Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mhmrcl1 Structured version   Visualization version   GIF version

Theorem mhmrcl1 18039
 Description: Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
Assertion
Ref Expression
mhmrcl1 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)

Proof of Theorem mhmrcl1
Dummy variables 𝑓 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mhm 18036 . 2 MndHom = (𝑠 ∈ Mnd, 𝑡 ∈ Mnd ↦ {𝑓 ∈ ((Base‘𝑡) ↑m (Base‘𝑠)) ∣ (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ∧ (𝑓‘(0g𝑠)) = (0g𝑡))})
21elmpocl1 7390 1 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074  ‘cfv 6340  (class class class)co 7156   ↑m cmap 8422  Basecbs 16555  +gcplusg 16637  0gc0g 16785  Mndcmnd 17991   MndHom cmhm 18034 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-xp 5534  df-dm 5538  df-iota 6299  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-mhm 18036 This theorem is referenced by:  mhmf1o  18046  resmhm2  18066  resmhm2b  18067  mhmco  18068  mhmeql  18070  pwsco2mhm  18077  gsumwmhm  18090  mhmmulg  18349  mhmvlin  21113  mhmhmeotmd  31411
 Copyright terms: Public domain W3C validator