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Theorem f1ghm0to0 19160
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 = (0g𝑅)
f1ghm0to0.0 0 = (0g𝑆)
Assertion
Ref Expression
f1ghm0to0 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = 0𝑋 = 𝑁))

Proof of Theorem f1ghm0to0
StepHypRef Expression
1 f1ghm0to0.n . . . . . 6 𝑁 = (0g𝑅)
2 f1ghm0to0.0 . . . . . 6 0 = (0g𝑆)
31, 2ghmid 19137 . . . . 5 (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹𝑁) = 0 )
433ad2ant1 1132 . . . 4 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → (𝐹𝑁) = 0 )
54eqeq2d 2742 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = (𝐹𝑁) ↔ (𝐹𝑋) = 0 ))
6 simp2 1136 . . . 4 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → 𝐹:𝐴1-1𝐵)
7 simp3 1137 . . . 4 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → 𝑋𝐴)
8 ghmgrp1 19133 . . . . . 6 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
9 f1ghm0to0.a . . . . . . 7 𝐴 = (Base‘𝑅)
109, 1grpidcl 18887 . . . . . 6 (𝑅 ∈ Grp → 𝑁𝐴)
118, 10syl 17 . . . . 5 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑁𝐴)
12113ad2ant1 1132 . . . 4 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → 𝑁𝐴)
13 f1veqaeq 7259 . . . 4 ((𝐹:𝐴1-1𝐵 ∧ (𝑋𝐴𝑁𝐴)) → ((𝐹𝑋) = (𝐹𝑁) → 𝑋 = 𝑁))
146, 7, 12, 13syl12anc 834 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = (𝐹𝑁) → 𝑋 = 𝑁))
155, 14sylbird 259 . 2 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = 0𝑋 = 𝑁))
16 fveq2 6892 . . . 4 (𝑋 = 𝑁 → (𝐹𝑋) = (𝐹𝑁))
1716, 4sylan9eqr 2793 . . 3 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑋 = 𝑁) → (𝐹𝑋) = 0 )
1817ex 412 . 2 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → (𝑋 = 𝑁 → (𝐹𝑋) = 0 ))
1915, 18impbid 211 1 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = 0𝑋 = 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1086   = wceq 1540  wcel 2105  1-1wf1 6541  cfv 6544  (class class class)co 7412  Basecbs 17149  0gc0g 17390  Grpcgrp 18856   GrpHom cghm 19128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-0g 17392  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-grp 18859  df-ghm 19129
This theorem is referenced by:  ghmf1  19161  kerf1ghm  19162  gim0to0  19184
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