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Theorem f1ghm0to0 20257
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 19083 . . . . 5 (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹0 ) = 𝑁)
433ad2ant1 1134 . . . 4 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → (𝐹0 ) = 𝑁)
54eqeq2d 2744 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = (𝐹0 ) ↔ (𝐹𝑋) = 𝑁))
6 simp2 1138 . . . 4 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → 𝐹:𝐴1-1𝐵)
7 simp3 1139 . . . 4 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → 𝑋𝐴)
8 ghmgrp1 19079 . . . . . 6 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
9 f1ghm0to0.a . . . . . . 7 𝐴 = (Base‘𝑅)
109, 1grpidcl 18837 . . . . . 6 (𝑅 ∈ Grp → 0𝐴)
118, 10syl 17 . . . . 5 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 0𝐴)
12113ad2ant1 1134 . . . 4 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → 0𝐴)
13 f1veqaeq 7243 . . . 4 ((𝐹:𝐴1-1𝐵 ∧ (𝑋𝐴0𝐴)) → ((𝐹𝑋) = (𝐹0 ) → 𝑋 = 0 ))
146, 7, 12, 13syl12anc 836 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = (𝐹0 ) → 𝑋 = 0 ))
155, 14sylbird 260 . 2 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = 𝑁𝑋 = 0 ))
16 fveq2 6881 . . . 4 (𝑋 = 0 → (𝐹𝑋) = (𝐹0 ))
1716, 4sylan9eqr 2795 . . 3 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑋 = 0 ) → (𝐹𝑋) = 𝑁)
1817ex 414 . 2 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → (𝑋 = 0 → (𝐹𝑋) = 𝑁))
1915, 18impbid 211 1 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = 𝑁𝑋 = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1088   = wceq 1542  wcel 2107  1-1wf1 6532  cfv 6535  (class class class)co 7396  Basecbs 17131  0gc0g 17372  Grpcgrp 18806   GrpHom cghm 19074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7352  df-ov 7399  df-oprab 7400  df-mpo 7401  df-0g 17374  df-mgm 18548  df-sgrp 18597  df-mnd 18613  df-grp 18809  df-ghm 19075
This theorem is referenced by:  gim0to0  20259  kerf1ghm  20260
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