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Theorem f1ghm0to0 19492
 Description: If a group homomorphism 𝐹 is injective, it maps the zero of one group (and only the zero) to the zero of the other group. (Contributed by AV, 24-Oct-2019.) (Revised by Thierry Arnoux, 13-May-2023.)
Hypotheses
Ref Expression
f1ghm0to0.a 𝐴 = (Base‘𝑅)
f1ghm0to0.b 𝐵 = (Base‘𝑆)
f1ghm0to0.n 𝑁 = (0g𝑆)
f1ghm0to0.1 0 = (0g𝑅)
Assertion
Ref Expression
f1ghm0to0 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = 𝑁𝑋 = 0 ))

Proof of Theorem f1ghm0to0
StepHypRef Expression
1 f1ghm0to0.1 . . . . . 6 0 = (0g𝑅)
2 f1ghm0to0.n . . . . . 6 𝑁 = (0g𝑆)
31, 2ghmid 18360 . . . . 5 (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹0 ) = 𝑁)
433ad2ant1 1130 . . . 4 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → (𝐹0 ) = 𝑁)
54eqeq2d 2809 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = (𝐹0 ) ↔ (𝐹𝑋) = 𝑁))
6 simp2 1134 . . . 4 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → 𝐹:𝐴1-1𝐵)
7 simp3 1135 . . . 4 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → 𝑋𝐴)
8 ghmgrp1 18356 . . . . . 6 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
9 f1ghm0to0.a . . . . . . 7 𝐴 = (Base‘𝑅)
109, 1grpidcl 18127 . . . . . 6 (𝑅 ∈ Grp → 0𝐴)
118, 10syl 17 . . . . 5 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 0𝐴)
12113ad2ant1 1130 . . . 4 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → 0𝐴)
13 f1veqaeq 6994 . . . 4 ((𝐹:𝐴1-1𝐵 ∧ (𝑋𝐴0𝐴)) → ((𝐹𝑋) = (𝐹0 ) → 𝑋 = 0 ))
146, 7, 12, 13syl12anc 835 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = (𝐹0 ) → 𝑋 = 0 ))
155, 14sylbird 263 . 2 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = 𝑁𝑋 = 0 ))
16 fveq2 6646 . . . 4 (𝑋 = 0 → (𝐹𝑋) = (𝐹0 ))
1716, 4sylan9eqr 2855 . . 3 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑋 = 0 ) → (𝐹𝑋) = 𝑁)
1817ex 416 . 2 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → (𝑋 = 0 → (𝐹𝑋) = 𝑁))
1915, 18impbid 215 1 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = 𝑁𝑋 = 0 ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  –1-1→wf1 6322  ‘cfv 6325  (class class class)co 7136  Basecbs 16478  0gc0g 16708  Grpcgrp 18098   GrpHom cghm 18351 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-grp 18101  df-ghm 18352 This theorem is referenced by:  gim0to0  19494  kerf1ghm  19495
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