Theorem fedgmullem1 30621

Description: Lemma for fedgmul 30623 (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 772	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑𝑚 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑𝑚 𝑋))
3		simplr 765	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑𝑚 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ 𝑌)
4	2, 3	ffvelrnd 6720	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑𝑚 𝑋)) ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝐺‘𝑗) ∈ ((Base‘(Scalar‘𝐶)) ↑𝑚 𝑋))
5		elmapi 8281	. . . . . . . . . . . 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 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑𝑚 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → 𝑖 ∈ 𝑋)
9	7, 8	ffvelrnd 6720	. . . . . . . . 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 19251	. . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉 ⊆ 𝑈))
19	18	simpld 495	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐸))
20		eqid 2794	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐸) = (Base‘𝐸)
21	20	subrgss 19226	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
22	19, 21	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐸))
23	12, 22	srasca 19643	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ↾s 𝑉) = (Scalar‘𝐴))
24	10, 23	syl5eq 2842	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐴))
25	18	simprd 496	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 ⊆ 𝑈)
26		ressabs 16392	. . . . . . . . . . . . . . 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 7029	. . . . . . . . . . . . . 14 ⊢ (𝐹 ↾s 𝑉) = ((𝐸 ↾s 𝑈) ↾s 𝑉)
29	27, 28, 10	3eqtr4g 2855	. . . . . . . . . . . . 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 2794	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐹) = (Base‘𝐹)
33	32	subrgss 19226	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
34	14, 33	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ⊆ (Base‘𝐹))
35	31, 34	srasca 19643	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ↾s 𝑉) = (Scalar‘𝐶))
36	29, 35	eqtr3d 2832	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 = (Scalar‘𝐶))
37	24, 36	eqtr3d 2832	. . . . . . . . . . 11 ⊢ (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
38	37	fveq2d 6545	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
39	38	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑𝑚 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
40	9, 39	eleqtrrd 2885	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑𝑚 𝑋)) ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
41	40	ralrimivva 3157	. . . . . . 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 7625	. . . . . . 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 6556	. . . . . . 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 7334	. . . . . . . 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 8273	. . . . . 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 683	. . . 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	ffvelrnda 6719	. . . . . . . . . . . 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 6372	. . . . . . . . . . 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 584	. . . . . . . . 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	ffvelrnd 6720	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
63	62	ralrimiva 3148	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ∀𝑖 ∈ 𝑋 ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐴)))
64	63	ralrimiva 3148	. . . . . 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 6389	. . . 4 ⊢ (𝜑 → Fun 𝐻)
67		fedgmul.1	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ DivRing)
68		drngring 19199	. . . . . 6 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
69	67, 68	syl 17	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Ring)
70		ringgrp 18992	. . . . 5 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Grp)
71		eqid 2794	. . . . . 6 ⊢ (0g‘𝐸) = (0g‘𝐸)
72	20, 71	grpidcl 17889	. . . . 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 8689	. . . . . 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 3873	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → 𝑗 ∈ 𝑌)
79		fedgmullem1.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿:𝑌⟶(Base‘(Scalar‘𝐵)))
80		ssidd 3913	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 supp (0g‘(Scalar‘𝐵))) ⊆ (𝐿 supp (0g‘(Scalar‘𝐵))))
81		fvexd 6556	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘(Scalar‘𝐵)) ∈ V)
82	79, 80, 46, 81	suppssr 7715	. . . . . . . . 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 19226	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
88	13, 87	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐸))
89	86, 88	srasca 19643	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ↾s 𝑈) = (Scalar‘𝐵))
90	15, 89	syl5eq 2842	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐵))
91	90	fveq2d 6545	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝐹) = (0g‘(Scalar‘𝐵)))
92		fedgmul.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ DivRing)
93	30, 92, 14	drgext0g 30588	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝐹) = (0g‘𝐶))
94	91, 93	eqtr3d 2832	. . . . . . . . . 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 2838	. . . . . . . 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 4967	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝐺‘𝑗) → (𝑔 finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶))))
99		fveq1 6540	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (𝐺‘𝑗) → (𝑔‘𝑖) = ((𝐺‘𝑗)‘𝑖))
100	99	oveq1d 7034	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝐺‘𝑗) → ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖) = (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))
101	100	mpteq2dv 5059	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝐺‘𝑗) → (𝑖 ∈ 𝑋 ↦ ((𝑔‘𝑖)( ·𝑠 ‘𝐶)𝑖)) = (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))
102	101	oveq2d 7035	. . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . 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 630	. . . . . . . . . . . 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 2798	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐺‘𝑗) → (𝑔 = (𝑋 × {(0g‘(Scalar‘𝐶))}) ↔ (𝐺‘𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))})))
106	104, 105	imbi12d 346	. . . . . . . . . . 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 2882	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 ↾s 𝑉) ∈ DivRing)
109		eqid 2794	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ↾s 𝑉) = (𝐹 ↾s 𝑉)
110	30, 109	sralvec 30586	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ DivRing ∧ (𝐹 ↾s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
111	92, 108, 14, 110	syl3anc 1364	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ LVec)
112		lveclmod 19568	. . . . . . . . . . . . . 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 2794	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐶) = (Base‘𝐶)
116		eqid 2794	. . . . . . . . . . . . . . 15 ⊢ (LBasis‘𝐶) = (LBasis‘𝐶)
117	115, 116	lbsss 19539	. . . . . . . . . . . . . 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 2794	. . . . . . . . . . . . . . . 16 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
121	115, 116, 120	islbs4 20658	. . . . . . . . . . . . . . 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 2794	. . . . . . . . . . . . . 14 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
126		eqid 2794	. . . . . . . . . . . . . 14 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
127		eqid 2794	. . . . . . . . . . . . . 14 ⊢ ( ·𝑠 ‘𝐶) = ( ·𝑠 ‘𝐶)
128		eqid 2794	. . . . . . . . . . . . . 14 ⊢ (0g‘𝐶) = (0g‘𝐶)
129		eqid 2794	. . . . . . . . . . . . . 14 ⊢ (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘𝐶))
130	115, 125, 126, 127, 128, 129	islinds5 30572	. . . . . . . . . . . . 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 834	. . . . . . . . . . 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 3563	. . . . . . . . . 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 691	. . . . . . . . 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 834	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑌 ∖ (𝐿 supp (0g‘(Scalar‘𝐵))))) → (𝐺‘𝑗) = (𝑋 × {(0g‘(Scalar‘𝐶))}))
137	1, 136	suppss 7714	. . . . . 6 ⊢ (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ (𝐿 supp (0g‘(Scalar‘𝐵))))
138	75, 137	ssfid 8590	. . . . 5 ⊢ (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ∈ Fin)
139		suppssdm 7697	. . . . . . . . . 10 ⊢ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ dom 𝐺
140	139, 1	fssdm 6401	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})) ⊆ 𝑌)
141	140	sselda 3891	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))) → 𝑤 ∈ 𝑌)
142		eleq1w 2864	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑤 → (𝑗 ∈ 𝑌 ↔ 𝑤 ∈ 𝑌))
143	142	anbi2d 628	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑤 → ((𝜑 ∧ 𝑗 ∈ 𝑌) ↔ (𝜑 ∧ 𝑤 ∈ 𝑌)))
144		fveq2 6541	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑤 → (𝐺‘𝑗) = (𝐺‘𝑤))
145	144	breq1d 4974	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑤 → ((𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺‘𝑤) finSupp (0g‘(Scalar‘𝐶))))
146	143, 145	imbi12d 346	. . . . . . . . . 10 ⊢ (𝑗 = 𝑤 → (((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶))) ↔ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝐺‘𝑤) finSupp (0g‘(Scalar‘𝐶)))))
147	146, 97	chvarv 2369	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝐺‘𝑤) finSupp (0g‘(Scalar‘𝐶)))
148	147	fsuppimpd 8689	. . . . . . . 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 3148	. . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))}))((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))) ∈ Fin)
151		iunfi 8661	. . . . . 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 8638	. . . . 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 6541	. . . . . . . . . 10 ⊢ (𝑣 = 𝑗 → (𝐺‘𝑣) = (𝐺‘𝑗))
156	155	fveq1d 6543	. . . . . . . . 9 ⊢ (𝑣 = 𝑗 → ((𝐺‘𝑣)‘𝑢) = ((𝐺‘𝑗)‘𝑢))
157	156	mpteq2dv 5059	. . . . . . . 8 ⊢ (𝑣 = 𝑗 → (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)) = (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑢)))
158		fveq2 6541	. . . . . . . . 9 ⊢ (𝑢 = 𝑖 → ((𝐺‘𝑗)‘𝑢) = ((𝐺‘𝑗)‘𝑖))
159	158	cbvmptv 5064	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑢)) = (𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖))
160	157, 159	syl6eq 2846	. . . . . . 7 ⊢ (𝑣 = 𝑗 → (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)) = (𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖)))
161	160	cbvmptv 5064	. . . . . 6 ⊢ (𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) = (𝑗 ∈ 𝑌 ↦ (𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖)))
162		fvexd 6556	. . . . . 6 ⊢ (𝜑 → (0g‘(Scalar‘𝐶)) ∈ V)
163		fvexd 6556	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → ((𝐺‘𝑗)‘𝑖) ∈ V)
164	42, 161, 46, 47, 162, 163	suppovss 30108	. . . . 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 19232	. . . . . . . 8 ⊢ (𝑉 ∈ (SubRing‘𝐸) → (0g‘𝐸) = (0g‘𝐾))
166	19, 165	syl 17	. . . . . . 7 ⊢ (𝜑 → (0g‘𝐸) = (0g‘𝐾))
167	36	fveq2d 6545	. . . . . . 7 ⊢ (𝜑 → (0g‘𝐾) = (0g‘(Scalar‘𝐶)))
168	166, 167	eqtr2d 2831	. . . . . 6 ⊢ (𝜑 → (0g‘(Scalar‘𝐶)) = (0g‘𝐸))
169	168	oveq2d 7035	. . . . 5 ⊢ (𝜑 → (𝐻 supp (0g‘(Scalar‘𝐶))) = (𝐻 supp (0g‘𝐸)))
170	1	feqmptd 6604	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝑌 ↦ (𝐺‘𝑣)))
171		eleq1w 2864	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑣 → (𝑗 ∈ 𝑌 ↔ 𝑣 ∈ 𝑌))
172	171	anbi2d 628	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑣 → ((𝜑 ∧ 𝑗 ∈ 𝑌) ↔ (𝜑 ∧ 𝑣 ∈ 𝑌)))
173		fveq2 6541	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑣 → (𝐺‘𝑗) = (𝐺‘𝑣))
174	173	feq1d 6370	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑣 → ((𝐺‘𝑗):𝑋⟶(Base‘𝐸) ↔ (𝐺‘𝑣):𝑋⟶(Base‘𝐸)))
175	172, 174	imbi12d 346	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑣 → (((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗):𝑋⟶(Base‘𝐸)) ↔ ((𝜑 ∧ 𝑣 ∈ 𝑌) → (𝐺‘𝑣):𝑋⟶(Base‘𝐸))))
176	10, 20	ressbas2 16384	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ⊆ (Base‘𝐸) → 𝑉 = (Base‘𝐾))
177	22, 176	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 = (Base‘𝐾))
178	36	fveq2d 6545	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝐶)))
179	177, 178	eqtrd 2830	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 = (Base‘(Scalar‘𝐶)))
180	179, 22	eqsstrrd 3929	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
181	180	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
182	56, 181	fssd 6399	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗):𝑋⟶(Base‘𝐸))
183	175, 182	chvarv 2369	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑌) → (𝐺‘𝑣):𝑋⟶(Base‘𝐸))
184	183	feqmptd 6604	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑌) → (𝐺‘𝑣) = (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))
185	184	mpteq2dva 5058	. . . . . . . 8 ⊢ (𝜑 → (𝑣 ∈ 𝑌 ↦ (𝐺‘𝑣)) = (𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))))
186	170, 185	eqtr2d 2831	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) = 𝐺)
187	186	oveq1d 7034	. . . . . 6 ⊢ (𝜑 → ((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢))) supp (𝑋 × {(0g‘(Scalar‘𝐶))})) = (𝐺 supp (𝑋 × {(0g‘(Scalar‘𝐶))})))
188	186	fveq1d 6543	. . . . . . . 8 ⊢ (𝜑 → ((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))‘𝑤) = (𝐺‘𝑤))
189	188	oveq1d 7034	. . . . . . 7 ⊢ (𝜑 → (((𝑣 ∈ 𝑌 ↦ (𝑢 ∈ 𝑋 ↦ ((𝐺‘𝑣)‘𝑢)))‘𝑤) supp (0g‘(Scalar‘𝐶))) = ((𝐺‘𝑤) supp (0g‘(Scalar‘𝐶))))
190	187, 189	iuneq12d 4854	. . . . . 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 5477	. . . . 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 3936	. . . 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 8697	. . . 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 1371	. . 3 ⊢ (𝜑 → 𝐻 finSupp (0g‘𝐸))
195	37	fveq2d 6545	. . . 4 ⊢ (𝜑 → (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐶)))
196	195, 168	eqtr2d 2831	. . 3 ⊢ (𝜑 → (0g‘𝐸) = (0g‘(Scalar‘𝐴)))
197	194, 196	breqtrd 4990	. 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 30593	. . 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 19231	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ∈ (SubGrp‘𝐸))
203		subgsubm 18055	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (SubGrp‘𝐸) → 𝑈 ∈ (SubMnd‘𝐸))
204	201, 202, 203	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑈 ∈ (SubMnd‘𝐸))
205	113	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐶 ∈ LMod)
206	56	ffvelrnda 6719	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘(Scalar‘𝐶)))
207	118	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑋 ⊆ (Base‘𝐶))
208	207, 61	sseldd 3892	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐶))
209	115, 126, 127, 125	lmodvscl 19341	. . . . . . . . . . . . . . 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 1364	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖) ∈ (Base‘𝐶))
211	15, 20	ressbas2 16384	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ⊆ (Base‘𝐸) → 𝑈 = (Base‘𝐹))
212	88, 211	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 = (Base‘𝐹))
213	31, 34	srabase 19640	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐹) = (Base‘𝐶))
214	212, 213	eqtrd 2830	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
215	214	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑈 = (Base‘𝐶))
216	210, 215	eleqtrrd 2885	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖) ∈ 𝑈)
217	216	fmpttd 6745	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)):𝑋⟶𝑈)
218	200, 204, 217, 15	gsumsubm 17812	. . . . . . . . . . 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 2794	. . . . . . . . . . . . . . . . . 18 ⊢ (.r‘𝐸) = (.r‘𝐸)
220	15, 219	ressmulr 16454	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (.r‘𝐸) = (.r‘𝐹))
221	13, 220	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (.r‘𝐸) = (.r‘𝐹))
222	31, 34	sravsca 19644	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (.r‘𝐹) = ( ·𝑠 ‘𝐶))
223	221, 222	eqtr2d 2831	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ( ·𝑠 ‘𝐶) = (.r‘𝐸))
224	223	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ( ·𝑠 ‘𝐶) = (.r‘𝐸))
225	224	oveqd 7036	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖) = (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖))
226	225	mpteq2dva 5058	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)) = (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)))
227	226	oveq2d 7035	. . . . . . . . . . 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 30593	. . . . . . . . . . . 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 2838	. . . . . . . . . 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 7034	. . . . . . . . 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 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ Ring)
233	180	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
234	233, 206	sseldd 3892	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ (Base‘𝐸))
235	214, 88	eqsstrrd 3929	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
236	118, 235	sstrd 3901	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝐸))
237	236	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑋 ⊆ (Base‘𝐸))
238	237, 61	sseldd 3892	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ (Base‘𝐸))
239		eqid 2794	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝐵) = (Base‘𝐵)
240		eqid 2794	. . . . . . . . . . . . . . . . . 18 ⊢ (LBasis‘𝐵) = (LBasis‘𝐵)
241	239, 240	lbsss 19539	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 ∈ (LBasis‘𝐵) → 𝑌 ⊆ (Base‘𝐵))
242	46, 241	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ⊆ (Base‘𝐵))
243	86, 88	srabase 19640	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐵))
244	242, 243	sseqtr4d 3931	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ⊆ (Base‘𝐸))
245	244	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑌 ⊆ (Base‘𝐸))
246		simplr 765	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ 𝑌)
247	245, 246	sseldd 3892	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ (Base‘𝐸))
248	20, 219	ringass 19004	. . . . . . . . . . . . 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 1365	. . . . . . . . . . . 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 5058	. . . . . . . . . . 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 7035	. . . . . . . . . 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		eqid 2794	. . . . . . . . . . 11 ⊢ (+g‘𝐸) = (+g‘𝐸)
253	69	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐸 ∈ Ring)
254	242	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑌 ⊆ (Base‘𝐵))
255	243	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (Base‘𝐸) = (Base‘𝐵))
256	254, 255	sseqtr4d 3931	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑌 ⊆ (Base‘𝐸))
257		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ 𝑌)
258	256, 257	sseldd 3892	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ (Base‘𝐸))
259	20, 219	ringcl 19001	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ Ring ∧ ((𝐺‘𝑗)‘𝑖) ∈ (Base‘𝐸) ∧ 𝑖 ∈ (Base‘𝐸)) → (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖) ∈ (Base‘𝐸))
260	232, 234, 238, 259	syl3anc 1364	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖) ∈ (Base‘𝐸))
261	168	breq2d 4976	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺‘𝑗) finSupp (0g‘𝐸)))
262	261	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶)) ↔ (𝐺‘𝑗) finSupp (0g‘𝐸)))
263	97, 262	mpbid 233	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) finSupp (0g‘𝐸))
264	20, 253, 200, 238, 182, 263	rmfsupp2 30512	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)𝑖)) finSupp (0g‘𝐸))
265	20, 71, 252, 219, 253, 200, 258, 260, 264	gsummulc1 19046	. . . . . . . . . 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))
266	251, 265	eqtr3d 2832	. . . . . . . . 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))
267	83	oveq1d 7034	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐿‘𝑗)(.r‘𝐸)𝑗) = ((𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))(.r‘𝐸)𝑗))
268	231, 266, 267	3eqtr4rd 2841	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐿‘𝑗)(.r‘𝐸)𝑗) = (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))))
269	86, 88	sravsca 19644	. . . . . . . . . 10 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘𝐵))
270	269	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (.r‘𝐸) = ( ·𝑠 ‘𝐵))
271	270	oveqd 7036	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐿‘𝑗)(.r‘𝐸)𝑗) = ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))
272		fvexd 6556	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → ((𝐺‘𝑗)‘𝑖) ∈ V)
273		ovexd 7053	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋) → (𝑖(.r‘𝐸)𝑗) ∈ V)
274	42	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐻 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖)))
275		fedgmullem.d	. . . . . . . . . . . . . . 15 ⊢ 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(.r‘𝐸)𝑗))
276	275	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(.r‘𝐸)𝑗)))
277	46, 47, 272, 273, 274, 276	offval22 7642	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐻 ∘𝑓 (.r‘𝐸)𝐷) = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗))))
278	277	oveqd 7036	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗(𝐻 ∘𝑓 (.r‘𝐸)𝐷)𝑖) = (𝑗(𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)))𝑖))
279	278	ad2antrr 722	. . . . . . . . . . 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))
280		ovexd 7053	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)) ∈ V)
281		eqid 2794	. . . . . . . . . . . . 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)))
282	281	ovmpt4g 7156	. . . . . . . . . . . 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)))
283	246, 61, 280, 282	syl3anc 1364	. . . . . . . . . . 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)))
284	279, 283	eqtr2d 2831	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗)) = (𝑗(𝐻 ∘𝑓 (.r‘𝐸)𝐷)𝑖))
285	284	mpteq2dva 5058	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)(.r‘𝐸)(𝑖(.r‘𝐸)𝑗))) = (𝑖 ∈ 𝑋 ↦ (𝑗(𝐻 ∘𝑓 (.r‘𝐸)𝐷)𝑖)))
286	285	oveq2d 7035	. . . . . . . 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))))
287	268, 271, 286	3eqtr3d 2838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗) = (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (𝑗(𝐻 ∘𝑓 (.r‘𝐸)𝐷)𝑖))))
288	287	mpteq2dva 5058	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗)) = (𝑗 ∈ 𝑌 ↦ (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (𝑗(𝐻 ∘𝑓 (.r‘𝐸)𝐷)𝑖)))))
289	288	oveq2d 7035	. . . . 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))))))
290		ringcmn 19021	. . . . . . 7 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ CMnd)
291	69, 290	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ CMnd)
292	69	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝐸 ∈ Ring)
293	38, 180	eqsstrd 3928	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸))
294	293	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸))
295		simprl 767	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑙 ∈ (Base‘(Scalar‘𝐴)))
296	294, 295	sseldd 3892	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑙 ∈ (Base‘𝐸))
297		simprr 769	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑘 ∈ (Base‘𝐴))
298	12, 22	srabase 19640	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐸) = (Base‘𝐴))
299	298	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (Base‘𝐸) = (Base‘𝐴))
300	297, 299	eleqtrrd 2885	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → 𝑘 ∈ (Base‘𝐸))
301	20, 219	ringcl 19001	. . . . . . . 8 ⊢ ((𝐸 ∈ Ring ∧ 𝑙 ∈ (Base‘𝐸) ∧ 𝑘 ∈ (Base‘𝐸)) → (𝑙(.r‘𝐸)𝑘) ∈ (Base‘𝐸))
302	292, 296, 300, 301	syl3anc 1364	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑙 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑘 ∈ (Base‘𝐴))) → (𝑙(.r‘𝐸)𝑘) ∈ (Base‘𝐸))
303	20, 219	ringcl 19001	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
304	232, 238, 247, 303	syl3anc 1364	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐸))
305	298	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (Base‘𝐸) = (Base‘𝐴))
306	304, 305	eleqtrd 2884	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴))
307	306	anasss 467	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋)) → (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴))
308	307	ralrimivva 3157	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴))
309	275	fmpo 7625	. . . . . . . 8 ⊢ (∀𝑗 ∈ 𝑌 ∀𝑖 ∈ 𝑋 (𝑖(.r‘𝐸)𝑗) ∈ (Base‘𝐴) ↔ 𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
310	308, 309	sylib 219	. . . . . . 7 ⊢ (𝜑 → 𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
311		inidm 4117	. . . . . . 7 ⊢ ((𝑌 × 𝑋) ∩ (𝑌 × 𝑋)) = (𝑌 × 𝑋)
312	302, 65, 310, 48, 48, 311	off 7285	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘𝑓 (.r‘𝐸)𝐷):(𝑌 × 𝑋)⟶(Base‘𝐸))
313	69	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → 𝐸 ∈ Ring)
314		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → 𝑢 ∈ (Base‘𝐴))
315	298	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → (Base‘𝐸) = (Base‘𝐴))
316	314, 315	eleqtrrd 2885	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → 𝑢 ∈ (Base‘𝐸))
317	20, 219, 71	ringlz 19027	. . . . . . . 8 ⊢ ((𝐸 ∈ Ring ∧ 𝑢 ∈ (Base‘𝐸)) → ((0g‘𝐸)(.r‘𝐸)𝑢) = (0g‘𝐸))
318	313, 316, 317	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (Base‘𝐴)) → ((0g‘𝐸)(.r‘𝐸)𝑢) = (0g‘𝐸))
319	48, 73, 73, 65, 310, 194, 318	offinsupp1 30143	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘𝑓 (.r‘𝐸)𝐷) finSupp (0g‘𝐸))
320	20, 71, 291, 46, 47, 312, 319	gsumxp 18816	. . . . 5 ⊢ (𝜑 → (𝐸 Σg (𝐻 ∘𝑓 (.r‘𝐸)𝐷)) = (𝐸 Σg (𝑗 ∈ 𝑌 ↦ (𝐸 Σg (𝑖 ∈ 𝑋 ↦ (𝑗(𝐻 ∘𝑓 (.r‘𝐸)𝐷)𝑖))))))
321	12, 22	sravsca 19644	. . . . . . . 8 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘𝐴))
322		ofeq 7272	. . . . . . . 8 ⊢ ((.r‘𝐸) = ( ·𝑠 ‘𝐴) → ∘𝑓 (.r‘𝐸) = ∘𝑓 ( ·𝑠 ‘𝐴))
323	321, 322	syl 17	. . . . . . 7 ⊢ (𝜑 → ∘𝑓 (.r‘𝐸) = ∘𝑓 ( ·𝑠 ‘𝐴))
324	323	oveqd 7036	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘𝑓 (.r‘𝐸)𝐷) = (𝐻 ∘𝑓 ( ·𝑠 ‘𝐴)𝐷))
325	324	oveq2d 7035	. . . . 5 ⊢ (𝜑 → (𝐸 Σg (𝐻 ∘𝑓 (.r‘𝐸)𝐷)) = (𝐸 Σg (𝐻 ∘𝑓 ( ·𝑠 ‘𝐴)𝐷)))
326	289, 320, 325	3eqtr2rd 2837	. . . 4 ⊢ (𝜑 → (𝐸 Σg (𝐻 ∘𝑓 ( ·𝑠 ‘𝐴)𝐷)) = (𝐸 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))))
327		ovexd 7053	. . . . 5 ⊢ (𝜑 → (𝐻 ∘𝑓 ( ·𝑠 ‘𝐴)𝐷) ∈ V)
328		fedgmullem1.a	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐴))
329	328	elfvexd 6575	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
330	11, 327, 67, 329, 22	gsumsra 30485	. . . 4 ⊢ (𝜑 → (𝐸 Σg (𝐻 ∘𝑓 ( ·𝑠 ‘𝐴)𝐷)) = (𝐴 Σg (𝐻 ∘𝑓 ( ·𝑠 ‘𝐴)𝐷)))
331	326, 330	eqtr3d 2832	. . 3 ⊢ (𝜑 → (𝐸 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))) = (𝐴 Σg (𝐻 ∘𝑓 ( ·𝑠 ‘𝐴)𝐷)))
332	198, 199, 331	3eqtr2d 2836	. 2 ⊢ (𝜑 → 𝑍 = (𝐴 Σg (𝐻 ∘𝑓 ( ·𝑠 ‘𝐴)𝐷)))
333	197, 332	jca 512	1 ⊢ (𝜑 → (𝐻 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑍 = (𝐴 Σg (𝐻 ∘𝑓 ( ·𝑠 ‘𝐴)𝐷))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2080  ∀wral 3104  Vcvv 3436   ∖ cdif 3858   ⊆ wss 3861  {csn 4474  ∪ ciun 4827   class class class wbr 4964   ↦ cmpt 5043   × cxp 5444  Fun wfun 6222  ⟶wf 6224  ‘cfv 6228  (class class class)co 7019   ∈ cmpo 7021   ∘_𝑓 cof 7268   supp csupp 7684   ↑_𝑚 cmap 8259  Fincfn 8360   finSupp cfsupp 8682  Basecbs 16312   ↾_s cress 16313  +_gcplusg 16394  ._rcmulr 16395  Scalarcsca 16397   ·_𝑠 cvsca 16398  0_gc0g 16542   Σ_g cgsu 16543  SubMndcsubmnd 17773  Grpcgrp 17861  SubGrpcsubg 18027  CMndccmn 18633  Ringcrg 18987  DivRingcdr 19192  SubRingcsubrg 19221  LModclmod 19324  LSpanclspn 19433  LBasisclbs 19536  LVecclvec 19564  subringAlg csra 19630  LIndSclinds 20631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-rep 5084  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322  ax-cnex 10442  ax-resscn 10443  ax-1cn 10444  ax-icn 10445  ax-addcl 10446  ax-addrcl 10447  ax-mulcl 10448  ax-mulrcl 10449  ax-mulcom 10450  ax-addass 10451  ax-mulass 10452  ax-distr 10453  ax-i2m1 10454  ax-1ne0 10455  ax-1rid 10456  ax-rnegex 10457  ax-rrecex 10458  ax-cnre 10459  ax-pre-lttri 10460  ax-pre-lttrn 10461  ax-pre-ltadd 10462  ax-pre-mulgt0 10463

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-nel 3090  df-ral 3109  df-rex 3110  df-reu 3111  df-rmo 3112  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-uni 4748  df-int 4785  df-iun 4829  df-iin 4830  df-br 4965  df-opab 5027  df-mpt 5044  df-tr 5067  df-id 5351  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-se 5406  df-we 5407  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-pred 6026  df-ord 6072  df-on 6073  df-lim 6074  df-suc 6075  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-isom 6237  df-riota 6980  df-ov 7022  df-oprab 7023  df-mpo 7024  df-of 7270  df-om 7440  df-1st 7548  df-2nd 7549  df-supp 7685  df-wrecs 7801  df-recs 7863  df-rdg 7901  df-1o 7956  df-oadd 7960  df-er 8142  df-map 8261  df-ixp 8314  df-en 8361  df-dom 8362  df-sdom 8363  df-fin 8364  df-fsupp 8683  df-sup 8755  df-oi 8823  df-card 9217  df-pnf 10526  df-mnf 10527  df-xr 10528  df-ltxr 10529  df-le 10530  df-sub 10721  df-neg 10722  df-nn 11489  df-2 11550  df-3 11551  df-4 11552  df-5 11553  df-6 11554  df-7 11555  df-8 11556  df-9 11557  df-n0 11748  df-z 11832  df-dec 11949  df-uz 12094  df-fz 12743  df-fzo 12884  df-seq 13220  df-hash 13541  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-mulr 16408  df-sca 16410  df-vsca 16411  df-ip 16412  df-tset 16413  df-ple 16414  df-ds 16416  df-hom 16418  df-cco 16419  df-0g 16544  df-gsum 16545  df-prds 16550  df-pws 16552  df-mre 16686  df-mrc 16687  df-acs 16689  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-mhm 17774  df-submnd 17775  df-grp 17864  df-minusg 17865  df-sbg 17866  df-mulg 17982  df-subg 18030  df-ghm 18097  df-cntz 18188  df-cmn 18635  df-abl 18636  df-mgp 18930  df-ur 18942  df-ring 18989  df-drng 19194  df-subrg 19223  df-lmod 19326  df-lss 19394  df-lsp 19434  df-lmhm 19484  df-lbs 19537  df-lvec 19565  df-sra 19634  df-rgmod 19635  df-nzr 19720  df-dsmm 20558  df-frlm 20573  df-uvc 20609  df-lindf 20632  df-linds 20633

This theorem is referenced by:  fedgmul  30623