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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 2622 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2622 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
5 | 1, 2, 3, 4 | isghm 17660 | . . 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 480 | . 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 475 | 1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋⟶𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 Grpcgrp 17422 GrpHom cghm 17657 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-ghm 17658 |
This theorem is referenced by: ghmid 17666 ghminv 17667 ghmsub 17668 ghmmhm 17670 ghmmulg 17672 ghmrn 17673 resghm 17676 ghmpreima 17682 ghmeql 17683 ghmnsgima 17684 ghmnsgpreima 17685 ghmeqker 17687 ghmf1 17689 ghmf1o 17690 gimcnv 17709 lactghmga 17824 frgpup3lem 18190 frgpup3 18191 ghmplusg 18249 rhmf 18726 isrhm2d 18728 lmhmf 19034 lmhmpropd 19073 evlslem2 19512 frgpcyg 19922 psgninv 19928 zrhpsgninv 19931 evpmss 19932 psgnevpmb 19933 psgnodpm 19934 zrhpsgnevpm 19937 zrhpsgnodpm 19938 nmoi 22532 nmoix 22533 nmoi2 22534 nmoleub 22535 nmoeq0 22540 nmoco 22541 nmotri 22543 nmods 22548 nghmcn 22549 isrnghmmul 41893 rnghmf 41899 |
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