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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 19088 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp) | |
2 | 1 | grpmndd 18828 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Mnd) |
3 | ghmgrp2 19089 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp) | |
4 | 3 | grpmndd 18828 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Mnd) |
5 | eqid 2733 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
6 | eqid 2733 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
7 | 5, 6 | ghmf 19090 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
8 | eqid 2733 | . . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
9 | eqid 2733 | . . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
10 | 5, 8, 9 | ghmlin 19091 | . . . . 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 1121 | . . . 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 3201 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))) |
13 | eqid 2733 | . . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
14 | eqid 2733 | . . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
15 | 13, 14 | ghmid 19092 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇)) |
16 | 7, 12, 15 | 3jca 1129 | . 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 18669 | . 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 1345 | 1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 +gcplusg 17193 0gc0g 17381 Mndcmnd 18621 MndHom cmhm 18665 GrpHom cghm 19083 |
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 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8818 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-grp 18818 df-ghm 19084 |
This theorem is referenced by: ghmmhmb 19097 ghmmulg 19098 resghm2 19103 ghmco 19106 ghmeql 19109 symgtrinv 19333 frgpup3lem 19638 gsummulglem 19801 gsumzinv 19805 gsuminv 19806 gsummulc1OLD 20116 gsummulc2OLD 20117 gsummulc1 20118 gsummulc2 20119 pwsco2rhm 20267 gsumvsmul 20524 zrhpsgnmhm 21121 evlslem2 21624 evlsgsumadd 21636 evls1gsumadd 21825 mat2pmatmul 22215 pm2mp 22309 cayhamlem4 22372 tsmsinv 23634 plypf1 25708 amgmlem 26474 lgseisenlem4 26861 gsumvsmul1 32181 rhmpreimaidl 32499 rhmcomulmpl 41074 rhmmpl 41075 selvcllem4 41103 selvvvval 41107 evlselv 41109 selvadd 41110 selvmul 41111 mendring 41867 amgmwlem 47751 amgmlemALT 47752 |
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