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Theorem ghmf1 19167
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 = (0g𝑅)
f1ghm0to0.0 0 = (0g𝑆)
Assertion
Ref Expression
ghmf1 (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴1-1𝐵 ↔ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 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.
StepHypRef Expression
1 f1ghm0to0.a . . . . . 6 𝐴 = (Base‘𝑅)
2 f1ghm0to0.b . . . . . 6 𝐵 = (Base‘𝑆)
3 f1ghm0to0.n . . . . . 6 𝑁 = (0g𝑅)
4 f1ghm0to0.0 . . . . . 6 0 = (0g𝑆)
51, 2, 3, 4f1ghm0to0 19166 . . . . 5 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑥𝐴) → ((𝐹𝑥) = 0𝑥 = 𝑁))
653expa 1115 . . . 4 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥𝐴) → ((𝐹𝑥) = 0𝑥 = 𝑁))
76biimpd 228 . . 3 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥𝐴) → ((𝐹𝑥) = 0𝑥 = 𝑁))
87ralrimiva 3138 . 2 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) → ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁))
91, 2ghmf 19141 . . . 4 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐴𝐵)
109adantr 480 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) → 𝐹:𝐴𝐵)
11 eqid 2724 . . . . . . . . . 10 (-g𝑅) = (-g𝑅)
12 eqid 2724 . . . . . . . . . 10 (-g𝑆) = (-g𝑆)
131, 11, 12ghmsub 19145 . . . . . . . . 9 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑦𝐴𝑧𝐴) → (𝐹‘(𝑦(-g𝑅)𝑧)) = ((𝐹𝑦)(-g𝑆)(𝐹𝑧)))
14133expb 1117 . . . . . . . 8 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (𝑦𝐴𝑧𝐴)) → (𝐹‘(𝑦(-g𝑅)𝑧)) = ((𝐹𝑦)(-g𝑆)(𝐹𝑧)))
1514adantlr 712 . . . . . . 7 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → (𝐹‘(𝑦(-g𝑅)𝑧)) = ((𝐹𝑦)(-g𝑆)(𝐹𝑧)))
1615eqeq1d 2726 . . . . . 6 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → ((𝐹‘(𝑦(-g𝑅)𝑧)) = 0 ↔ ((𝐹𝑦)(-g𝑆)(𝐹𝑧)) = 0 ))
17 fveqeq2 6891 . . . . . . . 8 (𝑥 = (𝑦(-g𝑅)𝑧) → ((𝐹𝑥) = 0 ↔ (𝐹‘(𝑦(-g𝑅)𝑧)) = 0 ))
18 eqeq1 2728 . . . . . . . 8 (𝑥 = (𝑦(-g𝑅)𝑧) → (𝑥 = 𝑁 ↔ (𝑦(-g𝑅)𝑧) = 𝑁))
1917, 18imbi12d 344 . . . . . . 7 (𝑥 = (𝑦(-g𝑅)𝑧) → (((𝐹𝑥) = 0𝑥 = 𝑁) ↔ ((𝐹‘(𝑦(-g𝑅)𝑧)) = 0 → (𝑦(-g𝑅)𝑧) = 𝑁)))
20 simplr 766 . . . . . . 7 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁))
21 ghmgrp1 19139 . . . . . . . . 9 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
2221adantr 480 . . . . . . . 8 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) → 𝑅 ∈ Grp)
231, 11grpsubcl 18944 . . . . . . . . 9 ((𝑅 ∈ Grp ∧ 𝑦𝐴𝑧𝐴) → (𝑦(-g𝑅)𝑧) ∈ 𝐴)
24233expb 1117 . . . . . . . 8 ((𝑅 ∈ Grp ∧ (𝑦𝐴𝑧𝐴)) → (𝑦(-g𝑅)𝑧) ∈ 𝐴)
2522, 24sylan 579 . . . . . . 7 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → (𝑦(-g𝑅)𝑧) ∈ 𝐴)
2619, 20, 25rspcdva 3605 . . . . . 6 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → ((𝐹‘(𝑦(-g𝑅)𝑧)) = 0 → (𝑦(-g𝑅)𝑧) = 𝑁))
2716, 26sylbird 260 . . . . 5 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → (((𝐹𝑦)(-g𝑆)(𝐹𝑧)) = 0 → (𝑦(-g𝑅)𝑧) = 𝑁))
28 ghmgrp2 19140 . . . . . . 7 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑆 ∈ Grp)
2928ad2antrr 723 . . . . . 6 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → 𝑆 ∈ Grp)
309ad2antrr 723 . . . . . . 7 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → 𝐹:𝐴𝐵)
31 simprl 768 . . . . . . 7 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → 𝑦𝐴)
3230, 31ffvelcdmd 7078 . . . . . 6 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → (𝐹𝑦) ∈ 𝐵)
33 simprr 770 . . . . . . 7 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → 𝑧𝐴)
3430, 33ffvelcdmd 7078 . . . . . 6 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → (𝐹𝑧) ∈ 𝐵)
352, 4, 12grpsubeq0 18950 . . . . . 6 ((𝑆 ∈ Grp ∧ (𝐹𝑦) ∈ 𝐵 ∧ (𝐹𝑧) ∈ 𝐵) → (((𝐹𝑦)(-g𝑆)(𝐹𝑧)) = 0 ↔ (𝐹𝑦) = (𝐹𝑧)))
3629, 32, 34, 35syl3anc 1368 . . . . 5 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → (((𝐹𝑦)(-g𝑆)(𝐹𝑧)) = 0 ↔ (𝐹𝑦) = (𝐹𝑧)))
3721ad2antrr 723 . . . . . 6 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → 𝑅 ∈ Grp)
381, 3, 11grpsubeq0 18950 . . . . . 6 ((𝑅 ∈ Grp ∧ 𝑦𝐴𝑧𝐴) → ((𝑦(-g𝑅)𝑧) = 𝑁𝑦 = 𝑧))
3937, 31, 33, 38syl3anc 1368 . . . . 5 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → ((𝑦(-g𝑅)𝑧) = 𝑁𝑦 = 𝑧))
4027, 36, 393imtr3d 293 . . . 4 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
4140ralrimivva 3192 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) → ∀𝑦𝐴𝑧𝐴 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
42 dff13 7247 . . 3 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐴𝑧𝐴 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧)))
4310, 41, 42sylanbrc 582 . 2 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) → 𝐹:𝐴1-1𝐵)
448, 43impbida 798 1 (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴1-1𝐵 ↔ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3053  wf 6530  1-1wf1 6531  cfv 6534  (class class class)co 7402  Basecbs 17149  0gc0g 17390  Grpcgrp 18859  -gcsg 18861   GrpHom cghm 19134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-0g 17392  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-grp 18862  df-minusg 18863  df-sbg 18864  df-ghm 19135
This theorem is referenced by:  cayleylem2  19329  fidomndrnglem  21215  islindf5  21723  pwssplit4  42381
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