Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mhmhmeotmd Structured version   Visualization version   GIF version

Theorem mhmhmeotmd 30571
Description: Deduce a Topological Monoid using mapping that is both a homeomorphism and a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)
Hypotheses
Ref Expression
mhmhmeotmd.m 𝐹 ∈ (𝑆 MndHom 𝑇)
mhmhmeotmd.h 𝐹 ∈ ((TopOpen‘𝑆)Homeo(TopOpen‘𝑇))
mhmhmeotmd.t 𝑆 ∈ TopMnd
mhmhmeotmd.s 𝑇 ∈ TopSp
Assertion
Ref Expression
mhmhmeotmd 𝑇 ∈ TopMnd

Proof of Theorem mhmhmeotmd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhmhmeotmd.m . . 3 𝐹 ∈ (𝑆 MndHom 𝑇)
2 mhmrcl2 17725 . . 3 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd)
31, 2ax-mp 5 . 2 𝑇 ∈ Mnd
4 mhmhmeotmd.s . 2 𝑇 ∈ TopSp
5 mhmhmeotmd.h . . 3 𝐹 ∈ ((TopOpen‘𝑆)Homeo(TopOpen‘𝑇))
6 mhmrcl1 17724 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
71, 6ax-mp 5 . . . 4 𝑆 ∈ Mnd
8 eqid 2778 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
9 eqid 2778 . . . . 5 (+𝑓𝑆) = (+𝑓𝑆)
108, 9mndplusf 17695 . . . 4 (𝑆 ∈ Mnd → (+𝑓𝑆):((Base‘𝑆) × (Base‘𝑆))⟶(Base‘𝑆))
117, 10ax-mp 5 . . 3 (+𝑓𝑆):((Base‘𝑆) × (Base‘𝑆))⟶(Base‘𝑆)
12 eqid 2778 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
13 eqid 2778 . . . . 5 (+𝑓𝑇) = (+𝑓𝑇)
1412, 13mndplusf 17695 . . . 4 (𝑇 ∈ Mnd → (+𝑓𝑇):((Base‘𝑇) × (Base‘𝑇))⟶(Base‘𝑇))
153, 14ax-mp 5 . . 3 (+𝑓𝑇):((Base‘𝑇) × (Base‘𝑇))⟶(Base‘𝑇)
16 mhmhmeotmd.t . . . 4 𝑆 ∈ TopMnd
17 eqid 2778 . . . . 5 (TopOpen‘𝑆) = (TopOpen‘𝑆)
1817, 8tmdtopon 22293 . . . 4 (𝑆 ∈ TopMnd → (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆)))
1916, 18ax-mp 5 . . 3 (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆))
20 eqid 2778 . . . . 5 (TopOpen‘𝑇) = (TopOpen‘𝑇)
2112, 20istps 21146 . . . 4 (𝑇 ∈ TopSp ↔ (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇)))
224, 21mpbi 222 . . 3 (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇))
23 eqid 2778 . . . . . 6 (+g𝑆) = (+g𝑆)
24 eqid 2778 . . . . . 6 (+g𝑇) = (+g𝑇)
258, 23, 24mhmlin 17728 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
261, 25mp3an1 1521 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
278, 23, 9plusfval 17634 . . . . 5 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+𝑓𝑆)𝑦) = (𝑥(+g𝑆)𝑦))
2827fveq2d 6450 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓𝑆)𝑦)) = (𝐹‘(𝑥(+g𝑆)𝑦)))
298, 12mhmf 17726 . . . . . . 7 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
301, 29ax-mp 5 . . . . . 6 𝐹:(Base‘𝑆)⟶(Base‘𝑇)
3130ffvelrni 6622 . . . . 5 (𝑥 ∈ (Base‘𝑆) → (𝐹𝑥) ∈ (Base‘𝑇))
3230ffvelrni 6622 . . . . 5 (𝑦 ∈ (Base‘𝑆) → (𝐹𝑦) ∈ (Base‘𝑇))
3312, 24, 13plusfval 17634 . . . . 5 (((𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐹𝑦) ∈ (Base‘𝑇)) → ((𝐹𝑥)(+𝑓𝑇)(𝐹𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3431, 32, 33syl2an 589 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑥)(+𝑓𝑇)(𝐹𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3526, 28, 343eqtr4d 2824 . . 3 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓𝑆)𝑦)) = ((𝐹𝑥)(+𝑓𝑇)(𝐹𝑦)))
3617, 9tmdcn 22295 . . . 4 (𝑆 ∈ TopMnd → (+𝑓𝑆) ∈ (((TopOpen‘𝑆) ×t (TopOpen‘𝑆)) Cn (TopOpen‘𝑆)))
3716, 36ax-mp 5 . . 3 (+𝑓𝑆) ∈ (((TopOpen‘𝑆) ×t (TopOpen‘𝑆)) Cn (TopOpen‘𝑆))
385, 11, 15, 19, 22, 35, 37mndpluscn 30570 . 2 (+𝑓𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))
3913, 20istmd 22286 . 2 (𝑇 ∈ TopMnd ↔ (𝑇 ∈ Mnd ∧ 𝑇 ∈ TopSp ∧ (+𝑓𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))))
403, 4, 38, 39mpbir3an 1398 1 𝑇 ∈ TopMnd
Colors of variables: wff setvar class
Syntax hints:  wa 386   = wceq 1601  wcel 2107   × cxp 5353  wf 6131  cfv 6135  (class class class)co 6922  Basecbs 16255  +gcplusg 16338  TopOpenctopn 16468  +𝑓cplusf 17625  Mndcmnd 17680   MndHom cmhm 17719  TopOnctopon 21122  TopSpctps 21144   Cn ccn 21436   ×t ctx 21772  Homeochmeo 21965  TopMndctmd 22282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-map 8142  df-topgen 16490  df-plusf 17627  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-mhm 17721  df-top 21106  df-topon 21123  df-topsp 21145  df-bases 21158  df-cn 21439  df-tx 21774  df-hmeo 21967  df-tmd 22284
This theorem is referenced by:  xrge0tmd  30590
  Copyright terms: Public domain W3C validator