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Theorem ghmf1 19199
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 19198 . . . . 5 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑥𝐴) → ((𝐹𝑥) = 0𝑥 = 𝑁))
653expa 1116 . . . 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 3143 . 2 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) → ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁))
91, 2ghmf 19173 . . . 4 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐴𝐵)
109adantr 480 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) → 𝐹:𝐴𝐵)
11 eqid 2728 . . . . . . . . . 10 (-g𝑅) = (-g𝑅)
12 eqid 2728 . . . . . . . . . 10 (-g𝑆) = (-g𝑆)
131, 11, 12ghmsub 19177 . . . . . . . . 9 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑦𝐴𝑧𝐴) → (𝐹‘(𝑦(-g𝑅)𝑧)) = ((𝐹𝑦)(-g𝑆)(𝐹𝑧)))
14133expb 1118 . . . . . . . 8 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (𝑦𝐴𝑧𝐴)) → (𝐹‘(𝑦(-g𝑅)𝑧)) = ((𝐹𝑦)(-g𝑆)(𝐹𝑧)))
1514adantlr 714 . . . . . . 7 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → (𝐹‘(𝑦(-g𝑅)𝑧)) = ((𝐹𝑦)(-g𝑆)(𝐹𝑧)))
1615eqeq1d 2730 . . . . . 6 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → ((𝐹‘(𝑦(-g𝑅)𝑧)) = 0 ↔ ((𝐹𝑦)(-g𝑆)(𝐹𝑧)) = 0 ))
17 fveqeq2 6906 . . . . . . . 8 (𝑥 = (𝑦(-g𝑅)𝑧) → ((𝐹𝑥) = 0 ↔ (𝐹‘(𝑦(-g𝑅)𝑧)) = 0 ))
18 eqeq1 2732 . . . . . . . 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 768 . . . . . . 7 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁))
21 ghmgrp1 19171 . . . . . . . . 9 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
2221adantr 480 . . . . . . . 8 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) → 𝑅 ∈ Grp)
231, 11grpsubcl 18975 . . . . . . . . 9 ((𝑅 ∈ Grp ∧ 𝑦𝐴𝑧𝐴) → (𝑦(-g𝑅)𝑧) ∈ 𝐴)
24233expb 1118 . . . . . . . 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 3610 . . . . . 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 19172 . . . . . . 7 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑆 ∈ Grp)
2928ad2antrr 725 . . . . . 6 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → 𝑆 ∈ Grp)
309ad2antrr 725 . . . . . . 7 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → 𝐹:𝐴𝐵)
31 simprl 770 . . . . . . 7 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → 𝑦𝐴)
3230, 31ffvelcdmd 7095 . . . . . 6 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → (𝐹𝑦) ∈ 𝐵)
33 simprr 772 . . . . . . 7 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → 𝑧𝐴)
3430, 33ffvelcdmd 7095 . . . . . 6 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → (𝐹𝑧) ∈ 𝐵)
352, 4, 12grpsubeq0 18981 . . . . . 6 ((𝑆 ∈ Grp ∧ (𝐹𝑦) ∈ 𝐵 ∧ (𝐹𝑧) ∈ 𝐵) → (((𝐹𝑦)(-g𝑆)(𝐹𝑧)) = 0 ↔ (𝐹𝑦) = (𝐹𝑧)))
3629, 32, 34, 35syl3anc 1369 . . . . 5 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → (((𝐹𝑦)(-g𝑆)(𝐹𝑧)) = 0 ↔ (𝐹𝑦) = (𝐹𝑧)))
3721ad2antrr 725 . . . . . 6 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) ∧ (𝑦𝐴𝑧𝐴)) → 𝑅 ∈ Grp)
381, 3, 11grpsubeq0 18981 . . . . . 6 ((𝑅 ∈ Grp ∧ 𝑦𝐴𝑧𝐴) → ((𝑦(-g𝑅)𝑧) = 𝑁𝑦 = 𝑧))
3937, 31, 33, 38syl3anc 1369 . . . . 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 3197 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐴 ((𝐹𝑥) = 0𝑥 = 𝑁)) → ∀𝑦𝐴𝑧𝐴 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
42 dff13 7265 . . 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 800 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 1534  wcel 2099  wral 3058  wf 6544  1-1wf1 6545  cfv 6548  (class class class)co 7420  Basecbs 17179  0gc0g 17420  Grpcgrp 18889  -gcsg 18891   GrpHom cghm 19166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-0g 17422  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18892  df-minusg 18893  df-sbg 18894  df-ghm 19167
This theorem is referenced by:  cayleylem2  19367  fidomndrnglem  21259  islindf5  21772  pwssplit4  42513
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