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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 17536 | . 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 7023 | 1 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∀wral 3061 {crab 3065 ‘cfv 6029 (class class class)co 6791 ↑𝑚 cmap 8007 Basecbs 16057 +gcplusg 16142 0gc0g 16301 Mndcmnd 17495 MndHom cmhm 17534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-xp 5255 df-dm 5259 df-iota 5992 df-fv 6037 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-mhm 17536 |
This theorem is referenced by: mhmf1o 17546 resmhm 17560 mhmco 17563 mhmima 17564 pwsco2mhm 17572 gsumwmhm 17583 mhmmulg 17784 mhmhmeotmd 30306 |
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