Theorem ghmf1 19360

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.) (Proof shortened by AV, 4-Apr-2025.)

Hypotheses

Ref	Expression
f1ghm0to0.a	⊢ 𝐴 = (Base‘𝑅)
f1ghm0to0.b	⊢ 𝐵 = (Base‘𝑆)
f1ghm0to0.n	⊢ 𝑁 = (0g‘𝑅)
f1ghm0to0.0	⊢ 0 = (0g‘𝑆)

Assertion

Ref	Expression
ghmf1	⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)))

Distinct variable groups:   𝑥, 0   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝑁   𝑥,𝑅   𝑥,𝑆

Proof of Theorem ghmf1

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

Step	Hyp	Ref	Expression
1		f1ghm0to0.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝑅)
2		f1ghm0to0.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
3		f1ghm0to0.n	. . . . . 6 ⊢ 𝑁 = (0g‘𝑅)
4		f1ghm0to0.0	. . . . . 6 ⊢ 0 = (0g‘𝑆)
5	1, 2, 3, 4	f1ghm0to0 19359	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = 𝑁))
6	5	3expa 1136	. . . 4 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = 𝑁))
7	6	biimpd 232	. . 3 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁))
8	7	ralrimiva 3159	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁))
9	1, 2	ghmf 19334	. . . 4 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐴⟶𝐵)
10	9	adantr 486	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) → 𝐹:𝐴⟶𝐵)
11		eqid 2765	. . . . . . . . . 10 ⊢ (-g‘𝑅) = (-g‘𝑅)
12		eqid 2765	. . . . . . . . . 10 ⊢ (-g‘𝑆) = (-g‘𝑆)
13	1, 11, 12	ghmsub 19338	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘(𝑦(-g‘𝑅)𝑧)) = ((𝐹‘𝑦)(-g‘𝑆)(𝐹‘𝑧)))
14	13	3expb 1138	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐹‘(𝑦(-g‘𝑅)𝑧)) = ((𝐹‘𝑦)(-g‘𝑆)(𝐹‘𝑧)))
15	14	adantlr 728	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐹‘(𝑦(-g‘𝑅)𝑧)) = ((𝐹‘𝑦)(-g‘𝑆)(𝐹‘𝑧)))
16	15	eqeq1d 2767	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘(𝑦(-g‘𝑅)𝑧)) = 0 ↔ ((𝐹‘𝑦)(-g‘𝑆)(𝐹‘𝑧)) = 0 ))
17		fveqeq2 6894	. . . . . . . 8 ⊢ (𝑥 = (𝑦(-g‘𝑅)𝑧) → ((𝐹‘𝑥) = 0 ↔ (𝐹‘(𝑦(-g‘𝑅)𝑧)) = 0 ))
18		eqeq1 2769	. . . . . . . 8 ⊢ (𝑥 = (𝑦(-g‘𝑅)𝑧) → (𝑥 = 𝑁 ↔ (𝑦(-g‘𝑅)𝑧) = 𝑁))
19	17, 18	imbi12d 347	. . . . . . 7 ⊢ (𝑥 = (𝑦(-g‘𝑅)𝑧) → (((𝐹‘𝑥) = 0 → 𝑥 = 𝑁) ↔ ((𝐹‘(𝑦(-g‘𝑅)𝑧)) = 0 → (𝑦(-g‘𝑅)𝑧) = 𝑁)))
20		simplr 781	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁))
21		ghmgrp1 19332	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
22	21	adantr 486	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) → 𝑅 ∈ Grp)
23	1, 11	grpsubcl 19130	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑦(-g‘𝑅)𝑧) ∈ 𝐴)
24	23	3expb 1138	. . . . . . . 8 ⊢ ((𝑅 ∈ Grp ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦(-g‘𝑅)𝑧) ∈ 𝐴)
25	22, 24	sylan 592	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦(-g‘𝑅)𝑧) ∈ 𝐴)
26	19, 20, 25	rspcdva 3584	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘(𝑦(-g‘𝑅)𝑧)) = 0 → (𝑦(-g‘𝑅)𝑧) = 𝑁))
27	16, 26	sylbird 263	. . . . 5 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝐹‘𝑦)(-g‘𝑆)(𝐹‘𝑧)) = 0 → (𝑦(-g‘𝑅)𝑧) = 𝑁))
28		ghmgrp2 19333	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑆 ∈ Grp)
29	28	ad2antrr 739	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑆 ∈ Grp)
30	9	ad2antrr 739	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐹:𝐴⟶𝐵)
31		simprl 783	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
32	30, 31	ffvelcdmd 7084	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐹‘𝑦) ∈ 𝐵)
33		simprr 785	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ 𝐴)
34	30, 33	ffvelcdmd 7084	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐹‘𝑧) ∈ 𝐵)
35	2, 4, 12	grpsubeq0 19136	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ (𝐹‘𝑦) ∈ 𝐵 ∧ (𝐹‘𝑧) ∈ 𝐵) → (((𝐹‘𝑦)(-g‘𝑆)(𝐹‘𝑧)) = 0 ↔ (𝐹‘𝑦) = (𝐹‘𝑧)))
36	29, 32, 34, 35	syl3anc 1398	. . . . 5 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝐹‘𝑦)(-g‘𝑆)(𝐹‘𝑧)) = 0 ↔ (𝐹‘𝑦) = (𝐹‘𝑧)))
37	21	ad2antrr 739	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑅 ∈ Grp)
38	1, 3, 11	grpsubeq0 19136	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑦(-g‘𝑅)𝑧) = 𝑁 ↔ 𝑦 = 𝑧))
39	37, 31, 33, 38	syl3anc 1398	. . . . 5 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑦(-g‘𝑅)𝑧) = 𝑁 ↔ 𝑦 = 𝑧))
40	27, 36, 39	3imtr3d 296	. . . 4 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
41	40	ralrimivva 3210	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
42		dff13 7257	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)))
43	10, 41, 42	sylanbrc 595	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) → 𝐹:𝐴–1-1→𝐵)
44	8, 43	impbida 813	1 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ⟶wf 6536  –1-1→wf1 6537  ‘cfv 6540  (class class class)co 7419  Basecbs 17291  0_gc0g 17514  Grpcgrp 19044  -_gcsg 19046   GrpHom cghm 19327

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-map 8832  df-0g 17516  df-mgm 18720  df-sgrp 18809  df-mnd 18825  df-grp 19047  df-minusg 19048  df-sbg 19049  df-ghm 19328

This theorem is used by:  cayleylem2  19527  fidomndrnglem  20926  islindf5  22039  asclf1  43357  pwssplit4  43874