Theorem pj1ghm 19617

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 7402	. . 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 17230	. . 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 19568	. . . 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 19043	. . 3 ⊢ ((𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺) → (𝐺 ↾s (𝑇 ⊕ 𝑈)) ∈ Grp)
16	14, 15	syl 17	. 2 ⊢ (𝜑 → (𝐺 ↾s (𝑇 ⊕ 𝑈)) ∈ Grp)
17		subgrcl 19045	. . 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 19611	. . . 4 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
23	2	subgss 19041	. . . . 5 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
24	8, 23	syl 17	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺))
25	22, 24	fssd 6687	. . 3 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶(Base‘𝐺))
26	4	subgbas 19044	. . . . 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 6654	. . 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 19613	. . . . . . . 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 19613	. . . . . . . 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 7387	. . . . . 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 18854	. . . . . . . 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 7035	. . . . . . . . 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 3944	. . . . . . 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 7035	. . . . . . . . 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 3944	. . . . . . 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 19041	. . . . . . . . 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 19612	. . . . . . . . 9 ⊢ (𝜑 → (𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
55		ffvelcdm 7035	. . . . . . . . 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 3944	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑥) ∈ (Base‘𝐺))
58		ffvelcdm 7035	. . . . . . . . 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 3944	. . . . . . 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 3944	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (𝑍‘𝑈))
63	5, 12	cntzi 19243	. . . . . . . 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 18657	. . . . . 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 19050	. . . . . . . 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 19050	. . . . . . 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 19050	. . . . . . 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 19614	. . . . 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 19139	1 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺 ↾s (𝑇 ⊕ 𝑈)) GrpHom 𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911  {csn 4585  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  Basecbs 17155   ↾_s cress 17176  +_gcplusg 17196  0_gc0g 17378  Mndcmnd 18643  Grpcgrp 18847  SubGrpcsubg 19034   GrpHom cghm 19126  Cntzccntz 19229  LSSumclsm 19548  proj₁cpj1 19549

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

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 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-0g 17380  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-submnd 18693  df-grp 18850  df-minusg 18851  df-sbg 18852  df-subg 19037  df-ghm 19127  df-cntz 19231  df-lsm 19550  df-pj1 19551

This theorem is referenced by:  pj1ghm2  19618  dpjghm  19979  pj1lmhm  21039