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| Mirrors > Home > MPE Home > Th. List > ghmid | Structured version Visualization version GIF version | ||
| Description: A homomorphism of groups preserves the identity. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
| Ref | Expression |
|---|---|
| ghmid.y | ⊢ 𝑌 = (0g‘𝑆) |
| ghmid.z | ⊢ 0 = (0g‘𝑇) |
| Ref | Expression |
|---|---|
| ghmid | ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘𝑌) = 0 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ghmgrp1 19236 | . . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp) | |
| 2 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
| 3 | ghmid.y | . . . . . . 7 ⊢ 𝑌 = (0g‘𝑆) | |
| 4 | 2, 3 | grpidcl 18983 | . . . . . 6 ⊢ (𝑆 ∈ Grp → 𝑌 ∈ (Base‘𝑆)) |
| 5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑌 ∈ (Base‘𝑆)) |
| 6 | eqid 2737 | . . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
| 7 | eqid 2737 | . . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
| 8 | 2, 6, 7 | ghmlin 19239 | . . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑌 ∈ (Base‘𝑆) ∧ 𝑌 ∈ (Base‘𝑆)) → (𝐹‘(𝑌(+g‘𝑆)𝑌)) = ((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌))) |
| 9 | 5, 5, 8 | mpd3an23 1465 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g‘𝑆)𝑌)) = ((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌))) |
| 10 | 2, 6, 3 | grplid 18985 | . . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑌 ∈ (Base‘𝑆)) → (𝑌(+g‘𝑆)𝑌) = 𝑌) |
| 11 | 1, 5, 10 | syl2anc 584 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑌(+g‘𝑆)𝑌) = 𝑌) |
| 12 | 11 | fveq2d 6910 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g‘𝑆)𝑌)) = (𝐹‘𝑌)) |
| 13 | 9, 12 | eqtr3d 2779 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)) = (𝐹‘𝑌)) |
| 14 | ghmgrp2 19237 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp) | |
| 15 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
| 16 | 2, 15 | ghmf 19238 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
| 17 | 16, 5 | ffvelcdmd 7105 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘𝑌) ∈ (Base‘𝑇)) |
| 18 | ghmid.z | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
| 19 | 15, 7, 18 | grpid 18993 | . . . 4 ⊢ ((𝑇 ∈ Grp ∧ (𝐹‘𝑌) ∈ (Base‘𝑇)) → (((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)) = (𝐹‘𝑌) ↔ 0 = (𝐹‘𝑌))) |
| 20 | 14, 17, 19 | syl2anc 584 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)) = (𝐹‘𝑌) ↔ 0 = (𝐹‘𝑌))) |
| 21 | 13, 20 | mpbid 232 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 0 = (𝐹‘𝑌)) |
| 22 | 21 | eqcomd 2743 | 1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘𝑌) = 0 ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Grpcgrp 18951 GrpHom cghm 19230 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-ghm 19231 |
| This theorem is referenced by: ghminv 19241 ghmmhm 19244 ghmpreima 19256 f1ghm0to0 19263 kerf1ghm 19265 ghmqusker 19305 lactghmga 19423 nrhmzr 20537 zrinitorngc 20642 imadrhmcl 20798 srng0 20855 islmhm2 21037 zrh0 21524 chrrhm 21546 zndvds0 21569 ip0l 21654 evlslem2 22103 evlslem3 22104 evlslem6 22105 rhmmpl 22387 rhmply1vr1 22391 0mat2pmat 22742 nmolb2d 24739 nmoi 24749 nmoix 24750 nmoleub 24752 nmoleub2lem2 25149 nmhmcn 25153 dchrptlem2 27309 psgnid 33117 dimkerim 33678 lvecendof1f1o 33684 ricdrng1 42538 rhmpsr 42562 |
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