Theorem pj1ghm 19640

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 2730	. 2 ⊢ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈))) = (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈)))
2		eqid 2730	. 2 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		ovex 7423	. . 3 ⊢ (𝑇 ⊕ 𝑈) ∈ V
4		eqid 2730	. . . 4 ⊢ (𝐺 ↾s (𝑇 ⊕ 𝑈)) = (𝐺 ↾s (𝑇 ⊕ 𝑈))
5		pj1eu.a	. . . 4 ⊢ + = (+g‘𝐺)
6	4, 5	ressplusg 17261	. . 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 19591	. . . 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 19068	. . 3 ⊢ ((𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺) → (𝐺 ↾s (𝑇 ⊕ 𝑈)) ∈ Grp)
16	14, 15	syl 17	. 2 ⊢ (𝜑 → (𝐺 ↾s (𝑇 ⊕ 𝑈)) ∈ Grp)
17		subgrcl 19070	. . 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 19634	. . . 4 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
23	2	subgss 19066	. . . . 5 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
24	8, 23	syl 17	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺))
25	22, 24	fssd 6708	. . 3 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶(Base‘𝐺))
26	4	subgbas 19069	. . . . 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 6675	. . 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 2815	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ 𝑥 ∈ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈)))))
31	27	eleq2d 2815	. . . . 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 19636	. . . . . . . 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 19636	. . . . . . . 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 7408	. . . . . 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 18879	. . . . . . . 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 7056	. . . . . . . . 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 3950	. . . . . . 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 7056	. . . . . . . . 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 3950	. . . . . . 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 19066	. . . . . . . . 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 19635	. . . . . . . . 9 ⊢ (𝜑 → (𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
55		ffvelcdm 7056	. . . . . . . . 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 3950	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑥) ∈ (Base‘𝐺))
58		ffvelcdm 7056	. . . . . . . . 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 3950	. . . . . . 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 3950	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (𝑍‘𝑈))
63	5, 12	cntzi 19268	. . . . . . . 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 18682	. . . . . 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 2768	. . . . 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 19075	. . . . . . . 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 19075	. . . . . . 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 19075	. . . . . . 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 19637	. . . . 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 19164	1 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺 ↾s (𝑇 ⊕ 𝑈)) GrpHom 𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∩ cin 3916   ⊆ wss 3917  {csn 4592  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Basecbs 17186   ↾_s cress 17207  +_gcplusg 17227  0_gc0g 17409  Mndcmnd 18668  Grpcgrp 18872  SubGrpcsubg 19059   GrpHom cghm 19151  Cntzccntz 19254  LSSumclsm 19571  proj₁cpj1 19572

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-0g 17411  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-submnd 18718  df-grp 18875  df-minusg 18876  df-sbg 18877  df-subg 19062  df-ghm 19152  df-cntz 19256  df-lsm 19573  df-pj1 19574

This theorem is referenced by:  pj1ghm2  19641  dpjghm  20002  pj1lmhm  21014