Theorem ghmf1 13229

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 13228	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = 𝑁))
6	5	3expa 1205	. . . 4 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = 𝑁))
7	6	biimpd 144	. . 3 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁))
8	7	ralrimiva 2563	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁))
9	1, 2	ghmf 13203	. . . 4 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐴⟶𝐵)
10	9	adantr 276	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) → 𝐹:𝐴⟶𝐵)
11		eqid 2189	. . . . . . . . . 10 ⊢ (-g‘𝑅) = (-g‘𝑅)
12		eqid 2189	. . . . . . . . . 10 ⊢ (-g‘𝑆) = (-g‘𝑆)
13	1, 11, 12	ghmsub 13207	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘(𝑦(-g‘𝑅)𝑧)) = ((𝐹‘𝑦)(-g‘𝑆)(𝐹‘𝑧)))
14	13	3expb 1206	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐹‘(𝑦(-g‘𝑅)𝑧)) = ((𝐹‘𝑦)(-g‘𝑆)(𝐹‘𝑧)))
15	14	adantlr 477	. . . . . . 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 2198	. . . . . 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 5543	. . . . . . . 8 ⊢ (𝑥 = (𝑦(-g‘𝑅)𝑧) → ((𝐹‘𝑥) = 0 ↔ (𝐹‘(𝑦(-g‘𝑅)𝑧)) = 0 ))
18		eqeq1 2196	. . . . . . . 8 ⊢ (𝑥 = (𝑦(-g‘𝑅)𝑧) → (𝑥 = 𝑁 ↔ (𝑦(-g‘𝑅)𝑧) = 𝑁))
19	17, 18	imbi12d 234	. . . . . . 7 ⊢ (𝑥 = (𝑦(-g‘𝑅)𝑧) → (((𝐹‘𝑥) = 0 → 𝑥 = 𝑁) ↔ ((𝐹‘(𝑦(-g‘𝑅)𝑧)) = 0 → (𝑦(-g‘𝑅)𝑧) = 𝑁)))
20		simplr 528	. . . . . . 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 13201	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
22	21	adantr 276	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) → 𝑅 ∈ Grp)
23	1, 11	grpsubcl 13039	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑦(-g‘𝑅)𝑧) ∈ 𝐴)
24	23	3expb 1206	. . . . . . . 8 ⊢ ((𝑅 ∈ Grp ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦(-g‘𝑅)𝑧) ∈ 𝐴)
25	22, 24	sylan 283	. . . . . . 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 2861	. . . . . 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 170	. . . . 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 13202	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑆 ∈ Grp)
29	28	ad2antrr 488	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑆 ∈ Grp)
30	9	ad2antrr 488	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐹:𝐴⟶𝐵)
31		simprl 529	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
32	30, 31	ffvelcdmd 5673	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐹‘𝑦) ∈ 𝐵)
33		simprr 531	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ 𝐴)
34	30, 33	ffvelcdmd 5673	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐹‘𝑧) ∈ 𝐵)
35	2, 4, 12	grpsubeq0 13045	. . . . . 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 1249	. . . . 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 488	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑅 ∈ Grp)
38	1, 3, 11	grpsubeq0 13045	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑦(-g‘𝑅)𝑧) = 𝑁 ↔ 𝑦 = 𝑧))
39	37, 31, 33, 38	syl3anc 1249	. . . . 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 202	. . . 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 2572	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
42		dff13 5790	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)))
43	10, 41, 42	sylanbrc 417	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) → 𝐹:𝐴–1-1→𝐵)
44	8, 43	impbida 596	1 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2160  ∀wral 2468  ⟶wf 5231  –1-1→wf1 5232  ‘cfv 5235  (class class class)co 5897  Basecbs 12515  0_gc0g 12764  Grpcgrp 12960  -_gcsg 12962   GrpHom cghm 13196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1re 7936  ax-addrcl 7939

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-inn 8951  df-2 9009  df-ndx 12518  df-slot 12519  df-base 12521  df-plusg 12605  df-0g 12766  df-mgm 12835  df-sgrp 12880  df-mnd 12893  df-grp 12963  df-minusg 12964  df-sbg 12965  df-ghm 13197

This theorem is referenced by: (None)