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Mirrors > Home > MPE Home > Th. List > mhmrcl2 | Structured version Visualization version GIF version |
Description: Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.) |
Ref | Expression |
---|---|
mhmrcl2 | ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mhm 17689 | . 2 ⊢ MndHom = (𝑠 ∈ Mnd, 𝑡 ∈ Mnd ↦ {𝑓 ∈ ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) ∣ (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑠)) = (0g‘𝑡))}) | |
2 | 1 | elmpt2cl2 7139 | 1 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∀wral 3118 {crab 3122 ‘cfv 6124 (class class class)co 6906 ↑𝑚 cmap 8123 Basecbs 16223 +gcplusg 16306 0gc0g 16454 Mndcmnd 17648 MndHom cmhm 17687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-xp 5349 df-dm 5353 df-iota 6087 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-mhm 17689 |
This theorem is referenced by: mhmf1o 17699 resmhm 17713 mhmco 17716 mhmima 17717 pwsco2mhm 17725 gsumwmhm 17737 mhmmulg 17935 mhmhmeotmd 30519 |
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