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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 19236 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp) | |
| 2 | 1 | grpmndd 18964 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Mnd) |
| 3 | ghmgrp2 19237 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp) | |
| 4 | 3 | grpmndd 18964 | . 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 19238 | . . 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 19239 | . . . . 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 3202 | . . 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 19240 | . . 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 18798 | . 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 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Mndcmnd 18747 MndHom cmhm 18794 GrpHom cghm 19230 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-ghm 19231 |
| This theorem is referenced by: ghmmhmb 19245 ghmmulg 19246 resghm2 19251 ghmco 19254 ghmeql 19257 symgtrinv 19490 frgpup3lem 19795 gsummulglem 19959 gsumzinv 19963 gsuminv 19964 gsummulc1OLD 20311 gsummulc2OLD 20312 gsummulc1 20313 gsummulc2 20314 pwsco2rhm 20503 gsumvsmul 20924 rhmpreimaidl 21287 zrhpsgnmhm 21602 evlslem2 22103 evlsgsumadd 22115 evls1gsumadd 22328 rhmcomulmpl 22386 rhmmpl 22387 rhmply1vsca 22392 mat2pmatmul 22737 pm2mp 22831 cayhamlem4 22894 tsmsinv 24156 plypf1 26251 amgmlem 27033 lgseisenlem4 27422 gsumvsmul1 33054 gsummulgc2 33063 algextdeglem8 33765 rhmcomulpsr 42561 rhmpsr 42562 selvcllem4 42591 selvvvval 42595 evlselv 42597 selvadd 42598 selvmul 42599 mendring 43200 amgmwlem 49321 amgmlemALT 49322 |
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