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Theorem mhmhmeotmd 32320
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 ∈ ((TopOpenβ€˜π‘†)Homeo(TopOpenβ€˜π‘‡))
mhmhmeotmd.t 𝑆 ∈ TopMnd
mhmhmeotmd.s 𝑇 ∈ TopSp
Assertion
Ref Expression
mhmhmeotmd 𝑇 ∈ TopMnd

Proof of Theorem mhmhmeotmd
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhmhmeotmd.m . . 3 𝐹 ∈ (𝑆 MndHom 𝑇)
2 mhmrcl2 18566 . . 3 (𝐹 ∈ (𝑆 MndHom 𝑇) β†’ 𝑇 ∈ Mnd)
31, 2ax-mp 5 . 2 𝑇 ∈ Mnd
4 mhmhmeotmd.s . 2 𝑇 ∈ TopSp
5 mhmhmeotmd.h . . 3 𝐹 ∈ ((TopOpenβ€˜π‘†)Homeo(TopOpenβ€˜π‘‡))
6 mhmrcl1 18565 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) β†’ 𝑆 ∈ Mnd)
71, 6ax-mp 5 . . . 4 𝑆 ∈ Mnd
8 eqid 2737 . . . . 5 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
9 eqid 2737 . . . . 5 (+π‘“β€˜π‘†) = (+π‘“β€˜π‘†)
108, 9mndplusf 18534 . . . 4 (𝑆 ∈ Mnd β†’ (+π‘“β€˜π‘†):((Baseβ€˜π‘†) Γ— (Baseβ€˜π‘†))⟢(Baseβ€˜π‘†))
117, 10ax-mp 5 . . 3 (+π‘“β€˜π‘†):((Baseβ€˜π‘†) Γ— (Baseβ€˜π‘†))⟢(Baseβ€˜π‘†)
12 eqid 2737 . . . . 5 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
13 eqid 2737 . . . . 5 (+π‘“β€˜π‘‡) = (+π‘“β€˜π‘‡)
1412, 13mndplusf 18534 . . . 4 (𝑇 ∈ Mnd β†’ (+π‘“β€˜π‘‡):((Baseβ€˜π‘‡) Γ— (Baseβ€˜π‘‡))⟢(Baseβ€˜π‘‡))
153, 14ax-mp 5 . . 3 (+π‘“β€˜π‘‡):((Baseβ€˜π‘‡) Γ— (Baseβ€˜π‘‡))⟢(Baseβ€˜π‘‡)
16 mhmhmeotmd.t . . . 4 𝑆 ∈ TopMnd
17 eqid 2737 . . . . 5 (TopOpenβ€˜π‘†) = (TopOpenβ€˜π‘†)
1817, 8tmdtopon 23384 . . . 4 (𝑆 ∈ TopMnd β†’ (TopOpenβ€˜π‘†) ∈ (TopOnβ€˜(Baseβ€˜π‘†)))
1916, 18ax-mp 5 . . 3 (TopOpenβ€˜π‘†) ∈ (TopOnβ€˜(Baseβ€˜π‘†))
20 eqid 2737 . . . . 5 (TopOpenβ€˜π‘‡) = (TopOpenβ€˜π‘‡)
2112, 20istps 22235 . . . 4 (𝑇 ∈ TopSp ↔ (TopOpenβ€˜π‘‡) ∈ (TopOnβ€˜(Baseβ€˜π‘‡)))
224, 21mpbi 229 . . 3 (TopOpenβ€˜π‘‡) ∈ (TopOnβ€˜(Baseβ€˜π‘‡))
23 eqid 2737 . . . . . 6 (+gβ€˜π‘†) = (+gβ€˜π‘†)
24 eqid 2737 . . . . . 6 (+gβ€˜π‘‡) = (+gβ€˜π‘‡)
258, 23, 24mhmlin 18569 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ (πΉβ€˜(π‘₯(+gβ€˜π‘†)𝑦)) = ((πΉβ€˜π‘₯)(+gβ€˜π‘‡)(πΉβ€˜π‘¦)))
261, 25mp3an1 1448 . . . 4 ((π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ (πΉβ€˜(π‘₯(+gβ€˜π‘†)𝑦)) = ((πΉβ€˜π‘₯)(+gβ€˜π‘‡)(πΉβ€˜π‘¦)))
278, 23, 9plusfval 18464 . . . . 5 ((π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ (π‘₯(+π‘“β€˜π‘†)𝑦) = (π‘₯(+gβ€˜π‘†)𝑦))
2827fveq2d 6843 . . . 4 ((π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ (πΉβ€˜(π‘₯(+π‘“β€˜π‘†)𝑦)) = (πΉβ€˜(π‘₯(+gβ€˜π‘†)𝑦)))
298, 12mhmf 18567 . . . . . . 7 (𝐹 ∈ (𝑆 MndHom 𝑇) β†’ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
301, 29ax-mp 5 . . . . . 6 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡)
3130ffvelcdmi 7030 . . . . 5 (π‘₯ ∈ (Baseβ€˜π‘†) β†’ (πΉβ€˜π‘₯) ∈ (Baseβ€˜π‘‡))
3230ffvelcdmi 7030 . . . . 5 (𝑦 ∈ (Baseβ€˜π‘†) β†’ (πΉβ€˜π‘¦) ∈ (Baseβ€˜π‘‡))
3312, 24, 13plusfval 18464 . . . . 5 (((πΉβ€˜π‘₯) ∈ (Baseβ€˜π‘‡) ∧ (πΉβ€˜π‘¦) ∈ (Baseβ€˜π‘‡)) β†’ ((πΉβ€˜π‘₯)(+π‘“β€˜π‘‡)(πΉβ€˜π‘¦)) = ((πΉβ€˜π‘₯)(+gβ€˜π‘‡)(πΉβ€˜π‘¦)))
3431, 32, 33syl2an 596 . . . 4 ((π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ ((πΉβ€˜π‘₯)(+π‘“β€˜π‘‡)(πΉβ€˜π‘¦)) = ((πΉβ€˜π‘₯)(+gβ€˜π‘‡)(πΉβ€˜π‘¦)))
3526, 28, 343eqtr4d 2787 . . 3 ((π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ (πΉβ€˜(π‘₯(+π‘“β€˜π‘†)𝑦)) = ((πΉβ€˜π‘₯)(+π‘“β€˜π‘‡)(πΉβ€˜π‘¦)))
3617, 9tmdcn 23386 . . . 4 (𝑆 ∈ TopMnd β†’ (+π‘“β€˜π‘†) ∈ (((TopOpenβ€˜π‘†) Γ—t (TopOpenβ€˜π‘†)) Cn (TopOpenβ€˜π‘†)))
3716, 36ax-mp 5 . . 3 (+π‘“β€˜π‘†) ∈ (((TopOpenβ€˜π‘†) Γ—t (TopOpenβ€˜π‘†)) Cn (TopOpenβ€˜π‘†))
385, 11, 15, 19, 22, 35, 37mndpluscn 32319 . 2 (+π‘“β€˜π‘‡) ∈ (((TopOpenβ€˜π‘‡) Γ—t (TopOpenβ€˜π‘‡)) Cn (TopOpenβ€˜π‘‡))
3913, 20istmd 23377 . 2 (𝑇 ∈ TopMnd ↔ (𝑇 ∈ Mnd ∧ 𝑇 ∈ TopSp ∧ (+π‘“β€˜π‘‡) ∈ (((TopOpenβ€˜π‘‡) Γ—t (TopOpenβ€˜π‘‡)) Cn (TopOpenβ€˜π‘‡))))
403, 4, 38, 39mpbir3an 1341 1 𝑇 ∈ TopMnd
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 396   = wceq 1541   ∈ wcel 2106   Γ— cxp 5629  βŸΆwf 6489  β€˜cfv 6493  (class class class)co 7351  Basecbs 17043  +gcplusg 17093  TopOpenctopn 17263  +𝑓cplusf 18454  Mndcmnd 18516   MndHom cmhm 18559  TopOnctopon 22211  TopSpctps 22233   Cn ccn 22527   Γ—t ctx 22863  Homeochmeo 23056  TopMndctmd 23373
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-map 8725  df-topgen 17285  df-plusf 18456  df-mgm 18457  df-sgrp 18506  df-mnd 18517  df-mhm 18561  df-top 22195  df-topon 22212  df-topsp 22234  df-bases 22248  df-cn 22530  df-tx 22865  df-hmeo 23058  df-tmd 23375
This theorem is referenced by:  xrge0tmd  32338
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