MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gim0to0 Structured version   Visualization version   GIF version

Theorem gim0to0 20273
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
gim0to0ALT.a 𝐴 = (Base‘𝑅)
gim0to0ALT.b 𝐵 = (Base‘𝑆)
gim0to0ALT.n 𝑁 = (0g𝑆)
gim0to0ALT.0 0 = (0g𝑅)
Assertion
Ref Expression
gim0to0 ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋𝐴) → ((𝐹𝑋) = 𝑁𝑋 = 0 ))

Proof of Theorem gim0to0
StepHypRef Expression
1 gimghm 19132 . . . . 5 (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
2 gim0to0ALT.a . . . . . . 7 𝐴 = (Base‘𝑅)
3 gim0to0ALT.b . . . . . . 7 𝐵 = (Base‘𝑆)
42, 3gimf1o 19131 . . . . . 6 (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐴1-1-onto𝐵)
5 f1of1 6829 . . . . . 6 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1𝐵)
64, 5syl 17 . . . . 5 (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐴1-1𝐵)
71, 6jca 512 . . . 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 233 . 2 ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋𝐴) → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴))
11 gim0to0ALT.n . . 3 𝑁 = (0g𝑆)
12 gim0to0ALT.0 . . 3 0 = (0g𝑅)
132, 3, 11, 12f1ghm0to0 20271 . 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 396  w3a 1087   = wceq 1541  wcel 2106  1-1wf1 6537  1-1-ontowf1o 6539  cfv 6540  (class class class)co 7405  Basecbs 17140  0gc0g 17381   GrpHom cghm 19083   GrpIso cgim 19125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-ghm 19084  df-gim 19127
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator