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Theorem gim0to0 19230
Description: A group isomorphism 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, 23-May-2023.)
Hypotheses
Ref Expression
gim0to0.a 𝐴 = (Base‘𝑅)
gim0to0.b 𝐵 = (Base‘𝑆)
gim0to0.n 𝑁 = (0g𝑆)
gim0to0.0 0 = (0g𝑅)
Assertion
Ref Expression
gim0to0 ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋𝐴) → ((𝐹𝑋) = 𝑁𝑋 = 0 ))

Proof of Theorem gim0to0
StepHypRef Expression
1 gimghm 19225 . . . . 5 (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
2 gim0to0.a . . . . . . 7 𝐴 = (Base‘𝑅)
3 gim0to0.b . . . . . . 7 𝐵 = (Base‘𝑆)
42, 3gimf1o 19224 . . . . . 6 (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐴1-1-onto𝐵)
5 f1of1 6843 . . . . . 6 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1𝐵)
64, 5syl 17 . . . . 5 (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐴1-1𝐵)
71, 6jca 510 . . . 4 (𝐹 ∈ (𝑅 GrpIso 𝑆) → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵))
87anim1i 613 . . 3 ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋𝐴) → ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑋𝐴))
9 df-3an 1086 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) ↔ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑋𝐴))
108, 9sylibr 233 . 2 ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋𝐴) → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴))
11 gim0to0.0 . . 3 0 = (0g𝑅)
12 gim0to0.n . . 3 𝑁 = (0g𝑆)
132, 3, 11, 12f1ghm0to0 19206 . 2 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = 𝑁𝑋 = 0 ))
1410, 13syl 17 1 ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋𝐴) → ((𝐹𝑋) = 𝑁𝑋 = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  1-1wf1 6550  1-1-ontowf1o 6552  cfv 6553  (class class class)co 7426  Basecbs 17187  0gc0g 17428   GrpHom cghm 19174   GrpIso cgim 19218
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-grp 18900  df-ghm 19175  df-gim 19220
This theorem is referenced by: (None)
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