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Theorem efmndtmd 24110
Description: The monoid of endofunctions on a set 𝐴 is a topological monoid. Formerly part of proof for symgtgp 24115. (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 18903 . 2 (𝐴𝑉𝑀 ∈ Mnd)
3 eqid 2736 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
41, 3efmndtopn 18897 . . . 4 (𝐴𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
5 distopon 23005 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
6 eqid 2736 . . . . . . 7 (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
76pttoponconst 23606 . . . . . 6 ((𝐴𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
85, 7mpdan 687 . . . . 5 (𝐴𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴m 𝐴)))
91, 3efmndbas 18885 . . . . . . . . 9 (Base‘𝑀) = (𝐴m 𝐴)
109eleq2i 2832 . . . . . . . 8 (𝑥 ∈ (Base‘𝑀) ↔ 𝑥 ∈ (𝐴m 𝐴))
1110biimpi 216 . . . . . . 7 (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ (𝐴m 𝐴))
1211a1i 11 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ (𝐴m 𝐴)))
1312ssrdv 3988 . . . . 5 (𝐴𝑉 → (Base‘𝑀) ⊆ (𝐴m 𝐴))
14 resttopon 23170 . . . . 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 2841 . . 3 (𝐴𝑉 → (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
17 eqid 2736 . . . 4 (TopOpen‘𝑀) = (TopOpen‘𝑀)
183, 17istps 22941 . . 3 (𝑀 ∈ TopSp ↔ (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
1916, 18sylibr 234 . 2 (𝐴𝑉𝑀 ∈ TopSp)
20 eqid 2736 . . . . . . 7 (+g𝑀) = (+g𝑀)
211, 3, 20efmndplusg 18894 . . . . . 6 (+g𝑀) = (𝑥 ∈ (Base‘𝑀), 𝑦 ∈ (Base‘𝑀) ↦ (𝑥𝑦))
22 eqid 2736 . . . . . . 7 ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) = ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀))
23 distop 23003 . . . . . . . 8 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
24 eqid 2736 . . . . . . . . 9 (𝒫 𝐴ko 𝒫 𝐴) = (𝒫 𝐴ko 𝒫 𝐴)
2524xkotopon 23609 . . . . . . . 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 23300 . . . . . . . . 9 ((𝐴𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴m 𝐴))
285, 27mpdan 687 . . . . . . . 8 (𝐴𝑉 → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴m 𝐴))
2913, 28sseqtrrd 4020 . . . . . . 7 (𝐴𝑉 → (Base‘𝑀) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴))
30 disllycmp 23507 . . . . . . . . 9 (𝐴𝑉 → 𝒫 𝐴 ∈ Locally Comp)
31 llynlly 23486 . . . . . . . . 9 (𝒫 𝐴 ∈ Locally Comp → 𝒫 𝐴 ∈ 𝑛-Locally Comp)
3230, 31syl 17 . . . . . . . 8 (𝐴𝑉 → 𝒫 𝐴 ∈ 𝑛-Locally Comp)
33 eqid 2736 . . . . . . . . 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 23669 . . . . . . . 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 1372 . . . . . . 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 23686 . . . . . 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 2844 . . . . 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 23664 . . . . . . . . . 10 ((𝒫 𝐴 ∈ Top ∧ 𝐴𝑉) → (𝒫 𝐴ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
3923, 38mpancom 688 . . . . . . . . 9 (𝐴𝑉 → (𝒫 𝐴ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
4039oveq1d 7447 . . . . . . . 8 (𝐴𝑉 → ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)))
4140, 4eqtrd 2776 . . . . . . 7 (𝐴𝑉 → ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
4241, 41oveq12d 7450 . . . . . 6 (𝐴𝑉 → (((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) ×t ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀))) = ((TopOpen‘𝑀) ×t (TopOpen‘𝑀)))
4342oveq1d 7447 . . . . 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 2842 . . . 4 (𝐴𝑉 → (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (𝒫 𝐴ko 𝒫 𝐴)))
45 vex 3483 . . . . . . . . . . 11 𝑥 ∈ V
46 vex 3483 . . . . . . . . . . 11 𝑦 ∈ V
4745, 46coex 7953 . . . . . . . . . 10 (𝑥𝑦) ∈ V
4821, 47fnmpoi 8096 . . . . . . . . 9 (+g𝑀) Fn ((Base‘𝑀) × (Base‘𝑀))
49 eqid 2736 . . . . . . . . . 10 (+𝑓𝑀) = (+𝑓𝑀)
503, 20, 49plusfeq 18662 . . . . . . . . 9 ((+g𝑀) Fn ((Base‘𝑀) × (Base‘𝑀)) → (+𝑓𝑀) = (+g𝑀))
5148, 50ax-mp 5 . . . . . . . 8 (+𝑓𝑀) = (+g𝑀)
5251eqcomi 2745 . . . . . . 7 (+g𝑀) = (+𝑓𝑀)
533, 52mndplusf 18766 . . . . . 6 (𝑀 ∈ Mnd → (+g𝑀):((Base‘𝑀) × (Base‘𝑀))⟶(Base‘𝑀))
54 frn 6742 . . . . . 6 ((+g𝑀):((Base‘𝑀) × (Base‘𝑀))⟶(Base‘𝑀) → ran (+g𝑀) ⊆ (Base‘𝑀))
552, 53, 543syl 18 . . . . 5 (𝐴𝑉 → ran (+g𝑀) ⊆ (Base‘𝑀))
56 cnrest2 23295 . . . . 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 1372 . . . 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 232 . . 3 (𝐴𝑉 → (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀))))
5941oveq2d 7448 . . 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 2842 . 2 (𝐴𝑉 → (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀)))
6152, 17istmd 24083 . 2 (𝑀 ∈ TopMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ TopSp ∧ (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀))))
622, 19, 60, 61syl3anbrc 1343 1 (𝐴𝑉𝑀 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  wcel 2107  wss 3950  𝒫 cpw 4599  {csn 4625   × cxp 5682  ran crn 5685  ccom 5688   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  cmpo 7434  m cmap 8867  Basecbs 17248  +gcplusg 17298  t crest 17466  TopOpenctopn 17467  tcpt 17484  +𝑓cplusf 18651  Mndcmnd 18748  EndoFMndcefmnd 18882  Topctop 22900  TopOnctopon 22917  TopSpctps 22939   Cn ccn 23233  Compccmp 23395  Locally clly 23473  𝑛-Locally cnlly 23474   ×t ctx 23569  ko cxko 23570  TopMndctmd 24079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fi 9452  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17249  df-plusg 17311  df-tset 17317  df-rest 17468  df-topn 17469  df-topgen 17489  df-pt 17490  df-plusf 18653  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-efmnd 18883  df-top 22901  df-topon 22918  df-topsp 22940  df-bases 22954  df-ntr 23029  df-nei 23107  df-cn 23236  df-cmp 23396  df-lly 23475  df-nlly 23476  df-tx 23571  df-xko 23572  df-tmd 24081
This theorem is referenced by:  symgtgp  24115
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