Theorem gim0to0 19192

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 = (0g‘𝑆)
gim0to0.0	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
gim0to0	⊢ ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 ))

Proof of Theorem gim0to0

Step	Hyp	Ref	Expression
1		gimghm 19187	. . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
2		gim0to0.a	. . . . . . 7 ⊢ 𝐴 = (Base‘𝑅)
3		gim0to0.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
4	2, 3	gimf1o 19186	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐴–1-1-onto→𝐵)
5		f1of1 6825	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵)
6	4, 5	syl 17	. . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐴–1-1→𝐵)
7	1, 6	jca 511	. . . 4 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵))
8	7	anim1i 614	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑋 ∈ 𝐴))
9		df-3an 1086	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ↔ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑋 ∈ 𝐴))
10	8, 9	sylibr 233	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋 ∈ 𝐴) → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴))
11		gim0to0.0	. . 3 ⊢ 0 = (0g‘𝑅)
12		gim0to0.n	. . 3 ⊢ 𝑁 = (0g‘𝑆)
13	2, 3, 11, 12	f1ghm0to0 19168	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 ))
14	10, 13	syl 17	1 ⊢ ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  –1-1→wf1 6533  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7404  Basecbs 17151  0_gc0g 17392   GrpHom cghm 19136   GrpIso cgim 19180

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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-0g 17394  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-grp 18864  df-ghm 19137  df-gim 19182

This theorem is referenced by: (None)