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Theorem efmndtmd 23335
Description: The monoid of endofunctions on a set 𝐴 is a topological monoid. Formerly part of proof for symgtgp 23340. (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 18604 . 2 (𝐴𝑉𝑀 ∈ Mnd)
3 eqid 2737 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
41, 3efmndtopn 18598 . . . 4 (𝐴𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
5 distopon 22230 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
6 eqid 2737 . . . . . . 7 (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
76pttoponconst 22831 . . . . . 6 ((𝐴𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
85, 7mpdan 684 . . . . 5 (𝐴𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
91, 3efmndbas 18586 . . . . . . . . 9 (Base‘𝑀) = (𝐴m 𝐴)
109eleq2i 2829 . . . . . . . 8 (𝑥 ∈ (Base‘𝑀) ↔ 𝑥 ∈ (𝐴m 𝐴))
1110biimpi 215 . . . . . . 7 (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ (𝐴m 𝐴))
1211a1i 11 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ (𝐴m 𝐴)))
1312ssrdv 3937 . . . . 5 (𝐴𝑉 → (Base‘𝑀) ⊆ (𝐴m 𝐴))
14 resttopon 22395 . . . . 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 584 . . . 4 (𝐴𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) ∈ (TopOn‘(Base‘𝑀)))
164, 15eqeltrrd 2839 . . 3 (𝐴𝑉 → (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
17 eqid 2737 . . . 4 (TopOpen‘𝑀) = (TopOpen‘𝑀)
183, 17istps 22166 . . 3 (𝑀 ∈ TopSp ↔ (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
1916, 18sylibr 233 . 2 (𝐴𝑉𝑀 ∈ TopSp)
20 eqid 2737 . . . . . . 7 (+g𝑀) = (+g𝑀)
211, 3, 20efmndplusg 18595 . . . . . 6 (+g𝑀) = (𝑥 ∈ (Base‘𝑀), 𝑦 ∈ (Base‘𝑀) ↦ (𝑥𝑦))
22 eqid 2737 . . . . . . 7 ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) = ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀))
23 distop 22228 . . . . . . . 8 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
24 eqid 2737 . . . . . . . . 9 (𝒫 𝐴ko 𝒫 𝐴) = (𝒫 𝐴ko 𝒫 𝐴)
2524xkotopon 22834 . . . . . . . 8 ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐴 ∈ Top) → (𝒫 𝐴ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
2623, 23, 25syl2anc 584 . . . . . . 7 (𝐴𝑉 → (𝒫 𝐴ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
27 cndis 22525 . . . . . . . . 9 ((𝐴𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴m 𝐴))
285, 27mpdan 684 . . . . . . . 8 (𝐴𝑉 → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴m 𝐴))
2913, 28sseqtrrd 3972 . . . . . . 7 (𝐴𝑉 → (Base‘𝑀) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴))
30 disllycmp 22732 . . . . . . . . 9 (𝐴𝑉 → 𝒫 𝐴 ∈ Locally Comp)
31 llynlly 22711 . . . . . . . . 9 (𝒫 𝐴 ∈ Locally Comp → 𝒫 𝐴 ∈ 𝑛-Locally Comp)
3230, 31syl 17 . . . . . . . 8 (𝐴𝑉 → 𝒫 𝐴 ∈ 𝑛-Locally Comp)
33 eqid 2737 . . . . . . . . 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 22894 . . . . . . . 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 1370 . . . . . . 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 22911 . . . . . 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 2842 . . . . 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 22889 . . . . . . . . . 10 ((𝒫 𝐴 ∈ Top ∧ 𝐴𝑉) → (𝒫 𝐴ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
3923, 38mpancom 685 . . . . . . . . 9 (𝐴𝑉 → (𝒫 𝐴ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
4039oveq1d 7332 . . . . . . . 8 (𝐴𝑉 → ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)))
4140, 4eqtrd 2777 . . . . . . 7 (𝐴𝑉 → ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
4241, 41oveq12d 7335 . . . . . 6 (𝐴𝑉 → (((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) ×t ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀))) = ((TopOpen‘𝑀) ×t (TopOpen‘𝑀)))
4342oveq1d 7332 . . . . 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 2840 . . . 4 (𝐴𝑉 → (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (𝒫 𝐴ko 𝒫 𝐴)))
45 vex 3445 . . . . . . . . . . 11 𝑥 ∈ V
46 vex 3445 . . . . . . . . . . 11 𝑦 ∈ V
4745, 46coex 7824 . . . . . . . . . 10 (𝑥𝑦) ∈ V
4821, 47fnmpoi 7957 . . . . . . . . 9 (+g𝑀) Fn ((Base‘𝑀) × (Base‘𝑀))
49 eqid 2737 . . . . . . . . . 10 (+𝑓𝑀) = (+𝑓𝑀)
503, 20, 49plusfeq 18411 . . . . . . . . 9 ((+g𝑀) Fn ((Base‘𝑀) × (Base‘𝑀)) → (+𝑓𝑀) = (+g𝑀))
5148, 50ax-mp 5 . . . . . . . 8 (+𝑓𝑀) = (+g𝑀)
5251eqcomi 2746 . . . . . . 7 (+g𝑀) = (+𝑓𝑀)
533, 52mndplusf 18480 . . . . . 6 (𝑀 ∈ Mnd → (+g𝑀):((Base‘𝑀) × (Base‘𝑀))⟶(Base‘𝑀))
54 frn 6645 . . . . . 6 ((+g𝑀):((Base‘𝑀) × (Base‘𝑀))⟶(Base‘𝑀) → ran (+g𝑀) ⊆ (Base‘𝑀))
552, 53, 543syl 18 . . . . 5 (𝐴𝑉 → ran (+g𝑀) ⊆ (Base‘𝑀))
56 cnrest2 22520 . . . . 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 1370 . . . 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 231 . . 3 (𝐴𝑉 → (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀))))
5941oveq2d 7333 . . 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 2840 . 2 (𝐴𝑉 → (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀)))
6152, 17istmd 23308 . 2 (𝑀 ∈ TopMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ TopSp ∧ (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀))))
622, 19, 60, 61syl3anbrc 1342 1 (𝐴𝑉𝑀 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1540  wcel 2105  wss 3897  𝒫 cpw 4545  {csn 4571   × cxp 5606  ran crn 5609  ccom 5612   Fn wfn 6461  wf 6462  cfv 6466  (class class class)co 7317  cmpo 7319  m cmap 8665  Basecbs 16989  +gcplusg 17039  t crest 17208  TopOpenctopn 17209  tcpt 17226  +𝑓cplusf 18400  Mndcmnd 18462  EndoFMndcefmnd 18583  Topctop 22125  TopOnctopon 22142  TopSpctps 22164   Cn ccn 22458  Compccmp 22620  Locally clly 22698  𝑛-Locally cnlly 22699   ×t ctx 22794  ko cxko 22795  TopMndctmd 23304
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 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7630  ax-cnex 11007  ax-resscn 11008  ax-1cn 11009  ax-icn 11010  ax-addcl 11011  ax-addrcl 11012  ax-mulcl 11013  ax-mulrcl 11014  ax-mulcom 11015  ax-addass 11016  ax-mulass 11017  ax-distr 11018  ax-i2m1 11019  ax-1ne0 11020  ax-1rid 11021  ax-rnegex 11022  ax-rrecex 11023  ax-cnre 11024  ax-pre-lttri 11025  ax-pre-lttrn 11026  ax-pre-ltadd 11027  ax-pre-mulgt0 11028
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4851  df-int 4893  df-iun 4939  df-iin 4940  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5563  df-we 5565  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-pred 6225  df-ord 6292  df-on 6293  df-lim 6294  df-suc 6295  df-iota 6418  df-fun 6468  df-fn 6469  df-f 6470  df-f1 6471  df-fo 6472  df-f1o 6473  df-fv 6474  df-riota 7274  df-ov 7320  df-oprab 7321  df-mpo 7322  df-om 7760  df-1st 7878  df-2nd 7879  df-frecs 8146  df-wrecs 8177  df-recs 8251  df-rdg 8290  df-1o 8346  df-er 8548  df-map 8667  df-ixp 8736  df-en 8784  df-dom 8785  df-sdom 8786  df-fin 8787  df-fi 9247  df-pnf 11091  df-mnf 11092  df-xr 11093  df-ltxr 11094  df-le 11095  df-sub 11287  df-neg 11288  df-nn 12054  df-2 12116  df-3 12117  df-4 12118  df-5 12119  df-6 12120  df-7 12121  df-8 12122  df-9 12123  df-n0 12314  df-z 12400  df-uz 12663  df-fz 13320  df-struct 16925  df-slot 16960  df-ndx 16972  df-base 16990  df-plusg 17052  df-tset 17058  df-rest 17210  df-topn 17211  df-topgen 17231  df-pt 17232  df-plusf 18402  df-mgm 18403  df-sgrp 18452  df-mnd 18463  df-efmnd 18584  df-top 22126  df-topon 22143  df-topsp 22165  df-bases 22179  df-ntr 22254  df-nei 22332  df-cn 22461  df-cmp 22621  df-lly 22700  df-nlly 22701  df-tx 22796  df-xko 22797  df-tmd 23306
This theorem is referenced by:  symgtgp  23340
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