Theorem gim0to0 19183

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 19178	. . . . 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 19177	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐴–1-1-onto→𝐵)
5		f1of1 6781	. . . . . 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 615	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑋 ∈ 𝐴))
9		df-3an 1088	. . 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 234	. 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 19159	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  –1-1→wf1 6496  –1-1-onto→wf1o 6498  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  0_gc0g 17378   GrpHom cghm 19126   GrpIso cgim 19171

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-map 8778  df-0g 17380  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-grp 18850  df-ghm 19127  df-gim 19173

This theorem is referenced by: (None)