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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 18360 | . . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp) | |
2 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
3 | ghmid.y | . . . . . . 7 ⊢ 𝑌 = (0g‘𝑆) | |
4 | 2, 3 | grpidcl 18131 | . . . . . 6 ⊢ (𝑆 ∈ Grp → 𝑌 ∈ (Base‘𝑆)) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑌 ∈ (Base‘𝑆)) |
6 | eqid 2821 | . . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
7 | eqid 2821 | . . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
8 | 2, 6, 7 | ghmlin 18363 | . . . . 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 18133 | . . . . . 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 6674 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g‘𝑆)𝑌)) = (𝐹‘𝑌)) |
13 | 9, 12 | eqtr3d 2858 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)) = (𝐹‘𝑌)) |
14 | ghmgrp2 18361 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp) | |
15 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
16 | 2, 15 | ghmf 18362 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
17 | 16, 5 | ffvelrnd 6852 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘𝑌) ∈ (Base‘𝑇)) |
18 | ghmid.z | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
19 | 15, 7, 18 | grpid 18139 | . . . 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 2827 | 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 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 0gc0g 16713 Grpcgrp 18103 GrpHom cghm 18355 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-ghm 18356 |
This theorem is referenced by: ghminv 18365 ghmmhm 18368 ghmpreima 18380 ghmf1 18387 lactghmga 18533 f1ghm0to0 19492 f1rhm0to0OLD 19493 f1rhm0to0ALT 19494 kerf1ghm 19497 kerf1hrmOLD 19498 srng0 19631 islmhm2 19810 evlslem2 20292 evlslem3 20293 evlslem6 20294 zrh0 20661 chrrhm 20678 zndvds0 20697 ip0l 20780 0mat2pmat 21344 nmolb2d 23327 nmoi 23337 nmoix 23338 nmoleub 23340 nmoleub2lem2 23720 nmhmcn 23724 dchrptlem2 25841 psgnid 30739 dimkerim 31023 nrhmzr 44164 zrinitorngc 44291 |
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