Theorem pj1ghm 19745

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 2740	. 2 ⊢ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈))) = (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈)))
2		eqid 2740	. 2 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		ovex 7481	. . 3 ⊢ (𝑇 ⊕ 𝑈) ∈ V
4		eqid 2740	. . . 4 ⊢ (𝐺 ↾s (𝑇 ⊕ 𝑈)) = (𝐺 ↾s (𝑇 ⊕ 𝑈))
5		pj1eu.a	. . . 4 ⊢ + = (+g‘𝐺)
6	4, 5	ressplusg 17349	. . 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 19696	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺))
14	8, 9, 10, 13	syl3anc 1371	. . 3 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺))
15	4	subggrp 19169	. . 3 ⊢ ((𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺) → (𝐺 ↾s (𝑇 ⊕ 𝑈)) ∈ Grp)
16	14, 15	syl 17	. 2 ⊢ (𝜑 → (𝐺 ↾s (𝑇 ⊕ 𝑈)) ∈ Grp)
17		subgrcl 19171	. . 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 19739	. . . 4 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
23	2	subgss 19167	. . . . 5 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
24	8, 23	syl 17	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺))
25	22, 24	fssd 6764	. . 3 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶(Base‘𝐺))
26	4	subgbas 19170	. . . . 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 6733	. . 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 2830	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ 𝑥 ∈ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈)))))
31	27	eleq2d 2830	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝑇 ⊕ 𝑈) ↔ 𝑦 ∈ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈)))))
32	30, 31	anbi12d 631	. . . 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 19741	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → 𝑥 = (((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)))
35	34	adantrr 716	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑥 = (((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)))
36	5, 11, 19, 12, 8, 9, 20, 10, 21	pj1id 19741	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦)))
37	36	adantrl 715	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦)))
38	35, 37	oveq12d 7466	. . . . . 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 18980	. . . . . . . 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 7115	. . . . . . . . 9 ⊢ (((𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇)
45	22, 43, 44	syl2an 595	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇)
46	42, 45	sseldd 4009	. . . . . . 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 7115	. . . . . . . . 9 ⊢ (((𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇 ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
49	22, 47, 48	syl2an 595	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
50	42, 49	sseldd 4009	. . . . . . 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 19167	. . . . . . . . 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 19740	. . . . . . . . 9 ⊢ (𝜑 → (𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
55		ffvelcdm 7115	. . . . . . . . 9 ⊢ (((𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈)
56	54, 43, 55	syl2an 595	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈)
57	53, 56	sseldd 4009	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑥) ∈ (Base‘𝐺))
58		ffvelcdm 7115	. . . . . . . . 9 ⊢ (((𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈 ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
59	54, 47, 58	syl2an 595	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
60	53, 59	sseldd 4009	. . . . . . 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 4009	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (𝑍‘𝑈))
63	5, 12	cntzi 19369	. . . . . . . 8 ⊢ ((((𝑇𝑃𝑈)‘𝑦) ∈ (𝑍‘𝑈) ∧ ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈) → (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑥)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
64	62, 56, 63	syl2anc 583	. . . . . . 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 18786	. . . . . 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 2783	. . . . 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 19176	. . . . . . . 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 579	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥 + 𝑦) ∈ (𝑇 ⊕ 𝑈))
71	5	subgcl 19176	. . . . . . 7 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇 ∧ ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇) → (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
72	39, 45, 49, 71	syl3anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
73	5	subgcl 19176	. . . . . . 7 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈 ∧ ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈) → (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦)) ∈ 𝑈)
74	51, 56, 59, 73	syl3anc 1371	. . . . . 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 19742	. . . . 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 590	. 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 19265	1 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺 ↾s (𝑇 ⊕ 𝑈)) GrpHom 𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  {csn 4648  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Basecbs 17258   ↾_s cress 17287  +_gcplusg 17311  0_gc0g 17499  Mndcmnd 18772  Grpcgrp 18973  SubGrpcsubg 19160   GrpHom cghm 19252  Cntzccntz 19355  LSSumclsm 19676  proj₁cpj1 19677

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-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-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  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-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-0g 17501  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-ghm 19253  df-cntz 19357  df-lsm 19678  df-pj1 19679

This theorem is referenced by:  pj1ghm2  19746  dpjghm  20107  pj1lmhm  21122