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Theorem mhmhmeotmd 29797
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 17279 . . 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 17278 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
71, 6ax-mp 5 . . . 4 𝑆 ∈ Mnd
8 eqid 2621 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
9 eqid 2621 . . . . 5 (+𝑓𝑆) = (+𝑓𝑆)
108, 9mndplusf 17249 . . . 4 (𝑆 ∈ Mnd → (+𝑓𝑆):((Base‘𝑆) × (Base‘𝑆))⟶(Base‘𝑆))
117, 10ax-mp 5 . . 3 (+𝑓𝑆):((Base‘𝑆) × (Base‘𝑆))⟶(Base‘𝑆)
12 eqid 2621 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
13 eqid 2621 . . . . 5 (+𝑓𝑇) = (+𝑓𝑇)
1412, 13mndplusf 17249 . . . 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 2621 . . . . 5 (TopOpen‘𝑆) = (TopOpen‘𝑆)
1817, 8tmdtopon 21825 . . . 4 (𝑆 ∈ TopMnd → (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆)))
1916, 18ax-mp 5 . . 3 (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆))
20 eqid 2621 . . . . 5 (TopOpen‘𝑇) = (TopOpen‘𝑇)
2112, 20istps 20678 . . . 4 (𝑇 ∈ TopSp ↔ (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇)))
224, 21mpbi 220 . . 3 (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇))
23 eqid 2621 . . . . . 6 (+g𝑆) = (+g𝑆)
24 eqid 2621 . . . . . 6 (+g𝑇) = (+g𝑇)
258, 23, 24mhmlin 17282 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
261, 25mp3an1 1408 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
278, 23, 9plusfval 17188 . . . . 5 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+𝑓𝑆)𝑦) = (𝑥(+g𝑆)𝑦))
2827fveq2d 6162 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓𝑆)𝑦)) = (𝐹‘(𝑥(+g𝑆)𝑦)))
298, 12mhmf 17280 . . . . . . 7 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
301, 29ax-mp 5 . . . . . 6 𝐹:(Base‘𝑆)⟶(Base‘𝑇)
3130ffvelrni 6324 . . . . 5 (𝑥 ∈ (Base‘𝑆) → (𝐹𝑥) ∈ (Base‘𝑇))
3230ffvelrni 6324 . . . . 5 (𝑦 ∈ (Base‘𝑆) → (𝐹𝑦) ∈ (Base‘𝑇))
3312, 24, 13plusfval 17188 . . . . 5 (((𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐹𝑦) ∈ (Base‘𝑇)) → ((𝐹𝑥)(+𝑓𝑇)(𝐹𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3431, 32, 33syl2an 494 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑥)(+𝑓𝑇)(𝐹𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3526, 28, 343eqtr4d 2665 . . 3 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓𝑆)𝑦)) = ((𝐹𝑥)(+𝑓𝑇)(𝐹𝑦)))
3617, 9tmdcn 21827 . . . 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 29796 . 2 (+𝑓𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))
3913, 20istmd 21818 . 2 (𝑇 ∈ TopMnd ↔ (𝑇 ∈ Mnd ∧ 𝑇 ∈ TopSp ∧ (+𝑓𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))))
403, 4, 38, 39mpbir3an 1242 1 𝑇 ∈ TopMnd
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987   × cxp 5082  wf 5853  cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  TopOpenctopn 16022  +𝑓cplusf 17179  Mndcmnd 17234   MndHom cmhm 17273  TopOnctopon 20655  TopSpctps 20676   Cn ccn 20968   ×t ctx 21303  Homeochmeo 21496  TopMndctmd 21814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-map 7819  df-topgen 16044  df-plusf 17181  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-mhm 17275  df-top 20639  df-topon 20656  df-topsp 20677  df-bases 20690  df-cn 20971  df-tx 21305  df-hmeo 21498  df-tmd 21816
This theorem is referenced by:  xrge0tmd  29816
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