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Mirrors > Home > MPE Home > Th. List > ghmmhm | Structured version Visualization version GIF version |
Description: A group homomorphism is a monoid homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
Ref | Expression |
---|---|
ghmmhm | ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ghmgrp1 19248 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp) | |
2 | 1 | grpmndd 18976 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Mnd) |
3 | ghmgrp2 19249 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp) | |
4 | 3 | grpmndd 18976 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Mnd) |
5 | eqid 2734 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
6 | eqid 2734 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
7 | 5, 6 | ghmf 19250 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
8 | eqid 2734 | . . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
9 | eqid 2734 | . . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
10 | 5, 8, 9 | ghmlin 19251 | . . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))) |
11 | 10 | 3expb 1119 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))) |
12 | 11 | ralrimivva 3199 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))) |
13 | eqid 2734 | . . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
14 | eqid 2734 | . . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
15 | 13, 14 | ghmid 19252 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇)) |
16 | 7, 12, 15 | 3jca 1127 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) |
17 | 5, 6, 8, 9, 13, 14 | ismhm 18810 | . 2 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇)))) |
18 | 2, 4, 16, 17 | syl21anbrc 1343 | 1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 +gcplusg 17297 0gc0g 17485 Mndcmnd 18759 MndHom cmhm 18806 GrpHom cghm 19242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-map 8866 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-grp 18966 df-ghm 19243 |
This theorem is referenced by: ghmmhmb 19257 ghmmulg 19258 resghm2 19263 ghmco 19266 ghmeql 19269 symgtrinv 19504 frgpup3lem 19809 gsummulglem 19973 gsumzinv 19977 gsuminv 19978 gsummulc1OLD 20327 gsummulc2OLD 20328 gsummulc1 20329 gsummulc2 20330 pwsco2rhm 20519 gsumvsmul 20940 rhmpreimaidl 21304 zrhpsgnmhm 21619 evlslem2 22120 evlsgsumadd 22132 evls1gsumadd 22343 rhmcomulmpl 22401 rhmmpl 22402 rhmply1vsca 22407 mat2pmatmul 22752 pm2mp 22846 cayhamlem4 22909 tsmsinv 24171 plypf1 26265 amgmlem 27047 lgseisenlem4 27436 gsumvsmul1 33036 gsummulgc2 33045 algextdeglem8 33729 rhmcomulpsr 42537 rhmpsr 42538 selvcllem4 42567 selvvvval 42571 evlselv 42573 selvadd 42574 selvmul 42575 mendring 43176 amgmwlem 49032 amgmlemALT 49033 |
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