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Theorem gim0to0 19287
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 19282 . . . . 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 19281 . . . . . 6 (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐴1-1-onto𝐵)
5 f1of1 6847 . . . . . 6 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1𝐵)
64, 5syl 17 . . . . 5 (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐴1-1𝐵)
71, 6jca 511 . . . 4 (𝐹 ∈ (𝑅 GrpIso 𝑆) → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵))
87anim1i 615 . . 3 ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋𝐴) → ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑋𝐴))
9 df-3an 1089 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) ↔ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑋𝐴))
108, 9sylibr 234 . 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 19263 . 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Basecbs 17247  0gc0g 17484   GrpHom cghm 19230   GrpIso cgim 19275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-ghm 19231  df-gim 19277
This theorem is referenced by: (None)
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