . 8 ⊢ ((𝐸 ∈ Ring ∧ 𝑙 ∈ (Base‘𝐸) ∧ 𝑘 ∈ (Base‘𝐸)) → (𝑙(.r‘𝐸)𝑘) ∈ (Base‘𝐸))
301	291, 295, 299, 300	syl3anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (𝑙(.r‘𝐸)𝑘) ∈ (Base‘𝐸))
302	20, 219	ringcl 20202	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
303	232, 238, 247, 302	syl3anc 1374	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
304	297	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (Base‘𝐸) = (Base‘𝐴))
305	303, 304	eleqtrd 2839	. . . . . . . . . 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 3181	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴))
308	274	fmpo 8024	. . . . . . . 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 4181	. . . . . . 7 ⊢ ((𝑌 × 𝑋) ∩ (𝑌 × 𝑋)) = (𝑌 × 𝑋)
311	301, 65, 309, 48, 48, 310	off 7652	. . . . . 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 2840	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → 𝑢 ∈ (Base‘𝐸))
316	20, 219, 71	ringlz 20245	. . . . . . . 8 ⊢ ((𝐸 ∈ Ring ∧ 𝑢 ∈ (Base‘𝐸)) → ((0g‘𝐸)(.r‘𝐸)𝑢) = (0g‘𝐸))
317	312, 315, 316	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → ((0g‘𝐸)(.r‘𝐸)𝑢) = (0g‘𝐸))
318	48, 73, 73, 65, 309, 194, 317	offinsupp1 32822	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘f (.r‘𝐸)𝐷) finSupp (0g‘𝐸))
319	20, 71, 290, 46, 47, 311, 318	gsumxp 19922	. . . . 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 21150	. . . . . . . 8 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘𝐴))
321	320	ofeqd 7636	. . . . . . 7 ⊢ (𝜑 → ∘f (.r‘𝐸) = ∘f ( ·𝑠 ‘𝐴))
322	321	oveqd 7387	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘f (.r‘𝐸)𝐷) = (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷))
323	322	oveq2d 7386	. . . . 5 ⊢ (𝜑 → (𝐸 Σg (𝐻 ∘f (.r‘𝐸)𝐷)) = (𝐸 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)))
324	288, 319, 323	3eqtr2rd 2779	. . . 4 ⊢ (𝜑 → (𝐸 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)) = (𝐸 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))))
325		ovexd 7405	. . . . 5 ⊢ (𝜑 → (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷) ∈ V)
326		fedgmullem1.a	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐴))
327	326	elfvexd 6880	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
328	11, 325, 67, 327, 22	gsumsra 33147	. . . 4 ⊢ (𝜑 → (𝐸 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)) = (𝐴 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)))
329	324, 328	eqtr3d 2774	. . 3 ⊢ (𝜑 → (𝐸 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))) = (𝐴 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)))
330	198, 199, 329	3eqtr2d 2778	. 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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ∖ cdif 3900   ⊆ wss 3903  {csn 4582  ∪ ciun 4948   class class class wbr 5100   ↦ cmpt 5181   × cxp 5632  Fun wfun 6496  ⟶wf 6498  ‘cfv 6502  (class class class)co 7370   ∈ cmpo 7372   ∘_f cof 7632   supp csupp 8114   ↑_m cmap 8777  Fincfn 8897   finSupp cfsupp 9278  Basecbs 17150   ↾_s cress 17171  ._rcmulr 17192  Scalarcsca 17194   ·_𝑠 cvsca 17195  0_gc0g 17373   Σ_g cgsu 17374  SubMndcsubmnd 18721  Grpcgrp 18880  SubGrpcsubg 19067  CMndccmn 19726  Ringcrg 20185  SubRingcsubrg 20519  DivRingcdr 20679  LModclmod 20828  LSpanclspn 20939  LBasisclbs 21043  LVecclvec 21071  subringAlg csra 21140  LIndSclinds 21777

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-of 7634  df-om 7821  df-1st 7945  df-2nd 7946  df-supp 8115  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-2o 8410  df-er 8647  df-map 8779  df-ixp 8850  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-fsupp 9279  df-sup 9359  df-oi 9429  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-3 12223  df-4 12224  df-5 12225  df-6 12226  df-7 12227  df-8 12228  df-9 12229  df-n0 12416  df-z 12503  df-dec 12622  df-uz 12766  df-fz 13438  df-fzo 13585  df-seq 13939  df-hash 14268  df-struct 17088  df-sets 17105  df-slot 17123  df-ndx 17135  df-base 17151  df-ress 17172  df-plusg 17204  df-mulr 17205  df-sca 17207  df-vsca 17208  df-ip 17209  df-tset 17210  df-ple 17211  df-ds 17213  df-hom 17215  df-cco 17216  df-0g 17375  df-gsum 17376  df-prds 17381  df-pws 17383  df-mre 17519  df-mrc 17520  df-acs 17522  df-mgm 18579  df-sgrp 18658  df-mnd 18674  df-mhm 18722  df-submnd 18723  df-grp 18883  df-minusg 18884  df-sbg 18885  df-mulg 19015  df-subg 19070  df-ghm 19159  df-cntz 19263  df-cmn 19728  df-abl 19729  df-mgp 20093  df-rng 20105  df-ur 20134  df-ring 20187  df-nzr 20463  df-subrg 20520  df-drng 20681  df-lmod 20830  df-lss 20900  df-lsp 20940  df-lmhm 20991  df-lbs 21044  df-lvec 21072  df-sra 21142  df-rgmod 21143  df-dsmm 21704  df-frlm 21719  df-uvc 21755  df-lindf 21778  df-linds 21779

This theorem is referenced by:  fedgmul  33815