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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 19011 | . . . . . 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 18779 | . . . . . 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 19014 | . . . . 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 1464 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g‘𝑆)𝑌)) = ((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌))) |
10 | 2, 6, 3 | grplid 18781 | . . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑌 ∈ (Base‘𝑆)) → (𝑌(+g‘𝑆)𝑌) = 𝑌) |
11 | 1, 5, 10 | syl2anc 585 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑌(+g‘𝑆)𝑌) = 𝑌) |
12 | 11 | fveq2d 6847 | . . . 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 19012 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp) | |
15 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
16 | 2, 15 | ghmf 19013 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
17 | 16, 5 | ffvelcdmd 7037 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘𝑌) ∈ (Base‘𝑇)) |
18 | ghmid.z | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
19 | 15, 7, 18 | grpid 18787 | . . . 4 ⊢ ((𝑇 ∈ Grp ∧ (𝐹‘𝑌) ∈ (Base‘𝑇)) → (((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)) = (𝐹‘𝑌) ↔ 0 = (𝐹‘𝑌))) |
20 | 14, 17, 19 | syl2anc 585 | . . 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 2743 | 1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘𝑌) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ‘cfv 6497 (class class class)co 7358 Basecbs 17084 +gcplusg 17134 0gc0g 17322 Grpcgrp 18749 GrpHom cghm 19006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-0g 17324 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-grp 18752 df-ghm 19007 |
This theorem is referenced by: ghminv 19016 ghmmhm 19019 ghmpreima 19031 ghmf1 19038 lactghmga 19188 f1ghm0to0 20175 f1rhm0to0ALT 20176 kerf1ghm 20178 srng0 20322 islmhm2 20502 zrh0 20917 chrrhm 20937 zndvds0 20960 ip0l 21043 evlslem2 21492 evlslem3 21493 evlslem6 21494 0mat2pmat 22088 nmolb2d 24085 nmoi 24095 nmoix 24096 nmoleub 24098 nmoleub2lem2 24482 nmhmcn 24486 dchrptlem2 26616 psgnid 31949 ghmqusker 32201 dimkerim 32325 imadrhmcl 40719 ricdrng1 40720 rhmmpl 40744 nrhmzr 46178 zrinitorngc 46305 |
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