Theorem f1ghm0to0 19157

Description: If a group homomorphism 𝐹 is injective, it 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, 13-May-2023.)

Hypotheses

Ref	Expression
f1ghm0to0.a	⊢ 𝐴 = (Base‘𝑅)
f1ghm0to0.b	⊢ 𝐵 = (Base‘𝑆)
f1ghm0to0.n	⊢ 𝑁 = (0g‘𝑅)
f1ghm0to0.0	⊢ 0 = (0g‘𝑆)

Assertion

Ref	Expression
f1ghm0to0	⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 0 ↔ 𝑋 = 𝑁))

Proof of Theorem f1ghm0to0

Step	Hyp	Ref	Expression
1		f1ghm0to0.n	. . . . . 6 ⊢ 𝑁 = (0g‘𝑅)
2		f1ghm0to0.0	. . . . . 6 ⊢ 0 = (0g‘𝑆)
3	1, 2	ghmid 19134	. . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘𝑁) = 0 )
4	3	3ad2ant1 1133	. . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑁) = 0 )
5	4	eqeq2d 2742	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = (𝐹‘𝑁) ↔ (𝐹‘𝑋) = 0 ))
6		simp2 1137	. . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → 𝐹:𝐴–1-1→𝐵)
7		simp3 1138	. . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
8		ghmgrp1 19130	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
9		f1ghm0to0.a	. . . . . . 7 ⊢ 𝐴 = (Base‘𝑅)
10	9, 1	grpidcl 18878	. . . . . 6 ⊢ (𝑅 ∈ Grp → 𝑁 ∈ 𝐴)
11	8, 10	syl 17	. . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑁 ∈ 𝐴)
12	11	3ad2ant1 1133	. . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → 𝑁 ∈ 𝐴)
13		f1veqaeq 7190	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑁 ∈ 𝐴)) → ((𝐹‘𝑋) = (𝐹‘𝑁) → 𝑋 = 𝑁))
14	6, 7, 12, 13	syl12anc 836	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = (𝐹‘𝑁) → 𝑋 = 𝑁))
15	5, 14	sylbird 260	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 0 → 𝑋 = 𝑁))
16		fveq2 6822	. . . 4 ⊢ (𝑋 = 𝑁 → (𝐹‘𝑋) = (𝐹‘𝑁))
17	16, 4	sylan9eqr 2788	. . 3 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = 𝑁) → (𝐹‘𝑋) = 0 )
18	17	ex 412	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → (𝑋 = 𝑁 → (𝐹‘𝑋) = 0 ))
19	15, 18	impbid 212	1 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 0 ↔ 𝑋 = 𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  –1-1→wf1 6478  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  0_gc0g 17343  Grpcgrp 18846   GrpHom cghm 19124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752  df-0g 17345  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-grp 18849  df-ghm 19125

This theorem is referenced by:  ghmf1  19158  kerf1ghm  19159  gim0to0  19181