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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 19009 | . . 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 498 | . 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 496 | 1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋⟶𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ⟶wf 6493 ‘cfv 6497 (class class class)co 7358 Basecbs 17084 +gcplusg 17134 Grpcgrp 18749 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-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-ov 7361 df-oprab 7362 df-mpo 7363 df-ghm 19007 |
This theorem is referenced by: ghmid 19015 ghminv 19016 ghmsub 19017 ghmmhm 19019 ghmmulg 19021 ghmrn 19022 resghm 19025 ghmpreima 19031 ghmeql 19032 ghmnsgima 19033 ghmnsgpreima 19034 ghmeqker 19036 ghmf1 19038 ghmf1o 19039 gimcnv 19058 lactghmga 19188 frgpup3lem 19560 frgpup3 19561 ghmplusg 19625 rhmf 20159 isrhm2d 20161 kerf1ghm 20178 lmhmf 20498 lmhmpropd 20537 frgpcyg 20983 psgninv 20989 zrhpsgninv 20992 evpmss 20993 psgnevpmb 20994 psgnodpm 20995 zrhpsgnevpm 20998 zrhpsgnodpm 20999 evlslem2 21492 nmoi 24095 nmoix 24096 nmoi2 24097 nmoleub 24098 nmoeq0 24103 nmoco 24104 nmotri 24106 nmods 24111 nghmcn 24112 ghmquskerlem1 32198 ghmquskerco 32199 ghmqusker 32201 isrnghmmul 46198 rnghmf 46204 |
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