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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 18298 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp) | |
2 | grpmnd 18048 | . . 3 ⊢ (𝑆 ∈ Grp → 𝑆 ∈ Mnd) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Mnd) |
4 | ghmgrp2 18299 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp) | |
5 | grpmnd 18048 | . . 3 ⊢ (𝑇 ∈ Grp → 𝑇 ∈ Mnd) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Mnd) |
7 | eqid 2818 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
8 | eqid 2818 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
9 | 7, 8 | ghmf 18300 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
10 | eqid 2818 | . . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
11 | eqid 2818 | . . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
12 | 7, 10, 11 | ghmlin 18301 | . . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))) |
13 | 12 | 3expb 1112 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))) |
14 | 13 | ralrimivva 3188 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))) |
15 | eqid 2818 | . . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
16 | eqid 2818 | . . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
17 | 15, 16 | ghmid 18302 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇)) |
18 | 9, 14, 17 | 3jca 1120 | . 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))) |
19 | 7, 8, 10, 11, 15, 16 | ismhm 17946 | . 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)))) |
20 | 3, 6, 18, 19 | syl21anbrc 1336 | 1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 0gc0g 16701 Mndcmnd 17899 MndHom cmhm 17942 Grpcgrp 18041 GrpHom cghm 18293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-grp 18044 df-ghm 18294 |
This theorem is referenced by: ghmmhmb 18307 ghmmulg 18308 resghm2 18313 ghmco 18316 ghmeql 18319 symgtrinv 18529 frgpup3lem 18832 gsummulglem 18990 gsumzinv 18994 gsuminv 18995 gsummulc1 19285 gsummulc2 19286 pwsco2rhm 19420 gsumvsmul 19627 evlslem2 20220 evlsgsumadd 20232 evls1gsumadd 20415 zrhpsgnmhm 20656 mat2pmatmul 21267 pm2mp 21361 cayhamlem4 21424 tsmsinv 22683 plypf1 24729 amgmlem 25494 lgseisenlem4 25881 gsumvsmul1 30616 mendring 39670 amgmwlem 44831 amgmlemALT 44832 |
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