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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 46484 | . 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 7643 | 1 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 {crab 3433 ‘cfv 6540 (class class class)co 7404 ↑m cmap 8816 Basecbs 17140 +gcplusg 17193 Mgmcmgm 18555 MgmHom cmgmhm 46482 |
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 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-xp 5681 df-dm 5685 df-iota 6492 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-mgmhm 46484 |
This theorem is referenced by: ismgmhm 46488 mgmhmf1o 46492 resmgmhm 46503 resmgmhm2 46504 resmgmhm2b 46505 mgmhmco 46506 mgmhmima 46507 mgmhmeql 46508 |
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