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Theorem efmndtmd 22720
 Description: The monoid of endofunctions on a set 𝐴 is a topological monoid. Formerly part of proof for symgtgp 22725. (Contributed by AV, 23-Feb-2024.)
Hypothesis
Ref Expression
efmndtmd.g 𝑀 = (EndoFMnd‘𝐴)
Assertion
Ref Expression
efmndtmd (𝐴𝑉𝑀 ∈ TopMnd)

Proof of Theorem efmndtmd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efmndtmd.g . . 3 𝑀 = (EndoFMnd‘𝐴)
21efmndmnd 18053 . 2 (𝐴𝑉𝑀 ∈ Mnd)
3 eqid 2798 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
41, 3efmndtopn 18047 . . . 4 (𝐴𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
5 distopon 21616 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
6 eqid 2798 . . . . . . 7 (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
76pttoponconst 22216 . . . . . 6 ((𝐴𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
85, 7mpdan 686 . . . . 5 (𝐴𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
91, 3efmndbas 18035 . . . . . . . . 9 (Base‘𝑀) = (𝐴m 𝐴)
109eleq2i 2881 . . . . . . . 8 (𝑥 ∈ (Base‘𝑀) ↔ 𝑥 ∈ (𝐴m 𝐴))
1110biimpi 219 . . . . . . 7 (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ (𝐴m 𝐴))
1211a1i 11 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ (𝐴m 𝐴)))
1312ssrdv 3921 . . . . 5 (𝐴𝑉 → (Base‘𝑀) ⊆ (𝐴m 𝐴))
14 resttopon 21780 . . . . 5 (((∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)) ∧ (Base‘𝑀) ⊆ (𝐴m 𝐴)) → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) ∈ (TopOn‘(Base‘𝑀)))
158, 13, 14syl2anc 587 . . . 4 (𝐴𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) ∈ (TopOn‘(Base‘𝑀)))
164, 15eqeltrrd 2891 . . 3 (𝐴𝑉 → (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
17 eqid 2798 . . . 4 (TopOpen‘𝑀) = (TopOpen‘𝑀)
183, 17istps 21553 . . 3 (𝑀 ∈ TopSp ↔ (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
1916, 18sylibr 237 . 2 (𝐴𝑉𝑀 ∈ TopSp)
20 eqid 2798 . . . . . . 7 (+g𝑀) = (+g𝑀)
211, 3, 20efmndplusg 18044 . . . . . 6 (+g𝑀) = (𝑥 ∈ (Base‘𝑀), 𝑦 ∈ (Base‘𝑀) ↦ (𝑥𝑦))
22 eqid 2798 . . . . . . 7 ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) = ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀))
23 distop 21614 . . . . . . . 8 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
24 eqid 2798 . . . . . . . . 9 (𝒫 𝐴ko 𝒫 𝐴) = (𝒫 𝐴ko 𝒫 𝐴)
2524xkotopon 22219 . . . . . . . 8 ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐴 ∈ Top) → (𝒫 𝐴ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
2623, 23, 25syl2anc 587 . . . . . . 7 (𝐴𝑉 → (𝒫 𝐴ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
27 cndis 21910 . . . . . . . . 9 ((𝐴𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴m 𝐴))
285, 27mpdan 686 . . . . . . . 8 (𝐴𝑉 → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴m 𝐴))
2913, 28sseqtrrd 3956 . . . . . . 7 (𝐴𝑉 → (Base‘𝑀) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴))
30 disllycmp 22117 . . . . . . . . 9 (𝐴𝑉 → 𝒫 𝐴 ∈ Locally Comp)
31 llynlly 22096 . . . . . . . . 9 (𝒫 𝐴 ∈ Locally Comp → 𝒫 𝐴 ∈ 𝑛-Locally Comp)
3230, 31syl 17 . . . . . . . 8 (𝐴𝑉 → 𝒫 𝐴 ∈ 𝑛-Locally Comp)
33 eqid 2798 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝐴 Cn 𝒫 𝐴), 𝑦 ∈ (𝒫 𝐴 Cn 𝒫 𝐴) ↦ (𝑥𝑦)) = (𝑥 ∈ (𝒫 𝐴 Cn 𝒫 𝐴), 𝑦 ∈ (𝒫 𝐴 Cn 𝒫 𝐴) ↦ (𝑥𝑦))
3433xkococn 22279 . . . . . . . 8 ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐴 ∈ 𝑛-Locally Comp ∧ 𝒫 𝐴 ∈ Top) → (𝑥 ∈ (𝒫 𝐴 Cn 𝒫 𝐴), 𝑦 ∈ (𝒫 𝐴 Cn 𝒫 𝐴) ↦ (𝑥𝑦)) ∈ (((𝒫 𝐴ko 𝒫 𝐴) ×t (𝒫 𝐴ko 𝒫 𝐴)) Cn (𝒫 𝐴ko 𝒫 𝐴)))
3523, 32, 23, 34syl3anc 1368 . . . . . . 7 (𝐴𝑉 → (𝑥 ∈ (𝒫 𝐴 Cn 𝒫 𝐴), 𝑦 ∈ (𝒫 𝐴 Cn 𝒫 𝐴) ↦ (𝑥𝑦)) ∈ (((𝒫 𝐴ko 𝒫 𝐴) ×t (𝒫 𝐴ko 𝒫 𝐴)) Cn (𝒫 𝐴ko 𝒫 𝐴)))
3622, 26, 29, 22, 26, 29, 35cnmpt2res 22296 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ (Base‘𝑀), 𝑦 ∈ (Base‘𝑀) ↦ (𝑥𝑦)) ∈ ((((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) ×t ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀))) Cn (𝒫 𝐴ko 𝒫 𝐴)))
3721, 36eqeltrid 2894 . . . . 5 (𝐴𝑉 → (+g𝑀) ∈ ((((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) ×t ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀))) Cn (𝒫 𝐴ko 𝒫 𝐴)))
38 xkopt 22274 . . . . . . . . . 10 ((𝒫 𝐴 ∈ Top ∧ 𝐴𝑉) → (𝒫 𝐴ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
3923, 38mpancom 687 . . . . . . . . 9 (𝐴𝑉 → (𝒫 𝐴ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
4039oveq1d 7155 . . . . . . . 8 (𝐴𝑉 → ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)))
4140, 4eqtrd 2833 . . . . . . 7 (𝐴𝑉 → ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
4241, 41oveq12d 7158 . . . . . 6 (𝐴𝑉 → (((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) ×t ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀))) = ((TopOpen‘𝑀) ×t (TopOpen‘𝑀)))
4342oveq1d 7155 . . . . 5 (𝐴𝑉 → ((((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) ×t ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀))) Cn (𝒫 𝐴ko 𝒫 𝐴)) = (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (𝒫 𝐴ko 𝒫 𝐴)))
4437, 43eleqtrd 2892 . . . 4 (𝐴𝑉 → (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (𝒫 𝐴ko 𝒫 𝐴)))
45 vex 3444 . . . . . . . . . . 11 𝑥 ∈ V
46 vex 3444 . . . . . . . . . . 11 𝑦 ∈ V
4745, 46coex 7624 . . . . . . . . . 10 (𝑥𝑦) ∈ V
4821, 47fnmpoi 7757 . . . . . . . . 9 (+g𝑀) Fn ((Base‘𝑀) × (Base‘𝑀))
49 eqid 2798 . . . . . . . . . 10 (+𝑓𝑀) = (+𝑓𝑀)
503, 20, 49plusfeq 17859 . . . . . . . . 9 ((+g𝑀) Fn ((Base‘𝑀) × (Base‘𝑀)) → (+𝑓𝑀) = (+g𝑀))
5148, 50ax-mp 5 . . . . . . . 8 (+𝑓𝑀) = (+g𝑀)
5251eqcomi 2807 . . . . . . 7 (+g𝑀) = (+𝑓𝑀)
533, 52mndplusf 17928 . . . . . 6 (𝑀 ∈ Mnd → (+g𝑀):((Base‘𝑀) × (Base‘𝑀))⟶(Base‘𝑀))
54 frn 6496 . . . . . 6 ((+g𝑀):((Base‘𝑀) × (Base‘𝑀))⟶(Base‘𝑀) → ran (+g𝑀) ⊆ (Base‘𝑀))
552, 53, 543syl 18 . . . . 5 (𝐴𝑉 → ran (+g𝑀) ⊆ (Base‘𝑀))
56 cnrest2 21905 . . . . 5 (((𝒫 𝐴ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)) ∧ ran (+g𝑀) ⊆ (Base‘𝑀) ∧ (Base‘𝑀) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴)) → ((+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (𝒫 𝐴ko 𝒫 𝐴)) ↔ (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)))))
5726, 55, 29, 56syl3anc 1368 . . . 4 (𝐴𝑉 → ((+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (𝒫 𝐴ko 𝒫 𝐴)) ↔ (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)))))
5844, 57mpbid 235 . . 3 (𝐴𝑉 → (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀))))
5941oveq2d 7156 . . 3 (𝐴𝑉 → (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀))) = (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀)))
6058, 59eleqtrd 2892 . 2 (𝐴𝑉 → (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀)))
6152, 17istmd 22693 . 2 (𝑀 ∈ TopMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ TopSp ∧ (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀))))
622, 19, 60, 61syl3anbrc 1340 1 (𝐴𝑉𝑀 ∈ TopMnd)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881  𝒫 cpw 4497  {csn 4525   × cxp 5518  ran crn 5521   ∘ ccom 5524   Fn wfn 6322  ⟶wf 6323  ‘cfv 6327  (class class class)co 7140   ∈ cmpo 7142   ↑m cmap 8396  Basecbs 16482  +gcplusg 16564   ↾t crest 16693  TopOpenctopn 16694  ∏tcpt 16711  +𝑓cplusf 17848  Mndcmnd 17910  EndoFMndcefmnd 18032  Topctop 21512  TopOnctopon 21529  TopSpctps 21551   Cn ccn 21843  Compccmp 22005  Locally clly 22083  𝑛-Locally cnlly 22084   ×t ctx 22179   ↑ko cxko 22180  TopMndctmd 22689 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-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448  ax-cnex 10589  ax-resscn 10590  ax-1cn 10591  ax-icn 10592  ax-addcl 10593  ax-addrcl 10594  ax-mulcl 10595  ax-mulrcl 10596  ax-mulcom 10597  ax-addass 10598  ax-mulass 10599  ax-distr 10600  ax-i2m1 10601  ax-1ne0 10602  ax-1rid 10603  ax-rnegex 10604  ax-rrecex 10605  ax-cnre 10606  ax-pre-lttri 10607  ax-pre-lttrn 10608  ax-pre-ltadd 10609  ax-pre-mulgt0 10610 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  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-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  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-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-iin 4885  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7568  df-1st 7678  df-2nd 7679  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-1o 8092  df-2o 8093  df-oadd 8096  df-er 8279  df-map 8398  df-ixp 8452  df-en 8500  df-dom 8501  df-sdom 8502  df-fin 8503  df-fi 8866  df-pnf 10673  df-mnf 10674  df-xr 10675  df-ltxr 10676  df-le 10677  df-sub 10868  df-neg 10869  df-nn 11633  df-2 11695  df-3 11696  df-4 11697  df-5 11698  df-6 11699  df-7 11700  df-8 11701  df-9 11702  df-n0 11893  df-z 11977  df-uz 12239  df-fz 12893  df-struct 16484  df-ndx 16485  df-slot 16486  df-base 16488  df-plusg 16577  df-tset 16583  df-rest 16695  df-topn 16696  df-topgen 16716  df-pt 16717  df-plusf 17850  df-mgm 17851  df-sgrp 17900  df-mnd 17911  df-efmnd 18033  df-top 21513  df-topon 21530  df-topsp 21552  df-bases 21565  df-ntr 21639  df-nei 21717  df-cn 21846  df-cmp 22006  df-lly 22085  df-nlly 22086  df-tx 22181  df-xko 22182  df-tmd 22691 This theorem is referenced by:  symgtgp  22725
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