Theorem lmhmqusker 33410

Description: A surjective module homomorphism 𝐹 from 𝐺 to 𝐻 induces an isomorphism 𝐽 from 𝑄 to 𝐻, where 𝑄 is the factor group of 𝐺 by 𝐹's kernel 𝐾. (Contributed by Thierry Arnoux, 25-Feb-2025.)

Hypotheses

Ref	Expression
lmhmqusker.1	⊢ 0 = (0g‘𝐻)
lmhmqusker.f	⊢ (𝜑 → 𝐹 ∈ (𝐺 LMHom 𝐻))
lmhmqusker.k	⊢ 𝐾 = (◡𝐹 “ { 0 })
lmhmqusker.q	⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
lmhmqusker.s	⊢ (𝜑 → ran 𝐹 = (Base‘𝐻))
lmhmqusker.j	⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))

Assertion

Ref	Expression
lmhmqusker	⊢ (𝜑 → 𝐽 ∈ (𝑄 LMIso 𝐻))

Distinct variable groups:   𝐹,𝑞   𝐺,𝑞   𝐻,𝑞   𝐽,𝑞   𝐾,𝑞   𝑄,𝑞   𝜑,𝑞

Allowed substitution hint:   0 (𝑞)

Proof of Theorem lmhmqusker

Dummy variables 𝑥 𝑘 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2740	. . 3 ⊢ (Base‘𝑄) = (Base‘𝑄)
2		eqid 2740	. . 3 ⊢ ( ·𝑠 ‘𝑄) = ( ·𝑠 ‘𝑄)
3		eqid 2740	. . 3 ⊢ ( ·𝑠 ‘𝐻) = ( ·𝑠 ‘𝐻)
4		eqid 2740	. . 3 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄)
5		eqid 2740	. . 3 ⊢ (Scalar‘𝐻) = (Scalar‘𝐻)
6		eqid 2740	. . 3 ⊢ (Base‘(Scalar‘𝑄)) = (Base‘(Scalar‘𝑄))
7		lmhmqusker.q	. . . 4 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
8		eqid 2740	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
9		lmhmqusker.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐺 LMHom 𝐻))
10		lmhmlmod1 21055	. . . . 5 ⊢ (𝐹 ∈ (𝐺 LMHom 𝐻) → 𝐺 ∈ LMod)
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ∈ LMod)
12		lmhmqusker.k	. . . . . 6 ⊢ 𝐾 = (◡𝐹 “ { 0 })
13		lmhmqusker.1	. . . . . 6 ⊢ 0 = (0g‘𝐻)
14		eqid 2740	. . . . . 6 ⊢ (LSubSp‘𝐺) = (LSubSp‘𝐺)
15	12, 13, 14	lmhmkerlss 21073	. . . . 5 ⊢ (𝐹 ∈ (𝐺 LMHom 𝐻) → 𝐾 ∈ (LSubSp‘𝐺))
16	9, 15	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (LSubSp‘𝐺))
17	7, 8, 11, 16	quslmod 33351	. . 3 ⊢ (𝜑 → 𝑄 ∈ LMod)
18		lmhmlmod2 21054	. . . 4 ⊢ (𝐹 ∈ (𝐺 LMHom 𝐻) → 𝐻 ∈ LMod)
19	9, 18	syl 17	. . 3 ⊢ (𝜑 → 𝐻 ∈ LMod)
20		eqid 2740	. . . . . 6 ⊢ (Scalar‘𝐺) = (Scalar‘𝐺)
21	20, 5	lmhmsca 21052	. . . . 5 ⊢ (𝐹 ∈ (𝐺 LMHom 𝐻) → (Scalar‘𝐻) = (Scalar‘𝐺))
22	9, 21	syl 17	. . . 4 ⊢ (𝜑 → (Scalar‘𝐻) = (Scalar‘𝐺))
23	7	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)))
24	8	a1i 11	. . . . 5 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
25		ovexd 7483	. . . . 5 ⊢ (𝜑 → (𝐺 ~QG 𝐾) ∈ V)
26	23, 24, 25, 11, 20	quss 17606	. . . 4 ⊢ (𝜑 → (Scalar‘𝐺) = (Scalar‘𝑄))
27	22, 26	eqtrd 2780	. . 3 ⊢ (𝜑 → (Scalar‘𝐻) = (Scalar‘𝑄))
28		lmghm 21053	. . . . . 6 ⊢ (𝐹 ∈ (𝐺 LMHom 𝐻) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
29	9, 28	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))
30		lmhmqusker.j	. . . . 5 ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
31		lmhmqusker.s	. . . . 5 ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻))
32	13, 29, 12, 7, 30, 31	ghmqusker 19327	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpIso 𝐻))
33		gimghm 19304	. . . 4 ⊢ (𝐽 ∈ (𝑄 GrpIso 𝐻) → 𝐽 ∈ (𝑄 GrpHom 𝐻))
34	32, 33	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))
35	13	ghmker 19282	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
36	29, 35	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
37	12, 36	eqeltrid 2848	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ (NrmSGrp‘𝐺))
38		nsgsubg 19198	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
39		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (𝐺 ~QG 𝐾) = (𝐺 ~QG 𝐾)
40	8, 39	eqger 19218	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
41	37, 38, 40	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
42	41	ad4antr 731	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
43		simpllr 775	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑟 ∈ (Base‘𝑄))
44	23, 24, 25, 11	qusbas 17605	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
45	44	ad4antr 731	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
46	43, 45	eleqtrrd 2847	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
47		simplr 768	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑥 ∈ 𝑟)
48		qsel 8854	. . . . . . . . . . 11 ⊢ (((𝐺 ~QG 𝐾) Er (Base‘𝐺) ∧ 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ∧ 𝑥 ∈ 𝑟) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
49	42, 46, 47, 48	syl3anc 1371	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
50	49	oveq2d 7464	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝑘( ·𝑠 ‘𝑄)𝑟) = (𝑘( ·𝑠 ‘𝑄)[𝑥](𝐺 ~QG 𝐾)))
51		eqid 2740	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝐺))
52		eqid 2740	. . . . . . . . . 10 ⊢ ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝐺)
53	11	ad4antr 731	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐺 ∈ LMod)
54	16	ad4antr 731	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐾 ∈ (LSubSp‘𝐺))
55		simp-4r 783	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑘 ∈ (Base‘(Scalar‘𝑄)))
56	26	fveq2d 6924	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝑄)))
57	56	ad4antr 731	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝑄)))
58	55, 57	eleqtrrd 2847	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑘 ∈ (Base‘(Scalar‘𝐺)))
59	41	qsss 8836	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ⊆ 𝒫 (Base‘𝐺))
60	44, 59	eqsstrrd 4048	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
61	60	sselda 4008	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
62	61	elpwid 4631	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
63	62	ad5ant13 756	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑟 ⊆ (Base‘𝐺))
64	63, 47	sseldd 4009	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑥 ∈ (Base‘𝐺))
65	8, 39, 51, 52, 53, 54, 58, 7, 2, 64	qusvsval 33345	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝑘( ·𝑠 ‘𝑄)[𝑥](𝐺 ~QG 𝐾)) = [(𝑘( ·𝑠 ‘𝐺)𝑥)](𝐺 ~QG 𝐾))
66	50, 65	eqtrd 2780	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝑘( ·𝑠 ‘𝑄)𝑟) = [(𝑘( ·𝑠 ‘𝐺)𝑥)](𝐺 ~QG 𝐾))
67	66	fveq2d 6924	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘(𝑘( ·𝑠 ‘𝑄)𝑟)) = (𝐽‘[(𝑘( ·𝑠 ‘𝐺)𝑥)](𝐺 ~QG 𝐾)))
68	29	ad4antr 731	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
69	8, 20, 52, 51	lmodvscl 20898	. . . . . . . . 9 ⊢ ((𝐺 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑘( ·𝑠 ‘𝐺)𝑥) ∈ (Base‘𝐺))
70	53, 58, 64, 69	syl3anc 1371	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝑘( ·𝑠 ‘𝐺)𝑥) ∈ (Base‘𝐺))
71	13, 68, 12, 7, 30, 70	ghmquskerlem1 19323	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘[(𝑘( ·𝑠 ‘𝐺)𝑥)](𝐺 ~QG 𝐾)) = (𝐹‘(𝑘( ·𝑠 ‘𝐺)𝑥)))
72	9	ad4antr 731	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐹 ∈ (𝐺 LMHom 𝐻))
73	20, 51, 8, 52, 3	lmhmlin 21057	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐺 LMHom 𝐻) ∧ 𝑘 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝐹‘(𝑘( ·𝑠 ‘𝐺)𝑥)) = (𝑘( ·𝑠 ‘𝐻)(𝐹‘𝑥)))
74	72, 58, 64, 73	syl3anc 1371	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐹‘(𝑘( ·𝑠 ‘𝐺)𝑥)) = (𝑘( ·𝑠 ‘𝐻)(𝐹‘𝑥)))
75	67, 71, 74	3eqtrd 2784	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘(𝑘( ·𝑠 ‘𝑄)𝑟)) = (𝑘( ·𝑠 ‘𝐻)(𝐹‘𝑥)))
76		simpr 484	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘𝑟) = (𝐹‘𝑥))
77	76	oveq2d 7464	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝑘( ·𝑠 ‘𝐻)(𝐽‘𝑟)) = (𝑘( ·𝑠 ‘𝐻)(𝐹‘𝑥)))
78	75, 77	eqtr4d 2783	. . . . 5 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘(𝑘( ·𝑠 ‘𝑄)𝑟)) = (𝑘( ·𝑠 ‘𝐻)(𝐽‘𝑟)))
79	29	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
80		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
81	13, 79, 12, 7, 30, 80	ghmquskerlem2 19325	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
82	78, 81	r19.29a 3168	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) → (𝐽‘(𝑘( ·𝑠 ‘𝑄)𝑟)) = (𝑘( ·𝑠 ‘𝐻)(𝐽‘𝑟)))
83	82	anasss 466	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑄)) ∧ 𝑟 ∈ (Base‘𝑄))) → (𝐽‘(𝑘( ·𝑠 ‘𝑄)𝑟)) = (𝑘( ·𝑠 ‘𝐻)(𝐽‘𝑟)))
84	1, 2, 3, 4, 5, 6, 17, 19, 27, 34, 83	islmhmd 21061	. 2 ⊢ (𝜑 → 𝐽 ∈ (𝑄 LMHom 𝐻))
85		eqid 2740	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
86	1, 85	gimf1o 19303	. . 3 ⊢ (𝐽 ∈ (𝑄 GrpIso 𝐻) → 𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻))
87	32, 86	syl 17	. 2 ⊢ (𝜑 → 𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻))
88	1, 85	islmim 21084	. 2 ⊢ (𝐽 ∈ (𝑄 LMIso 𝐻) ↔ (𝐽 ∈ (𝑄 LMHom 𝐻) ∧ 𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻)))
89	84, 87, 88	sylanbrc 582	1 ⊢ (𝜑 → 𝐽 ∈ (𝑄 LMIso 𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622  {csn 4648  ∪ cuni 4931   ↦ cmpt 5249  ^◡ccnv 5699  ran crn 5701   “ cima 5703  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448   Er wer 8760  [cec 8761   / cqs 8762  Basecbs 17258  Scalarcsca 17314   ·_𝑠 cvsca 17315  0_gc0g 17499   /_s cqus 17565  SubGrpcsubg 19160  NrmSGrpcnsg 19161   ~_QG cqg 19162   GrpHom cghm 19252   GrpIso cgim 19297  LModclmod 20880  LSubSpclss 20952   LMHom clmhm 21041   LMIso clmim 21042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-ec 8765  df-qs 8769  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-0g 17501  df-imas 17568  df-qus 17569  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-subg 19163  df-nsg 19164  df-eqg 19165  df-ghm 19253  df-gim 19299  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-lmod 20882  df-lss 20953  df-lmhm 21044  df-lmim 21045

This theorem is referenced by:  lmicqusker  33411  algextdeglem4  33711