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Mirrors > Home > MPE Home > Th. List > mhmrcl1 | Structured version Visualization version GIF version |
Description: Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.) |
Ref | Expression |
---|---|
mhmrcl1 | ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mhm 17335 | . 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 | elmpt2cl1 6877 | 1 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 Basecbs 15857 +gcplusg 15941 0gc0g 16100 Mndcmnd 17294 MndHom cmhm 17333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-dm 5124 df-iota 5851 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-mhm 17335 |
This theorem is referenced by: mhmf1o 17345 resmhm2 17360 resmhm2b 17361 mhmco 17362 mhmeql 17364 pwsco2mhm 17371 gsumwmhm 17382 mhmmulg 17583 mhmvlin 20203 mhmhmeotmd 29973 |
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