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Theorem efmndtmd 22998
Description: The monoid of endofunctions on a set 𝐴 is a topological monoid. Formerly part of proof for symgtgp 23003. (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 18316 . 2 (𝐴𝑉𝑀 ∈ Mnd)
3 eqid 2737 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
41, 3efmndtopn 18310 . . . 4 (𝐴𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
5 distopon 21894 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
6 eqid 2737 . . . . . . 7 (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
76pttoponconst 22494 . . . . . 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 18298 . . . . . . . . 9 (Base‘𝑀) = (𝐴m 𝐴)
109eleq2i 2829 . . . . . . . 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 3907 . . . . 5 (𝐴𝑉 → (Base‘𝑀) ⊆ (𝐴m 𝐴))
14 resttopon 22058 . . . . 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 2839 . . 3 (𝐴𝑉 → (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
17 eqid 2737 . . . 4 (TopOpen‘𝑀) = (TopOpen‘𝑀)
183, 17istps 21831 . . 3 (𝑀 ∈ TopSp ↔ (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
1916, 18sylibr 237 . 2 (𝐴𝑉𝑀 ∈ TopSp)
20 eqid 2737 . . . . . . 7 (+g𝑀) = (+g𝑀)
211, 3, 20efmndplusg 18307 . . . . . 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 21892 . . . . . . . 8 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
24 eqid 2737 . . . . . . . . 9 (𝒫 𝐴ko 𝒫 𝐴) = (𝒫 𝐴ko 𝒫 𝐴)
2524xkotopon 22497 . . . . . . . 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 22188 . . . . . . . . 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 3942 . . . . . . 7 (𝐴𝑉 → (Base‘𝑀) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴))
30 disllycmp 22395 . . . . . . . . 9 (𝐴𝑉 → 𝒫 𝐴 ∈ Locally Comp)
31 llynlly 22374 . . . . . . . . 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 22557 . . . . . . . 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 1373 . . . . . . 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 22574 . . . . . 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 22552 . . . . . . . . . 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 7228 . . . . . . . 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 7231 . . . . . 6 (𝐴𝑉 → (((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) ×t ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀))) = ((TopOpen‘𝑀) ×t (TopOpen‘𝑀)))
4342oveq1d 7228 . . . . 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 3412 . . . . . . . . . . 11 𝑥 ∈ V
46 vex 3412 . . . . . . . . . . 11 𝑦 ∈ V
4745, 46coex 7708 . . . . . . . . . 10 (𝑥𝑦) ∈ V
4821, 47fnmpoi 7840 . . . . . . . . 9 (+g𝑀) Fn ((Base‘𝑀) × (Base‘𝑀))
49 eqid 2737 . . . . . . . . . 10 (+𝑓𝑀) = (+𝑓𝑀)
503, 20, 49plusfeq 18122 . . . . . . . . 9 ((+g𝑀) Fn ((Base‘𝑀) × (Base‘𝑀)) → (+𝑓𝑀) = (+g𝑀))
5148, 50ax-mp 5 . . . . . . . 8 (+𝑓𝑀) = (+g𝑀)
5251eqcomi 2746 . . . . . . 7 (+g𝑀) = (+𝑓𝑀)
533, 52mndplusf 18191 . . . . . 6 (𝑀 ∈ Mnd → (+g𝑀):((Base‘𝑀) × (Base‘𝑀))⟶(Base‘𝑀))
54 frn 6552 . . . . . 6 ((+g𝑀):((Base‘𝑀) × (Base‘𝑀))⟶(Base‘𝑀) → ran (+g𝑀) ⊆ (Base‘𝑀))
552, 53, 543syl 18 . . . . 5 (𝐴𝑉 → ran (+g𝑀) ⊆ (Base‘𝑀))
56 cnrest2 22183 . . . . 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 1373 . . . 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 7229 . . 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 22971 . 2 (𝑀 ∈ TopMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ TopSp ∧ (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀))))
622, 19, 60, 61syl3anbrc 1345 1 (𝐴𝑉𝑀 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wcel 2110  wss 3866  𝒫 cpw 4513  {csn 4541   × cxp 5549  ran crn 5552  ccom 5555   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  cmpo 7215  m cmap 8508  Basecbs 16760  +gcplusg 16802  t crest 16925  TopOpenctopn 16926  tcpt 16943  +𝑓cplusf 18111  Mndcmnd 18173  EndoFMndcefmnd 18295  Topctop 21790  TopOnctopon 21807  TopSpctps 21829   Cn ccn 22121  Compccmp 22283  Locally clly 22361  𝑛-Locally cnlly 22362   ×t ctx 22457  ko cxko 22458  TopMndctmd 22967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-map 8510  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fi 9027  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-uz 12439  df-fz 13096  df-struct 16700  df-slot 16735  df-ndx 16745  df-base 16761  df-plusg 16815  df-tset 16821  df-rest 16927  df-topn 16928  df-topgen 16948  df-pt 16949  df-plusf 18113  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-efmnd 18296  df-top 21791  df-topon 21808  df-topsp 21830  df-bases 21843  df-ntr 21917  df-nei 21995  df-cn 22124  df-cmp 22284  df-lly 22363  df-nlly 22364  df-tx 22459  df-xko 22460  df-tmd 22969
This theorem is referenced by:  symgtgp  23003
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