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Theorem mhmhmeotmd 30303
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 17560 . . 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 17559 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
71, 6ax-mp 5 . . . 4 𝑆 ∈ Mnd
8 eqid 2760 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
9 eqid 2760 . . . . 5 (+𝑓𝑆) = (+𝑓𝑆)
108, 9mndplusf 17530 . . . 4 (𝑆 ∈ Mnd → (+𝑓𝑆):((Base‘𝑆) × (Base‘𝑆))⟶(Base‘𝑆))
117, 10ax-mp 5 . . 3 (+𝑓𝑆):((Base‘𝑆) × (Base‘𝑆))⟶(Base‘𝑆)
12 eqid 2760 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
13 eqid 2760 . . . . 5 (+𝑓𝑇) = (+𝑓𝑇)
1412, 13mndplusf 17530 . . . 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 2760 . . . . 5 (TopOpen‘𝑆) = (TopOpen‘𝑆)
1817, 8tmdtopon 22106 . . . 4 (𝑆 ∈ TopMnd → (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆)))
1916, 18ax-mp 5 . . 3 (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆))
20 eqid 2760 . . . . 5 (TopOpen‘𝑇) = (TopOpen‘𝑇)
2112, 20istps 20960 . . . 4 (𝑇 ∈ TopSp ↔ (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇)))
224, 21mpbi 220 . . 3 (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇))
23 eqid 2760 . . . . . 6 (+g𝑆) = (+g𝑆)
24 eqid 2760 . . . . . 6 (+g𝑇) = (+g𝑇)
258, 23, 24mhmlin 17563 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
261, 25mp3an1 1560 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
278, 23, 9plusfval 17469 . . . . 5 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+𝑓𝑆)𝑦) = (𝑥(+g𝑆)𝑦))
2827fveq2d 6357 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓𝑆)𝑦)) = (𝐹‘(𝑥(+g𝑆)𝑦)))
298, 12mhmf 17561 . . . . . . 7 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
301, 29ax-mp 5 . . . . . 6 𝐹:(Base‘𝑆)⟶(Base‘𝑇)
3130ffvelrni 6522 . . . . 5 (𝑥 ∈ (Base‘𝑆) → (𝐹𝑥) ∈ (Base‘𝑇))
3230ffvelrni 6522 . . . . 5 (𝑦 ∈ (Base‘𝑆) → (𝐹𝑦) ∈ (Base‘𝑇))
3312, 24, 13plusfval 17469 . . . . 5 (((𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐹𝑦) ∈ (Base‘𝑇)) → ((𝐹𝑥)(+𝑓𝑇)(𝐹𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3431, 32, 33syl2an 495 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑥)(+𝑓𝑇)(𝐹𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3526, 28, 343eqtr4d 2804 . . 3 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓𝑆)𝑦)) = ((𝐹𝑥)(+𝑓𝑇)(𝐹𝑦)))
3617, 9tmdcn 22108 . . . 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 30302 . 2 (+𝑓𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))
3913, 20istmd 22099 . 2 (𝑇 ∈ TopMnd ↔ (𝑇 ∈ Mnd ∧ 𝑇 ∈ TopSp ∧ (+𝑓𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))))
403, 4, 38, 39mpbir3an 1427 1 𝑇 ∈ TopMnd
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wcel 2139   × cxp 5264  wf 6045  cfv 6049  (class class class)co 6814  Basecbs 16079  +gcplusg 16163  TopOpenctopn 16304  +𝑓cplusf 17460  Mndcmnd 17515   MndHom cmhm 17554  TopOnctopon 20937  TopSpctps 20958   Cn ccn 21250   ×t ctx 21585  Homeochmeo 21778  TopMndctmd 22095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027  df-topgen 16326  df-plusf 17462  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-mhm 17556  df-top 20921  df-topon 20938  df-topsp 20959  df-bases 20972  df-cn 21253  df-tx 21587  df-hmeo 21780  df-tmd 22097
This theorem is referenced by:  xrge0tmd  30322
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