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Mirrors > Home > MPE Home > Th. List > Mathboxes > mgmhmrcl | Structured version Visualization version GIF version |
Description: Reverse closure of a magma homomorphism. (Contributed by AV, 24-Feb-2020.) |
Ref | Expression |
---|---|
mgmhmrcl | ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mgmhm 44039 | . 2 ⊢ MgmHom = (𝑠 ∈ Mgm, 𝑡 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑡) ↑m (Base‘𝑠)) ∣ ∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦))}) | |
2 | 1 | elmpocl 7381 | 1 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {crab 3142 ‘cfv 6350 (class class class)co 7150 ↑m cmap 8400 Basecbs 16477 +gcplusg 16559 Mgmcmgm 17844 MgmHom cmgmhm 44037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-xp 5556 df-dm 5560 df-iota 6309 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-mgmhm 44039 |
This theorem is referenced by: ismgmhm 44043 mgmhmf1o 44047 resmgmhm 44058 resmgmhm2 44059 resmgmhm2b 44060 mgmhmco 44061 mgmhmima 44062 mgmhmeql 44063 |
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