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| Mirrors > Home > MPE Home > Th. List > 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 18705 | . 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 7674 | 1 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 Basecbs 17247 +gcplusg 17297 Mgmcmgm 18651 MgmHom cmgmhm 18703 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-dm 5695 df-iota 6514 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-mgmhm 18705 |
| This theorem is referenced by: ismgmhm 18709 mgmhmf1o 18713 resmgmhm 18724 resmgmhm2 18725 resmgmhm2b 18726 mgmhmco 18727 mgmhmima 18728 mgmhmeql 18729 |
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