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Theorem mhmhmeotmd 33370
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 18716 . . 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 18715 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
71, 6ax-mp 5 . . . 4 𝑆 ∈ Mnd
8 eqid 2731 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
9 eqid 2731 . . . . 5 (+𝑓𝑆) = (+𝑓𝑆)
108, 9mndplusf 18683 . . . 4 (𝑆 ∈ Mnd → (+𝑓𝑆):((Base‘𝑆) × (Base‘𝑆))⟶(Base‘𝑆))
117, 10ax-mp 5 . . 3 (+𝑓𝑆):((Base‘𝑆) × (Base‘𝑆))⟶(Base‘𝑆)
12 eqid 2731 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
13 eqid 2731 . . . . 5 (+𝑓𝑇) = (+𝑓𝑇)
1412, 13mndplusf 18683 . . . 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 2731 . . . . 5 (TopOpen‘𝑆) = (TopOpen‘𝑆)
1817, 8tmdtopon 23904 . . . 4 (𝑆 ∈ TopMnd → (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆)))
1916, 18ax-mp 5 . . 3 (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆))
20 eqid 2731 . . . . 5 (TopOpen‘𝑇) = (TopOpen‘𝑇)
2112, 20istps 22755 . . . 4 (𝑇 ∈ TopSp ↔ (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇)))
224, 21mpbi 229 . . 3 (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇))
23 eqid 2731 . . . . . 6 (+g𝑆) = (+g𝑆)
24 eqid 2731 . . . . . 6 (+g𝑇) = (+g𝑇)
258, 23, 24mhmlin 18721 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
261, 25mp3an1 1447 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
278, 23, 9plusfval 18578 . . . . 5 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+𝑓𝑆)𝑦) = (𝑥(+g𝑆)𝑦))
2827fveq2d 6895 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓𝑆)𝑦)) = (𝐹‘(𝑥(+g𝑆)𝑦)))
298, 12mhmf 18717 . . . . . . 7 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
301, 29ax-mp 5 . . . . . 6 𝐹:(Base‘𝑆)⟶(Base‘𝑇)
3130ffvelcdmi 7085 . . . . 5 (𝑥 ∈ (Base‘𝑆) → (𝐹𝑥) ∈ (Base‘𝑇))
3230ffvelcdmi 7085 . . . . 5 (𝑦 ∈ (Base‘𝑆) → (𝐹𝑦) ∈ (Base‘𝑇))
3312, 24, 13plusfval 18578 . . . . 5 (((𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐹𝑦) ∈ (Base‘𝑇)) → ((𝐹𝑥)(+𝑓𝑇)(𝐹𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3431, 32, 33syl2an 595 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑥)(+𝑓𝑇)(𝐹𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3526, 28, 343eqtr4d 2781 . . 3 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓𝑆)𝑦)) = ((𝐹𝑥)(+𝑓𝑇)(𝐹𝑦)))
3617, 9tmdcn 23906 . . . 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 33369 . 2 (+𝑓𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))
3913, 20istmd 23897 . 2 (𝑇 ∈ TopMnd ↔ (𝑇 ∈ Mnd ∧ 𝑇 ∈ TopSp ∧ (+𝑓𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))))
403, 4, 38, 39mpbir3an 1340 1 𝑇 ∈ TopMnd
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2105   × cxp 5674  wf 6539  cfv 6543  (class class class)co 7412  Basecbs 17151  +gcplusg 17204  TopOpenctopn 17374  +𝑓cplusf 18568  Mndcmnd 18665   MndHom cmhm 18709  TopOnctopon 22731  TopSpctps 22753   Cn ccn 23047   ×t ctx 23383  Homeochmeo 23576  TopMndctmd 23893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-map 8828  df-topgen 17396  df-plusf 18570  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-mhm 18711  df-top 22715  df-topon 22732  df-topsp 22754  df-bases 22768  df-cn 23050  df-tx 23385  df-hmeo 23578  df-tmd 23895
This theorem is referenced by:  xrge0tmd  33388
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