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| Mirrors > Home > MPE Home > Th. List > f1ghm0to0 | Structured version Visualization version GIF version | ||
| 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.) | 
| Ref | Expression | 
|---|---|
| f1ghm0to0.a | ⊢ 𝐴 = (Base‘𝑅) | 
| f1ghm0to0.b | ⊢ 𝐵 = (Base‘𝑆) | 
| f1ghm0to0.n | ⊢ 𝑁 = (0g‘𝑅) | 
| f1ghm0to0.0 | ⊢ 0 = (0g‘𝑆) | 
| Ref | Expression | 
|---|---|
| f1ghm0to0 | ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 0 ↔ 𝑋 = 𝑁)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | f1ghm0to0.n | . . . . . 6 ⊢ 𝑁 = (0g‘𝑅) | |
| 2 | f1ghm0to0.0 | . . . . . 6 ⊢ 0 = (0g‘𝑆) | |
| 3 | 1, 2 | ghmid 19241 | . . . . 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 2747 | . . 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 19237 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp) | |
| 9 | f1ghm0to0.a | . . . . . . 7 ⊢ 𝐴 = (Base‘𝑅) | |
| 10 | 9, 1 | grpidcl 18984 | . . . . . 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 7278 | . . . 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 6905 | . . . 4 ⊢ (𝑋 = 𝑁 → (𝐹‘𝑋) = (𝐹‘𝑁)) | |
| 17 | 16, 4 | sylan9eqr 2798 | . . 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)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 –1-1→wf1 6557 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 0gc0g 17485 Grpcgrp 18952 GrpHom cghm 19231 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-map 8869 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-ghm 19232 | 
| This theorem is referenced by: ghmf1 19265 kerf1ghm 19266 gim0to0 19288 | 
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