Theorem lmhmqusker 33380

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 2731	. . 3 ⊢ (Base‘𝑄) = (Base‘𝑄)
2		eqid 2731	. . 3 ⊢ ( ·𝑠 ‘𝑄) = ( ·𝑠 ‘𝑄)
3		eqid 2731	. . 3 ⊢ ( ·𝑠 ‘𝐻) = ( ·𝑠 ‘𝐻)
4		eqid 2731	. . 3 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄)
5		eqid 2731	. . 3 ⊢ (Scalar‘𝐻) = (Scalar‘𝐻)
6		eqid 2731	. . 3 ⊢ (Base‘(Scalar‘𝑄)) = (Base‘(Scalar‘𝑄))
7		lmhmqusker.q	. . . 4 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
8		eqid 2731	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
9		lmhmqusker.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐺 LMHom 𝐻))
10		lmhmlmod1 20968	. . . . 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 2731	. . . . . 6 ⊢ (LSubSp‘𝐺) = (LSubSp‘𝐺)
15	12, 13, 14	lmhmkerlss 20986	. . . . 5 ⊢ (𝐹 ∈ (𝐺 LMHom 𝐻) → 𝐾 ∈ (LSubSp‘𝐺))
16	9, 15	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (LSubSp‘𝐺))
17	7, 8, 11, 16	quslmod 33321	. . 3 ⊢ (𝜑 → 𝑄 ∈ LMod)
18		lmhmlmod2 20967	. . . 4 ⊢ (𝐹 ∈ (𝐺 LMHom 𝐻) → 𝐻 ∈ LMod)
19	9, 18	syl 17	. . 3 ⊢ (𝜑 → 𝐻 ∈ LMod)
20		eqid 2731	. . . . . 6 ⊢ (Scalar‘𝐺) = (Scalar‘𝐺)
21	20, 5	lmhmsca 20965	. . . . 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 7381	. . . . 5 ⊢ (𝜑 → (𝐺 ~QG 𝐾) ∈ V)
26	23, 24, 25, 11, 20	quss 17450	. . . 4 ⊢ (𝜑 → (Scalar‘𝐺) = (Scalar‘𝑄))
27	22, 26	eqtrd 2766	. . 3 ⊢ (𝜑 → (Scalar‘𝐻) = (Scalar‘𝑄))
28		lmghm 20966	. . . . . 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 19200	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpIso 𝐻))
33		gimghm 19177	. . . 4 ⊢ (𝐽 ∈ (𝑄 GrpIso 𝐻) → 𝐽 ∈ (𝑄 GrpHom 𝐻))
34	32, 33	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))
35	13	ghmker 19155	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
36	29, 35	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
37	12, 36	eqeltrid 2835	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ (NrmSGrp‘𝐺))
38		nsgsubg 19071	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
39		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (𝐺 ~QG 𝐾) = (𝐺 ~QG 𝐾)
40	8, 39	eqger 19091	. . . . . . . . . . . . 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 732	. . . . . . . . . . 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 17449	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
45	44	ad4antr 732	. . . . . . . . . . . 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 2834	. . . . . . . . . . 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 8720	. . . . . . . . . . 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 1373	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
50	49	oveq2d 7362	. . . . . . . . 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 2731	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝐺))
52		eqid 2731	. . . . . . . . . 10 ⊢ ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝐺)
53	11	ad4antr 732	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐺 ∈ LMod)
54	16	ad4antr 732	. . . . . . . . . 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 6826	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘(Scalar‘𝐺)) = (Base‘(Scalar‘𝑄)))
57	56	ad4antr 732	. . . . . . . . . . 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 2834	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑘 ∈ (Base‘(Scalar‘𝐺)))
59	41	qsss 8700	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ⊆ 𝒫 (Base‘𝐺))
60	44, 59	eqsstrrd 3970	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
61	60	sselda 3934	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
62	61	elpwid 4559	. . . . . . . . . . . 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 3935	. . . . . . . . . 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 33315	. . . . . . . . 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 2766	. . . . . . . 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 6826	. . . . . . 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 732	. . . . . . . 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 20812	. . . . . . . . 9 ⊢ ((𝐺 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝐺)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑘( ·𝑠 ‘𝐺)𝑥) ∈ (Base‘𝐺))
70	53, 58, 64, 69	syl3anc 1373	. . . . . . . 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 19196	. . . . . . 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 732	. . . . . . . 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 20970	. . . . . . . 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 1373	. . . . . . 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 2770	. . . . . 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 7362	. . . . . 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 2769	. . . . 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 726	. . . . . 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 19198	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑄))) ∧ 𝑟 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
82	78, 81	r19.29a 3140	. . . 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 20974	. 2 ⊢ (𝜑 → 𝐽 ∈ (𝑄 LMHom 𝐻))
85		eqid 2731	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
86	1, 85	gimf1o 19176	. . 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 20997	. 2 ⊢ (𝐽 ∈ (𝑄 LMIso 𝐻) ↔ (𝐽 ∈ (𝑄 LMHom 𝐻) ∧ 𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻)))
89	84, 87, 88	sylanbrc 583	1 ⊢ (𝜑 → 𝐽 ∈ (𝑄 LMIso 𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ⊆ wss 3902  𝒫 cpw 4550  {csn 4576  ∪ cuni 4859   ↦ cmpt 5172  ^◡ccnv 5615  ran crn 5617   “ cima 5619  –1-1-onto→wf1o 6480  ‘cfv 6481  (class class class)co 7346   Er wer 8619  [cec 8620   / cqs 8621  Basecbs 17120  Scalarcsca 17164   ·_𝑠 cvsca 17165  0_gc0g 17343   /_s cqus 17409  SubGrpcsubg 19033  NrmSGrpcnsg 19034   ~_QG cqg 19035   GrpHom cghm 19125   GrpIso cgim 19170  LModclmod 20794  LSubSpclss 20865   LMHom clmhm 20954   LMIso clmim 20955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-ec 8624  df-qs 8628  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-inf 9327  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-z 12469  df-dec 12589  df-uz 12733  df-fz 13408  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-0g 17345  df-imas 17412  df-qus 17413  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-submnd 18692  df-grp 18849  df-minusg 18850  df-sbg 18851  df-subg 19036  df-nsg 19037  df-eqg 19038  df-ghm 19126  df-gim 19172  df-cmn 19695  df-abl 19696  df-mgp 20060  df-rng 20072  df-ur 20101  df-ring 20154  df-lmod 20796  df-lss 20866  df-lmhm 20957  df-lmim 20958

This theorem is referenced by:  lmicqusker  33381  algextdeglem4  33731