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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 44896 | . 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 7405 | 1 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∀wral 3053 {crab 3057 ‘cfv 6339 (class class class)co 7172 ↑m cmap 8439 Basecbs 16588 +gcplusg 16670 Mgmcmgm 17968 MgmHom cmgmhm 44894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-xp 5531 df-dm 5535 df-iota 6297 df-fv 6347 df-ov 7175 df-oprab 7176 df-mpo 7177 df-mgmhm 44896 |
This theorem is referenced by: ismgmhm 44900 mgmhmf1o 44904 resmgmhm 44915 resmgmhm2 44916 resmgmhm2b 44917 mgmhmco 44918 mgmhmima 44919 mgmhmeql 44920 |
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