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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 18362 | . . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp) | |
2 | eqid 2823 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
3 | ghmid.y | . . . . . . 7 ⊢ 𝑌 = (0g‘𝑆) | |
4 | 2, 3 | grpidcl 18133 | . . . . . 6 ⊢ (𝑆 ∈ Grp → 𝑌 ∈ (Base‘𝑆)) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑌 ∈ (Base‘𝑆)) |
6 | eqid 2823 | . . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
7 | eqid 2823 | . . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
8 | 2, 6, 7 | ghmlin 18365 | . . . . 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 18135 | . . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑌 ∈ (Base‘𝑆)) → (𝑌(+g‘𝑆)𝑌) = 𝑌) |
11 | 1, 5, 10 | syl2anc 586 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑌(+g‘𝑆)𝑌) = 𝑌) |
12 | 11 | fveq2d 6676 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g‘𝑆)𝑌)) = (𝐹‘𝑌)) |
13 | 9, 12 | eqtr3d 2860 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)) = (𝐹‘𝑌)) |
14 | ghmgrp2 18363 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp) | |
15 | eqid 2823 | . . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
16 | 2, 15 | ghmf 18364 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
17 | 16, 5 | ffvelrnd 6854 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘𝑌) ∈ (Base‘𝑇)) |
18 | ghmid.z | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
19 | 15, 7, 18 | grpid 18141 | . . . 4 ⊢ ((𝑇 ∈ Grp ∧ (𝐹‘𝑌) ∈ (Base‘𝑇)) → (((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)) = (𝐹‘𝑌) ↔ 0 = (𝐹‘𝑌))) |
20 | 14, 17, 19 | syl2anc 586 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)) = (𝐹‘𝑌) ↔ 0 = (𝐹‘𝑌))) |
21 | 13, 20 | mpbid 234 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 0 = (𝐹‘𝑌)) |
22 | 21 | eqcomd 2829 | 1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘𝑌) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 0gc0g 16715 Grpcgrp 18105 GrpHom cghm 18357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-ghm 18358 |
This theorem is referenced by: ghminv 18367 ghmmhm 18370 ghmpreima 18382 ghmf1 18389 lactghmga 18535 f1ghm0to0 19494 f1rhm0to0OLD 19495 f1rhm0to0ALT 19496 kerf1ghm 19499 kerf1hrmOLD 19500 srng0 19633 islmhm2 19812 evlslem2 20294 evlslem3 20295 evlslem6 20296 zrh0 20663 chrrhm 20680 zndvds0 20699 ip0l 20782 0mat2pmat 21346 nmolb2d 23329 nmoi 23339 nmoix 23340 nmoleub 23342 nmoleub2lem2 23722 nmhmcn 23726 dchrptlem2 25843 psgnid 30741 dimkerim 31025 nrhmzr 44151 zrinitorngc 44278 |
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