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| Mirrors > Home > ILE Home > Th. List > f1ghm0to0 | 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 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)) |
| Colors of variables: wff set class |
| 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 0gc0g 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 |
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