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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 17485 | . . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp) | |
2 | eqid 2610 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
3 | ghmid.y | . . . . . . 7 ⊢ 𝑌 = (0g‘𝑆) | |
4 | 2, 3 | grpidcl 17273 | . . . . . 6 ⊢ (𝑆 ∈ Grp → 𝑌 ∈ (Base‘𝑆)) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑌 ∈ (Base‘𝑆)) |
6 | eqid 2610 | . . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
7 | eqid 2610 | . . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
8 | 2, 6, 7 | ghmlin 17488 | . . . . 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 1418 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g‘𝑆)𝑌)) = ((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌))) |
10 | 2, 6, 3 | grplid 17275 | . . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑌 ∈ (Base‘𝑆)) → (𝑌(+g‘𝑆)𝑌) = 𝑌) |
11 | 1, 5, 10 | syl2anc 691 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑌(+g‘𝑆)𝑌) = 𝑌) |
12 | 11 | fveq2d 6107 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g‘𝑆)𝑌)) = (𝐹‘𝑌)) |
13 | 9, 12 | eqtr3d 2646 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)) = (𝐹‘𝑌)) |
14 | ghmgrp2 17486 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp) | |
15 | eqid 2610 | . . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
16 | 2, 15 | ghmf 17487 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
17 | 16, 5 | ffvelrnd 6268 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘𝑌) ∈ (Base‘𝑇)) |
18 | ghmid.z | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
19 | 15, 7, 18 | grpid 17280 | . . . 4 ⊢ ((𝑇 ∈ Grp ∧ (𝐹‘𝑌) ∈ (Base‘𝑇)) → (((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)) = (𝐹‘𝑌) ↔ 0 = (𝐹‘𝑌))) |
20 | 14, 17, 19 | syl2anc 691 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)) = (𝐹‘𝑌) ↔ 0 = (𝐹‘𝑌))) |
21 | 13, 20 | mpbid 221 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 0 = (𝐹‘𝑌)) |
22 | 21 | eqcomd 2616 | 1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘𝑌) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Grpcgrp 17245 GrpHom cghm 17480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-ghm 17481 |
This theorem is referenced by: ghminv 17490 ghmmhm 17493 ghmpreima 17505 ghmf1 17512 lactghmga 17647 f1rhm0to0 18563 f1rhm0to0ALT 18564 kerf1hrm 18566 srng0 18683 islmhm2 18859 evlslem2 19333 evlslem6 19334 evlslem3 19335 zrh0 19681 chrrhm 19698 zndvds0 19718 ip0l 19800 0mat2pmat 20360 nmolb2d 22332 nmoi 22342 nmoix 22343 nmoleub 22345 nmoleub2lem2 22724 nmhmcn 22728 dchrptlem2 24790 psgnid 29178 nrhmzr 41663 zrinitorngc 41792 |
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