Theorem ghmf1 17689

Description: Two ways of saying a group homomorphism is 1-1 into its codomain. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)

Hypotheses

Ref	Expression
ghmf1.x	⊢ 𝑋 = (Base‘𝑆)
ghmf1.y	⊢ 𝑌 = (Base‘𝑇)
ghmf1.z	⊢ 0 = (0g‘𝑆)
ghmf1.u	⊢ 𝑈 = (0g‘𝑇)

Assertion

Ref	Expression
ghmf1	⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋–1-1→𝑌 ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )))

Distinct variable groups:   𝑥,𝐹   𝑥,𝑆   𝑥,𝑇   𝑥,𝑈   𝑥,𝑋   𝑥,𝑌   𝑥, 0

Proof of Theorem ghmf1

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

Step	Hyp	Ref	Expression
1		ghmf1.z	. . . . . . . 8 ⊢ 0 = (0g‘𝑆)
2		ghmf1.u	. . . . . . . 8 ⊢ 𝑈 = (0g‘𝑇)
3	1, 2	ghmid 17666	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘ 0 ) = 𝑈)
4	3	ad2antrr 762	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → (𝐹‘ 0 ) = 𝑈)
5	4	eqeq2d 2632	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = (𝐹‘ 0 ) ↔ (𝐹‘𝑥) = 𝑈))
6		simplr 792	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐹:𝑋–1-1→𝑌)
7		simpr 477	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
8		ghmgrp1 17662	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
9	8	ad2antrr 762	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑆 ∈ Grp)
10		ghmf1.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝑆)
11	10, 1	grpidcl 17450	. . . . . . 7 ⊢ (𝑆 ∈ Grp → 0 ∈ 𝑋)
12	9, 11	syl 17	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → 0 ∈ 𝑋)
13		f1fveq 6519	. . . . . 6 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝑥 ∈ 𝑋 ∧ 0 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘ 0 ) ↔ 𝑥 = 0 ))
14	6, 7, 12, 13	syl12anc 1324	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = (𝐹‘ 0 ) ↔ 𝑥 = 0 ))
15	5, 14	bitr3d 270	. . . 4 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = 𝑈 ↔ 𝑥 = 0 ))
16	15	biimpd 219	. . 3 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 ))
17	16	ralrimiva 2966	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) → ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 ))
18		ghmf1.y	. . . . 5 ⊢ 𝑌 = (Base‘𝑇)
19	10, 18	ghmf 17664	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋⟶𝑌)
20	19	adantr 481	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) → 𝐹:𝑋⟶𝑌)
21		eqid 2622	. . . . . . . . . 10 ⊢ (-g‘𝑆) = (-g‘𝑆)
22		eqid 2622	. . . . . . . . . 10 ⊢ (-g‘𝑇) = (-g‘𝑇)
23	10, 21, 22	ghmsub 17668	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝐹‘(𝑦(-g‘𝑆)𝑧)) = ((𝐹‘𝑦)(-g‘𝑇)(𝐹‘𝑧)))
24	23	3expb 1266	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(-g‘𝑆)𝑧)) = ((𝐹‘𝑦)(-g‘𝑇)(𝐹‘𝑧)))
25	24	adantlr 751	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(-g‘𝑆)𝑧)) = ((𝐹‘𝑦)(-g‘𝑇)(𝐹‘𝑧)))
26	25	eqeq1d 2624	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐹‘(𝑦(-g‘𝑆)𝑧)) = 𝑈 ↔ ((𝐹‘𝑦)(-g‘𝑇)(𝐹‘𝑧)) = 𝑈))
27	8	adantr 481	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) → 𝑆 ∈ Grp)
28	10, 21	grpsubcl 17495	. . . . . . . . 9 ⊢ ((𝑆 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(-g‘𝑆)𝑧) ∈ 𝑋)
29	28	3expb 1266	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(-g‘𝑆)𝑧) ∈ 𝑋)
30	27, 29	sylan 488	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(-g‘𝑆)𝑧) ∈ 𝑋)
31		simplr 792	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 ))
32		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦(-g‘𝑆)𝑧) → (𝐹‘𝑥) = (𝐹‘(𝑦(-g‘𝑆)𝑧)))
33	32	eqeq1d 2624	. . . . . . . . 9 ⊢ (𝑥 = (𝑦(-g‘𝑆)𝑧) → ((𝐹‘𝑥) = 𝑈 ↔ (𝐹‘(𝑦(-g‘𝑆)𝑧)) = 𝑈))
34		eqeq1 2626	. . . . . . . . 9 ⊢ (𝑥 = (𝑦(-g‘𝑆)𝑧) → (𝑥 = 0 ↔ (𝑦(-g‘𝑆)𝑧) = 0 ))
35	33, 34	imbi12d 334	. . . . . . . 8 ⊢ (𝑥 = (𝑦(-g‘𝑆)𝑧) → (((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 ) ↔ ((𝐹‘(𝑦(-g‘𝑆)𝑧)) = 𝑈 → (𝑦(-g‘𝑆)𝑧) = 0 )))
36	35	rspcv 3305	. . . . . . 7 ⊢ ((𝑦(-g‘𝑆)𝑧) ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 ) → ((𝐹‘(𝑦(-g‘𝑆)𝑧)) = 𝑈 → (𝑦(-g‘𝑆)𝑧) = 0 )))
37	30, 31, 36	sylc 65	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐹‘(𝑦(-g‘𝑆)𝑧)) = 𝑈 → (𝑦(-g‘𝑆)𝑧) = 0 ))
38	26, 37	sylbird 250	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((𝐹‘𝑦)(-g‘𝑇)(𝐹‘𝑧)) = 𝑈 → (𝑦(-g‘𝑆)𝑧) = 0 ))
39		ghmgrp2 17663	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
40	39	ad2antrr 762	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑇 ∈ Grp)
41	19	ad2antrr 762	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝐹:𝑋⟶𝑌)
42		simprl 794	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
43	41, 42	ffvelrnd 6360	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑦) ∈ 𝑌)
44		simprr 796	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
45	41, 44	ffvelrnd 6360	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑧) ∈ 𝑌)
46	18, 2, 22	grpsubeq0 17501	. . . . . 6 ⊢ ((𝑇 ∈ Grp ∧ (𝐹‘𝑦) ∈ 𝑌 ∧ (𝐹‘𝑧) ∈ 𝑌) → (((𝐹‘𝑦)(-g‘𝑇)(𝐹‘𝑧)) = 𝑈 ↔ (𝐹‘𝑦) = (𝐹‘𝑧)))
47	40, 43, 45, 46	syl3anc 1326	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((𝐹‘𝑦)(-g‘𝑇)(𝐹‘𝑧)) = 𝑈 ↔ (𝐹‘𝑦) = (𝐹‘𝑧)))
48	8	ad2antrr 762	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑆 ∈ Grp)
49	10, 1, 21	grpsubeq0 17501	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑦(-g‘𝑆)𝑧) = 0 ↔ 𝑦 = 𝑧))
50	48, 42, 44, 49	syl3anc 1326	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦(-g‘𝑆)𝑧) = 0 ↔ 𝑦 = 𝑧))
51	38, 47, 50	3imtr3d 282	. . . 4 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
52	51	ralrimivva 2971	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
53		dff13 6512	. . 3 ⊢ (𝐹:𝑋–1-1→𝑌 ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)))
54	20, 52, 53	sylanbrc 698	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) → 𝐹:𝑋–1-1→𝑌)
55	17, 54	impbida 877	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋–1-1→𝑌 ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ⟶wf 5884  –1-1→wf1 5885  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  0_gc0g 16100  Grpcgrp 17422  -_gcsg 17424   GrpHom cghm 17657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-ghm 17658

This theorem is referenced by:  cayleylem2  17833  f1rhm0to0ALT  18741  fidomndrnglem  19306  islindf5  20178  pwssplit4  37659