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Theorem gim0to0 19300
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 19295 . . . . 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 19294 . . . . . 6 (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐴1-1-onto𝐵)
5 f1of1 6848 . . . . . 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 1088 . . 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 19276 . 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 1086   = wceq 1537  wcel 2106  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  Basecbs 17245  0gc0g 17486   GrpHom cghm 19243   GrpIso cgim 19288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-ghm 19244  df-gim 19290
This theorem is referenced by: (None)
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