Theorem pj1ghm 19582

Description: The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
pj1eu.a	⊢ + = (+g‘𝐺)
pj1eu.s	⊢ ⊕ = (LSSum‘𝐺)
pj1eu.o	⊢ 0 = (0g‘𝐺)
pj1eu.z	⊢ 𝑍 = (Cntz‘𝐺)
pj1eu.2	⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
pj1eu.3	⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
pj1eu.4	⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })
pj1eu.5	⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))
pj1f.p	⊢ 𝑃 = (proj1‘𝐺)

Assertion

Ref	Expression
pj1ghm	⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺 ↾s (𝑇 ⊕ 𝑈)) GrpHom 𝐺))

Proof of Theorem pj1ghm

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. 2 ⊢ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈))) = (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈)))
2		eqid 2729	. 2 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		ovex 7382	. . 3 ⊢ (𝑇 ⊕ 𝑈) ∈ V
4		eqid 2729	. . . 4 ⊢ (𝐺 ↾s (𝑇 ⊕ 𝑈)) = (𝐺 ↾s (𝑇 ⊕ 𝑈))
5		pj1eu.a	. . . 4 ⊢ + = (+g‘𝐺)
6	4, 5	ressplusg 17195	. . 3 ⊢ ((𝑇 ⊕ 𝑈) ∈ V → + = (+g‘(𝐺 ↾s (𝑇 ⊕ 𝑈))))
7	3, 6	ax-mp 5	. 2 ⊢ + = (+g‘(𝐺 ↾s (𝑇 ⊕ 𝑈)))
8		pj1eu.2	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
9		pj1eu.3	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
10		pj1eu.5	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))
11		pj1eu.s	. . . . 5 ⊢ ⊕ = (LSSum‘𝐺)
12		pj1eu.z	. . . . 5 ⊢ 𝑍 = (Cntz‘𝐺)
13	11, 12	lsmsubg 19533	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺))
14	8, 9, 10, 13	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺))
15	4	subggrp 19008	. . 3 ⊢ ((𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺) → (𝐺 ↾s (𝑇 ⊕ 𝑈)) ∈ Grp)
16	14, 15	syl 17	. 2 ⊢ (𝜑 → (𝐺 ↾s (𝑇 ⊕ 𝑈)) ∈ Grp)
17		subgrcl 19010	. . 3 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
18	8, 17	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
19		pj1eu.o	. . . . 5 ⊢ 0 = (0g‘𝐺)
20		pj1eu.4	. . . . 5 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })
21		pj1f.p	. . . . 5 ⊢ 𝑃 = (proj1‘𝐺)
22	5, 11, 19, 12, 8, 9, 20, 10, 21	pj1f 19576	. . . 4 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
23	2	subgss 19006	. . . . 5 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
24	8, 23	syl 17	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺))
25	22, 24	fssd 6669	. . 3 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶(Base‘𝐺))
26	4	subgbas 19009	. . . . 5 ⊢ ((𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺) → (𝑇 ⊕ 𝑈) = (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈))))
27	14, 26	syl 17	. . . 4 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) = (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈))))
28	27	feq2d 6636	. . 3 ⊢ (𝜑 → ((𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶(Base‘𝐺) ↔ (𝑇𝑃𝑈):(Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈)))⟶(Base‘𝐺)))
29	25, 28	mpbid 232	. 2 ⊢ (𝜑 → (𝑇𝑃𝑈):(Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈)))⟶(Base‘𝐺))
30	27	eleq2d 2814	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ 𝑥 ∈ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈)))))
31	27	eleq2d 2814	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝑇 ⊕ 𝑈) ↔ 𝑦 ∈ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈)))))
32	30, 31	anbi12d 632	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) ↔ (𝑥 ∈ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈))) ∧ 𝑦 ∈ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈))))))
33	32	biimpar 477	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈))) ∧ 𝑦 ∈ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈))))) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)))
34	5, 11, 19, 12, 8, 9, 20, 10, 21	pj1id 19578	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → 𝑥 = (((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)))
35	34	adantrr 717	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑥 = (((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)))
36	5, 11, 19, 12, 8, 9, 20, 10, 21	pj1id 19578	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦)))
37	36	adantrl 716	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦)))
38	35, 37	oveq12d 7367	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥 + 𝑦) = ((((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)) + (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦))))
39	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑇 ∈ (SubGrp‘𝐺))
40		grpmnd 18819	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
41	39, 17, 40	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝐺 ∈ Mnd)
42	39, 23	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑇 ⊆ (Base‘𝐺))
43		simpl 482	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → 𝑥 ∈ (𝑇 ⊕ 𝑈))
44		ffvelcdm 7015	. . . . . . . . 9 ⊢ (((𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇)
45	22, 43, 44	syl2an 596	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇)
46	42, 45	sseldd 3936	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑥) ∈ (Base‘𝐺))
47		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → 𝑦 ∈ (𝑇 ⊕ 𝑈))
48		ffvelcdm 7015	. . . . . . . . 9 ⊢ (((𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇 ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
49	22, 47, 48	syl2an 596	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
50	42, 49	sseldd 3936	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (Base‘𝐺))
51	9	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑈 ∈ (SubGrp‘𝐺))
52	2	subgss 19006	. . . . . . . . 9 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
53	51, 52	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑈 ⊆ (Base‘𝐺))
54	5, 11, 19, 12, 8, 9, 20, 10, 21	pj2f 19577	. . . . . . . . 9 ⊢ (𝜑 → (𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
55		ffvelcdm 7015	. . . . . . . . 9 ⊢ (((𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈)
56	54, 43, 55	syl2an 596	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈)
57	53, 56	sseldd 3936	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑥) ∈ (Base‘𝐺))
58		ffvelcdm 7015	. . . . . . . . 9 ⊢ (((𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈 ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
59	54, 47, 58	syl2an 596	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
60	53, 59	sseldd 3936	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ (Base‘𝐺))
61	10	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑇 ⊆ (𝑍‘𝑈))
62	61, 49	sseldd 3936	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (𝑍‘𝑈))
63	5, 12	cntzi 19208	. . . . . . . 8 ⊢ ((((𝑇𝑃𝑈)‘𝑦) ∈ (𝑍‘𝑈) ∧ ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈) → (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑥)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
64	62, 56, 63	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑥)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
65	2, 5, 41, 46, 50, 57, 60, 64	mnd4g 18622	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) + (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦))) = ((((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)) + (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦))))
66	38, 65	eqtr4d 2767	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥 + 𝑦) = ((((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) + (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦))))
67	20	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑇 ∩ 𝑈) = { 0 })
68	5	subgcl 19015	. . . . . . . 8 ⊢ (((𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → (𝑥 + 𝑦) ∈ (𝑇 ⊕ 𝑈))
69	68	3expb 1120	. . . . . . 7 ⊢ (((𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥 + 𝑦) ∈ (𝑇 ⊕ 𝑈))
70	14, 69	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥 + 𝑦) ∈ (𝑇 ⊕ 𝑈))
71	5	subgcl 19015	. . . . . . 7 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇 ∧ ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇) → (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
72	39, 45, 49, 71	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
73	5	subgcl 19015	. . . . . . 7 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈 ∧ ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈) → (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦)) ∈ 𝑈)
74	51, 56, 59, 73	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦)) ∈ 𝑈)
75	5, 11, 19, 12, 39, 51, 67, 61, 21, 70, 72, 74	pj1eq 19579	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑥 + 𝑦) = ((((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) + (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦))) ↔ (((𝑇𝑃𝑈)‘(𝑥 + 𝑦)) = (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∧ ((𝑈𝑃𝑇)‘(𝑥 + 𝑦)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦)))))
76	66, 75	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (((𝑇𝑃𝑈)‘(𝑥 + 𝑦)) = (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∧ ((𝑈𝑃𝑇)‘(𝑥 + 𝑦)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦))))
77	76	simpld 494	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘(𝑥 + 𝑦)) = (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
78	33, 77	syldan 591	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈))) ∧ 𝑦 ∈ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈))))) → ((𝑇𝑃𝑈)‘(𝑥 + 𝑦)) = (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
79	1, 2, 7, 5, 16, 18, 29, 78	isghmd 19104	1 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺 ↾s (𝑇 ⊕ 𝑈)) GrpHom 𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436   ∩ cin 3902   ⊆ wss 3903  {csn 4577  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  Basecbs 17120   ↾_s cress 17141  +_gcplusg 17161  0_gc0g 17343  Mndcmnd 18608  Grpcgrp 18812  SubGrpcsubg 18999   GrpHom cghm 19091  Cntzccntz 19194  LSSumclsm 19513  proj₁cpj1 19514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-0g 17345  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-submnd 18658  df-grp 18815  df-minusg 18816  df-sbg 18817  df-subg 19002  df-ghm 19092  df-cntz 19196  df-lsm 19515  df-pj1 19516

This theorem is referenced by:  pj1ghm2  19583  dpjghm  19944  pj1lmhm  21004