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| Mirrors > Home > MPE Home > Th. List > ghmf | Structured version Visualization version GIF version | ||
| Description: A group homomorphism is a function. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
| Ref | Expression |
|---|---|
| ghmf.x | ⊢ 𝑋 = (Base‘𝑆) |
| ghmf.y | ⊢ 𝑌 = (Base‘𝑇) |
| Ref | Expression |
|---|---|
| ghmf | ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋⟶𝑌) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ghmf.x | . . . 4 ⊢ 𝑋 = (Base‘𝑆) | |
| 2 | ghmf.y | . . . 4 ⊢ 𝑌 = (Base‘𝑇) | |
| 3 | eqid 2737 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
| 4 | eqid 2737 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
| 5 | 1, 2, 3, 4 | isghm 19233 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝐹‘(𝑦(+g‘𝑆)𝑥)) = ((𝐹‘𝑦)(+g‘𝑇)(𝐹‘𝑥))))) |
| 6 | 5 | simprbi 496 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝐹‘(𝑦(+g‘𝑆)𝑥)) = ((𝐹‘𝑦)(+g‘𝑇)(𝐹‘𝑥)))) |
| 7 | 6 | simpld 494 | 1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋⟶𝑌) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Grpcgrp 18951 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-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-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-ghm 19231 |
| This theorem is referenced by: ghmid 19240 ghminv 19241 ghmsub 19242 ghmmhm 19244 ghmmulg 19246 ghmrn 19247 resghm 19250 ghmpreima 19256 ghmeql 19257 ghmnsgima 19258 ghmnsgpreima 19259 ghmeqker 19261 ghmf1 19264 kerf1ghm 19265 ghmf1o 19266 gimcnv 19285 ghmqusnsglem1 19298 ghmqusnsg 19300 ghmquskerlem1 19301 ghmquskerco 19302 ghmquskerlem3 19304 ghmqusker 19305 lactghmga 19423 frgpup3lem 19795 frgpup3 19796 ghmplusg 19864 isrnghmmul 20442 rnghmf 20448 rhmf 20485 isrhm2d 20487 lmhmf 21033 lmhmpropd 21072 frgpcyg 21592 psgninv 21600 zrhpsgninv 21603 evpmss 21604 psgnevpmb 21605 psgnodpm 21606 zrhpsgnevpm 21609 zrhpsgnodpm 21610 evlslem2 22103 nmoi 24749 nmoix 24750 nmoi2 24751 nmoleub 24752 nmoeq0 24757 nmoco 24758 nmotri 24760 nmods 24765 nghmcn 24766 aks6d1c1p2 42110 aks6d1c1p3 42111 aks6d1c5lem1 42137 |
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