Theorem pj1ghm 18556

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 2795	. 2 ⊢ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈))) = (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈)))
2		eqid 2795	. 2 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		ovex 7048	. . 3 ⊢ (𝑇 ⊕ 𝑈) ∈ V
4		eqid 2795	. . . 4 ⊢ (𝐺 ↾s (𝑇 ⊕ 𝑈)) = (𝐺 ↾s (𝑇 ⊕ 𝑈))
5		pj1eu.a	. . . 4 ⊢ + = (+g‘𝐺)
6	4, 5	ressplusg 16441	. . 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 18509	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺))
14	8, 9, 10, 13	syl3anc 1364	. . 3 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺))
15	4	subggrp 18036	. . 3 ⊢ ((𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺) → (𝐺 ↾s (𝑇 ⊕ 𝑈)) ∈ Grp)
16	14, 15	syl 17	. 2 ⊢ (𝜑 → (𝐺 ↾s (𝑇 ⊕ 𝑈)) ∈ Grp)
17		subgrcl 18038	. . 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 18550	. . . 4 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
23	2	subgss 18034	. . . . 5 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
24	8, 23	syl 17	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺))
25	22, 24	fssd 6396	. . 3 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶(Base‘𝐺))
26	4	subgbas 18037	. . . . 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 6368	. . 3 ⊢ (𝜑 → ((𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶(Base‘𝐺) ↔ (𝑇𝑃𝑈):(Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈)))⟶(Base‘𝐺)))
29	25, 28	mpbid 233	. 2 ⊢ (𝜑 → (𝑇𝑃𝑈):(Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈)))⟶(Base‘𝐺))
30	27	eleq2d 2868	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ 𝑥 ∈ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈)))))
31	27	eleq2d 2868	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝑇 ⊕ 𝑈) ↔ 𝑦 ∈ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈)))))
32	30, 31	anbi12d 630	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) ↔ (𝑥 ∈ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈))) ∧ 𝑦 ∈ (Base‘(𝐺 ↾s (𝑇 ⊕ 𝑈))))))
33	32	biimpar 478	. . 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 18552	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → 𝑥 = (((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)))
35	34	adantrr 713	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑥 = (((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)))
36	5, 11, 19, 12, 8, 9, 20, 10, 21	pj1id 18552	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦)))
37	36	adantrl 712	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦)))
38	35, 37	oveq12d 7034	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥 + 𝑦) = ((((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)) + (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦))))
39	8	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑇 ∈ (SubGrp‘𝐺))
40		grpmnd 17868	. . . . . . . 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 483	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → 𝑥 ∈ (𝑇 ⊕ 𝑈))
44		ffvelrn 6714	. . . . . . . . 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 3890	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑥) ∈ (Base‘𝐺))
47		simpr 485	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → 𝑦 ∈ (𝑇 ⊕ 𝑈))
48		ffvelrn 6714	. . . . . . . . 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 3890	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (Base‘𝐺))
51	9	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑈 ∈ (SubGrp‘𝐺))
52	2	subgss 18034	. . . . . . . . 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 18551	. . . . . . . . 9 ⊢ (𝜑 → (𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
55		ffvelrn 6714	. . . . . . . . 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 3890	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑥) ∈ (Base‘𝐺))
58		ffvelrn 6714	. . . . . . . . 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 3890	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ (Base‘𝐺))
61	10	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑇 ⊆ (𝑍‘𝑈))
62	61, 49	sseldd 3890	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (𝑍‘𝑈))
63	5, 12	cntzi 18200	. . . . . . . 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 17746	. . . . . 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 2834	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥 + 𝑦) = ((((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) + (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦))))
67	20	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑇 ∩ 𝑈) = { 0 })
68	5	subgcl 18043	. . . . . . . 8 ⊢ (((𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → (𝑥 + 𝑦) ∈ (𝑇 ⊕ 𝑈))
69	68	3expb 1113	. . . . . . 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 18043	. . . . . . 7 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇 ∧ ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇) → (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
72	39, 45, 49, 71	syl3anc 1364	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
73	5	subgcl 18043	. . . . . . 7 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈 ∧ ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈) → (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦)) ∈ 𝑈)
74	51, 56, 59, 73	syl3anc 1364	. . . . . 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 18553	. . . . 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 233	. . . 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 495	. . 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 18108	1 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺 ↾s (𝑇 ⊕ 𝑈)) GrpHom 𝐺))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081  Vcvv 3437   ∩ cin 3858   ⊆ wss 3859  {csn 4472  ⟶wf 6221  ‘cfv 6225  (class class class)co 7016  Basecbs 16312   ↾_s cress 16313  +_gcplusg 16394  0_gc0g 16542  Mndcmnd 17733  Grpcgrp 17861  SubGrpcsubg 18027   GrpHom cghm 18096  Cntzccntz 18186  LSSumclsm 18489  proj₁cpj1 18490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-0g 16544  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-submnd 17775  df-grp 17864  df-minusg 17865  df-sbg 17866  df-subg 18030  df-ghm 18097  df-cntz 18188  df-lsm 18491  df-pj1 18492

This theorem is referenced by:  pj1ghm2  18557  dpjghm  18902  pj1lmhm  19562