Theorem lmhmqusker 33950

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 2761	. . 3 ⊢ (Base‘𝑄) = (Base‘𝑄)
2		eqid 2761	. . 3 ⊢ ( ·𝑠 ‘𝑄) = ( ·𝑠 ‘𝑄)
3		eqid 2761	. . 3 ⊢ ( ·𝑠 ‘𝐻) = ( ·𝑠 ‘𝐻)
4		eqid 2761	. . 3 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄)
5		eqid 2761	. . 3 ⊢ (Scalar‘𝐻) = (Scalar‘𝐻)
6		eqid 2761	. . 3 ⊢ (Base‘(Scalar‘𝑄)) = (Base‘(Scalar‘𝑄))
7		lmhmqusker.q	. . . 4 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
8		eqid 2761	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
9		lmhmqusker.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐺 LMHom 𝐻))
10		lmhmlmod1 21288	. . . . 5 ⊢ (𝐹 ∈ (𝐺 LMHom 𝐻) → 𝐺 ∈ LMod)
11	9, 10	syl 18	. . . 4 ⊢ (𝜑 → 𝐺 ∈ LMod)
12		lmhmqusker.k	. . . . . 6 ⊢ 𝐾 = (◡𝐹 “ { 0 })
13		lmhmqusker.1	. . . . . 6 ⊢ 0 = (0g‘𝐻)
14		eqid 2761	. . . . . 6 ⊢ (LSubSp‘𝐺) = (LSubSp‘𝐺)
15	12, 13, 14	lmhmkerlss 21306	. . . . 5 ⊢ (𝐹 ∈ (𝐺 LMHom 𝐻) → 𝐾 ∈ (LSubSp‘𝐺))
16	9, 15	syl 18	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (LSubSp‘𝐺))
17	7, 8, 11, 16	quslmod 33901	. . 3 ⊢ (𝜑 → 𝑄 ∈ LMod)
18		lmhmlmod2 21287	. . . 4 ⊢ (𝐹 ∈ (𝐺 LMHom 𝐻) → 𝐻 ∈ LMod)
19	9, 18	syl 18	. . 3 ⊢ (𝜑 → 𝐻 ∈ LMod)
20		eqid 2761	. . . . . 6 ⊢ (Scalar‘𝐺) = (Scalar‘𝐺)
21	20, 5	lmhmsca 21285	. . . . 5 ⊢ (𝐹 ∈ (𝐺 LMHom 𝐻) → (Scalar‘𝐻) = (Scalar‘𝐺))
22	9, 21	syl 18	. . . 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 7447	. . . . 5 ⊢ (𝜑 → (𝐺 ~QG 𝐾) ∈ V)
26	23, 24, 25, 11, 20	quss 17698	. . . 4 ⊢ (𝜑 → (Scalar‘𝐺) = (Scalar‘𝑄))
27	22, 26	eqtrd 2796	. . 3 ⊢ (𝜑 → (Scalar‘𝐻) = (Scalar‘𝑄))
28		lmghm 21286	. . . . . 6 ⊢ (𝐹 ∈ (𝐺 LMHom 𝐻) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
29	9, 28	syl 18	. . . . 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 19481	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpIso 𝐻))
33		gimghm 19458	. . . 4 ⊢ (𝐽 ∈ (𝑄 GrpIso 𝐻) → 𝐽 ∈ (𝑄 GrpHom 𝐻))
34	32, 33	syl 18	. . 3 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))
35	13	ghmker 19436	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
36	29, 35	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
37	12, 36	eqeltrid 2865	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ (NrmSGrp‘𝐺))
38		nsgsubg 19348	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
39		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (𝐺 ~QG 𝐾) = (𝐺 ~QG 𝐾)
40	8, 39	eqger 19370	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
41	37, 38, 40	3syl 19	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
42	41	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
43		simpllr 788	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑟 ∈ (Base‘𝑄))
44	23, 24, 25, 11	qusbas 17697	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
45	44	ad4antr 745	. . . . . . . . . . . 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 2864	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
47		simplr 781	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑥 ∈ 𝑟)
48		qsel 8801	. . . . . . . . . . 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 1398	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
50	49	oveq2d 7428	. . . . . . . . 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 2761	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝐺))
52		eqid 2761	. . . . . . . . . 10 ⊢ ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝐺)
53	11	ad4antr 745	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐺 ∈ LMod)
54	16	ad4antr 745	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐾 ∈ (LSubSp‘𝐺))
55		simp-4r 796	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑘 ∈ (Base‘(Scalar‘𝑄)))
56	26	fveq2d 6881	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝑄)))
57	56	ad4antr 745	. . . . . . . . . . 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 2864	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑘 ∈ (Base‘(Scalar‘𝐺)))
59	41	qsss 8780	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ⊆ 𝒫 (Base‘𝐺))
60	44, 59	eqsstrrd 3966	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
61	60	sselda 3931	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
62	61	elpwid 4566	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
63	62	ad5ant13 769	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑟 ⊆ (Base‘𝐺))
64	63, 47	sseldd 3932	. . . . . . . . . 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 33895	. . . . . . . . 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 2796	. . . . . . . 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 6881	. . . . . . 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 745	. . . . . . . 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 21133	. . . . . . . . 9 ⊢ ((𝐺 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑘( ·𝑠 ‘𝐺)𝑥) ∈ (Base‘𝐺))
70	53, 58, 64, 69	syl3anc 1398	. . . . . . . 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 19477	. . . . . . 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 745	. . . . . . . 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 21290	. . . . . . . 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 1398	. . . . . . 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 2800	. . . . . 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 490	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘𝑟) = (𝐹‘𝑥))
77	76	oveq2d 7428	. . . . . 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 2799	. . . . 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 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
80		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
81	13, 79, 12, 7, 30, 80	ghmquskerlem2 19479	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
82	78, 81	r19.29a 3171	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) → (𝐽‘(𝑘( ·𝑠 ‘𝑄)𝑟)) = (𝑘( ·𝑠 ‘𝐻)(𝐽‘𝑟)))
83	82	anasss 472	. . 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 21294	. 2 ⊢ (𝜑 → 𝐽 ∈ (𝑄 LMHom 𝐻))
85		eqid 2761	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
86	1, 85	gimf1o 19457	. . 3 ⊢ (𝐽 ∈ (𝑄 GrpIso 𝐻) → 𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻))
87	32, 86	syl 18	. 2 ⊢ (𝜑 → 𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻))
88	1, 85	islmim 21317	. 2 ⊢ (𝐽 ∈ (𝑄 LMIso 𝐻) ↔ (𝐽 ∈ (𝑄 LMHom 𝐻) ∧ 𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻)))
89	84, 87, 88	sylanbrc 595	1 ⊢ (𝜑 → 𝐽 ∈ (𝑄 LMIso 𝐻))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ⊆ wss 3899  𝒫 cpw 4557  {csn 4584  ∪ cuni 4867   ↦ cmpt 5186  ^◡ccnv 5650  ran crn 5652   “ cima 5654  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7412   Er wer 8698  [cec 8699   / cqs 8700  Basecbs 17367  Scalarcsca 17411   ·_𝑠 cvsca 17412  0_gc0g 17590   /_s cqus 17657  SubGrpcsubg 19310  NrmSGrpcnsg 19311   ~_QG cqg 19312   GrpHom cghm 19407   GrpIso cgim 19451  LModclmod 21115  LSubSpclss 21186   LMHom clmhm 21274   LMIso clmim 21275

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-ec 8703  df-qs 8707  df-map 8833  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9418  df-inf 9419  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-3 12387  df-4 12388  df-5 12389  df-6 12390  df-7 12391  df-8 12392  df-9 12393  df-n0 12588  df-z 12675  df-dec 12796  df-uz 12947  df-fz 13621  df-struct 17305  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-ress 17389  df-plusg 17421  df-mulr 17422  df-sca 17424  df-vsca 17425  df-ip 17426  df-tset 17427  df-ple 17428  df-ds 17430  df-0g 17592  df-imas 17660  df-qus 17661  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-submnd 18959  df-grp 19127  df-minusg 19128  df-sbg 19129  df-subg 19313  df-nsg 19314  df-eqg 19315  df-ghm 19408  df-gim 19453  df-cmn 19976  df-abl 19977  df-mgp 20341  df-rng 20355  df-ur 20388  df-ring 20441  df-lmod 21117  df-lss 21187  df-lmhm 21277  df-lmim 21278

This theorem is used by:  lmicqusker  33951  algextdeglem4  34334