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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 19207 | . . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp) | |
2 | eqid 2725 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
3 | ghmid.y | . . . . . . 7 ⊢ 𝑌 = (0g‘𝑆) | |
4 | 2, 3 | grpidcl 18955 | . . . . . 6 ⊢ (𝑆 ∈ Grp → 𝑌 ∈ (Base‘𝑆)) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑌 ∈ (Base‘𝑆)) |
6 | eqid 2725 | . . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
7 | eqid 2725 | . . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
8 | 2, 6, 7 | ghmlin 19210 | . . . . 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 1459 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g‘𝑆)𝑌)) = ((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌))) |
10 | 2, 6, 3 | grplid 18957 | . . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑌 ∈ (Base‘𝑆)) → (𝑌(+g‘𝑆)𝑌) = 𝑌) |
11 | 1, 5, 10 | syl2anc 582 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑌(+g‘𝑆)𝑌) = 𝑌) |
12 | 11 | fveq2d 6904 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g‘𝑆)𝑌)) = (𝐹‘𝑌)) |
13 | 9, 12 | eqtr3d 2767 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)) = (𝐹‘𝑌)) |
14 | ghmgrp2 19208 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp) | |
15 | eqid 2725 | . . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
16 | 2, 15 | ghmf 19209 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
17 | 16, 5 | ffvelcdmd 7098 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘𝑌) ∈ (Base‘𝑇)) |
18 | ghmid.z | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
19 | 15, 7, 18 | grpid 18965 | . . . 4 ⊢ ((𝑇 ∈ Grp ∧ (𝐹‘𝑌) ∈ (Base‘𝑇)) → (((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)) = (𝐹‘𝑌) ↔ 0 = (𝐹‘𝑌))) |
20 | 14, 17, 19 | syl2anc 582 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)) = (𝐹‘𝑌) ↔ 0 = (𝐹‘𝑌))) |
21 | 13, 20 | mpbid 231 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 0 = (𝐹‘𝑌)) |
22 | 21 | eqcomd 2731 | 1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘𝑌) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7423 Basecbs 17208 +gcplusg 17261 0gc0g 17449 Grpcgrp 18923 GrpHom cghm 19201 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-1st 8002 df-2nd 8003 df-map 8856 df-0g 17451 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-grp 18926 df-ghm 19202 |
This theorem is referenced by: ghminv 19212 ghmmhm 19215 ghmpreima 19227 f1ghm0to0 19234 kerf1ghm 19236 ghmqusker 19276 lactghmga 19398 nrhmzr 20514 zrinitorngc 20615 imadrhmcl 20725 srng0 20780 islmhm2 20963 zrh0 21495 chrrhm 21517 zndvds0 21540 ip0l 21624 evlslem2 22086 evlslem3 22087 evlslem6 22088 rhmmpl 22366 rhmply1vr1 22370 0mat2pmat 22721 nmolb2d 24718 nmoi 24728 nmoix 24729 nmoleub 24731 nmoleub2lem2 25126 nmhmcn 25130 dchrptlem2 27286 psgnid 32952 dimkerim 33494 ricdrng1 41948 rhmpsr 41966 |
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