![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 19011 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp) | |
2 | 1 | grpmndd 18761 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Mnd) |
3 | ghmgrp2 19012 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp) | |
4 | 3 | grpmndd 18761 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Mnd) |
5 | eqid 2737 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
6 | eqid 2737 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
7 | 5, 6 | ghmf 19013 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
8 | eqid 2737 | . . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
9 | eqid 2737 | . . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
10 | 5, 8, 9 | ghmlin 19014 | . . . . 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 3198 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))) |
13 | eqid 2737 | . . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
14 | eqid 2737 | . . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
15 | 13, 14 | ghmid 19015 | . . 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 18604 | . 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 3065 ⟶wf 6493 ‘cfv 6497 (class class class)co 7358 Basecbs 17084 +gcplusg 17134 0gc0g 17322 Mndcmnd 18557 MndHom cmhm 18600 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-map 8768 df-0g 17324 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-mhm 18602 df-grp 18752 df-ghm 19007 |
This theorem is referenced by: ghmmhmb 19020 ghmmulg 19021 resghm2 19026 ghmco 19029 ghmeql 19032 symgtrinv 19255 frgpup3lem 19560 gsummulglem 19719 gsumzinv 19723 gsuminv 19724 gsummulc1 20031 gsummulc2 20032 pwsco2rhm 20174 gsumvsmul 20389 zrhpsgnmhm 20991 evlslem2 21492 evlsgsumadd 21504 evls1gsumadd 21693 mat2pmatmul 22083 pm2mp 22177 cayhamlem4 22240 tsmsinv 23502 plypf1 25576 amgmlem 26342 lgseisenlem4 26729 gsumvsmul1 31896 rhmpreimaidl 32203 rhmcomulmpl 40743 rhmmpl 40744 selvcllem4 40762 selvadd 40766 selvmul 40767 mendring 41522 amgmwlem 47256 amgmlemALT 47257 |
Copyright terms: Public domain | W3C validator |