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Theorem mhmhmeotmd 31292
 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 17954 . . 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 17953 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
71, 6ax-mp 5 . . . 4 𝑆 ∈ Mnd
8 eqid 2798 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
9 eqid 2798 . . . . 5 (+𝑓𝑆) = (+𝑓𝑆)
108, 9mndplusf 17923 . . . 4 (𝑆 ∈ Mnd → (+𝑓𝑆):((Base‘𝑆) × (Base‘𝑆))⟶(Base‘𝑆))
117, 10ax-mp 5 . . 3 (+𝑓𝑆):((Base‘𝑆) × (Base‘𝑆))⟶(Base‘𝑆)
12 eqid 2798 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
13 eqid 2798 . . . . 5 (+𝑓𝑇) = (+𝑓𝑇)
1412, 13mndplusf 17923 . . . 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 2798 . . . . 5 (TopOpen‘𝑆) = (TopOpen‘𝑆)
1817, 8tmdtopon 22693 . . . 4 (𝑆 ∈ TopMnd → (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆)))
1916, 18ax-mp 5 . . 3 (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆))
20 eqid 2798 . . . . 5 (TopOpen‘𝑇) = (TopOpen‘𝑇)
2112, 20istps 21546 . . . 4 (𝑇 ∈ TopSp ↔ (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇)))
224, 21mpbi 233 . . 3 (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇))
23 eqid 2798 . . . . . 6 (+g𝑆) = (+g𝑆)
24 eqid 2798 . . . . . 6 (+g𝑇) = (+g𝑇)
258, 23, 24mhmlin 17957 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
261, 25mp3an1 1445 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
278, 23, 9plusfval 17853 . . . . 5 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+𝑓𝑆)𝑦) = (𝑥(+g𝑆)𝑦))
2827fveq2d 6649 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓𝑆)𝑦)) = (𝐹‘(𝑥(+g𝑆)𝑦)))
298, 12mhmf 17955 . . . . . . 7 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
301, 29ax-mp 5 . . . . . 6 𝐹:(Base‘𝑆)⟶(Base‘𝑇)
3130ffvelrni 6827 . . . . 5 (𝑥 ∈ (Base‘𝑆) → (𝐹𝑥) ∈ (Base‘𝑇))
3230ffvelrni 6827 . . . . 5 (𝑦 ∈ (Base‘𝑆) → (𝐹𝑦) ∈ (Base‘𝑇))
3312, 24, 13plusfval 17853 . . . . 5 (((𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐹𝑦) ∈ (Base‘𝑇)) → ((𝐹𝑥)(+𝑓𝑇)(𝐹𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3431, 32, 33syl2an 598 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑥)(+𝑓𝑇)(𝐹𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3526, 28, 343eqtr4d 2843 . . 3 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓𝑆)𝑦)) = ((𝐹𝑥)(+𝑓𝑇)(𝐹𝑦)))
3617, 9tmdcn 22695 . . . 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 31291 . 2 (+𝑓𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))
3913, 20istmd 22686 . 2 (𝑇 ∈ TopMnd ↔ (𝑇 ∈ Mnd ∧ 𝑇 ∈ TopSp ∧ (+𝑓𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))))
403, 4, 38, 39mpbir3an 1338 1 𝑇 ∈ TopMnd
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2111   × cxp 5517  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  Basecbs 16477  +gcplusg 16559  TopOpenctopn 16689  +𝑓cplusf 17843  Mndcmnd 17905   MndHom cmhm 17948  TopOnctopon 21522  TopSpctps 21544   Cn ccn 21836   ×t ctx 22172  Homeochmeo 22365  TopMndctmd 22682 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 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7673  df-2nd 7674  df-map 8393  df-topgen 16711  df-plusf 17845  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-mhm 17950  df-top 21506  df-topon 21523  df-topsp 21545  df-bases 21558  df-cn 21839  df-tx 22174  df-hmeo 22367  df-tmd 22684 This theorem is referenced by:  xrge0tmd  31310
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