Theorem f1ghm0to0 13550

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 13527	. . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘𝑁) = 0 )
4	3	3ad2ant1 1020	. . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑁) = 0 )
5	4	eqeq2d 2216	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = (𝐹‘𝑁) ↔ (𝐹‘𝑋) = 0 ))
6		simp2 1000	. . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → 𝐹:𝐴–1-1→𝐵)
7		simp3 1001	. . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
8		ghmgrp1 13523	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
9		f1ghm0to0.a	. . . . . . 7 ⊢ 𝐴 = (Base‘𝑅)
10	9, 1	grpidcl 13303	. . . . . 6 ⊢ (𝑅 ∈ Grp → 𝑁 ∈ 𝐴)
11	8, 10	syl 14	. . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑁 ∈ 𝐴)
12	11	3ad2ant1 1020	. . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → 𝑁 ∈ 𝐴)
13		f1veqaeq 5837	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑁 ∈ 𝐴)) → ((𝐹‘𝑋) = (𝐹‘𝑁) → 𝑋 = 𝑁))
14	6, 7, 12, 13	syl12anc 1247	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = (𝐹‘𝑁) → 𝑋 = 𝑁))
15	5, 14	sylbird 170	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 0 → 𝑋 = 𝑁))
16		fveq2 5575	. . . 4 ⊢ (𝑋 = 𝑁 → (𝐹‘𝑋) = (𝐹‘𝑁))
17	16, 4	sylan9eqr 2259	. . 3 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = 𝑁) → (𝐹‘𝑋) = 0 )
18	17	ex 115	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → (𝑋 = 𝑁 → (𝐹‘𝑋) = 0 ))
19	15, 18	impbid 129	1 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 0 ↔ 𝑋 = 𝑁))

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 980   = wceq 1372   ∈ wcel 2175  –1-1→wf1 5267  ‘cfv 5270  (class class class)co 5943  Basecbs 12774  0_gc0g 13030  Grpcgrp 13274   GrpHom cghm 13518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1re 8018  ax-addrcl 8021

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-inn 9036  df-2 9094  df-ndx 12777  df-slot 12778  df-base 12780  df-plusg 12864  df-0g 13032  df-mgm 13130  df-sgrp 13176  df-mnd 13191  df-grp 13277  df-ghm 13519

This theorem is referenced by:  ghmf1  13551  kerf1ghm  13552