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Theorem efmndtmd 24125
Description: The monoid of endofunctions on a set 𝐴 is a topological monoid. Formerly part of proof for symgtgp 24130. (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 18915 . 2 (𝐴𝑉𝑀 ∈ Mnd)
3 eqid 2735 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
41, 3efmndtopn 18909 . . . 4 (𝐴𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
5 distopon 23020 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
6 eqid 2735 . . . . . . 7 (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
76pttoponconst 23621 . . . . . 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 18897 . . . . . . . . 9 (Base‘𝑀) = (𝐴m 𝐴)
109eleq2i 2831 . . . . . . . 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 4001 . . . . 5 (𝐴𝑉 → (Base‘𝑀) ⊆ (𝐴m 𝐴))
14 resttopon 23185 . . . . 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 2840 . . 3 (𝐴𝑉 → (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
17 eqid 2735 . . . 4 (TopOpen‘𝑀) = (TopOpen‘𝑀)
183, 17istps 22956 . . 3 (𝑀 ∈ TopSp ↔ (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
1916, 18sylibr 234 . 2 (𝐴𝑉𝑀 ∈ TopSp)
20 eqid 2735 . . . . . . 7 (+g𝑀) = (+g𝑀)
211, 3, 20efmndplusg 18906 . . . . . 6 (+g𝑀) = (𝑥 ∈ (Base‘𝑀), 𝑦 ∈ (Base‘𝑀) ↦ (𝑥𝑦))
22 eqid 2735 . . . . . . 7 ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) = ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀))
23 distop 23018 . . . . . . . 8 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
24 eqid 2735 . . . . . . . . 9 (𝒫 𝐴ko 𝒫 𝐴) = (𝒫 𝐴ko 𝒫 𝐴)
2524xkotopon 23624 . . . . . . . 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 23315 . . . . . . . . 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 4037 . . . . . . 7 (𝐴𝑉 → (Base‘𝑀) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴))
30 disllycmp 23522 . . . . . . . . 9 (𝐴𝑉 → 𝒫 𝐴 ∈ Locally Comp)
31 llynlly 23501 . . . . . . . . 9 (𝒫 𝐴 ∈ Locally Comp → 𝒫 𝐴 ∈ 𝑛-Locally Comp)
3230, 31syl 17 . . . . . . . 8 (𝐴𝑉 → 𝒫 𝐴 ∈ 𝑛-Locally Comp)
33 eqid 2735 . . . . . . . . 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 23684 . . . . . . . 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 23701 . . . . . 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 2843 . . . . 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 23679 . . . . . . . . . 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 7446 . . . . . . . 8 (𝐴𝑉 → ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)))
4140, 4eqtrd 2775 . . . . . . 7 (𝐴𝑉 → ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
4241, 41oveq12d 7449 . . . . . 6 (𝐴𝑉 → (((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀)) ×t ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀))) = ((TopOpen‘𝑀) ×t (TopOpen‘𝑀)))
4342oveq1d 7446 . . . . 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 2841 . . . 4 (𝐴𝑉 → (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (𝒫 𝐴ko 𝒫 𝐴)))
45 vex 3482 . . . . . . . . . . 11 𝑥 ∈ V
46 vex 3482 . . . . . . . . . . 11 𝑦 ∈ V
4745, 46coex 7953 . . . . . . . . . 10 (𝑥𝑦) ∈ V
4821, 47fnmpoi 8094 . . . . . . . . 9 (+g𝑀) Fn ((Base‘𝑀) × (Base‘𝑀))
49 eqid 2735 . . . . . . . . . 10 (+𝑓𝑀) = (+𝑓𝑀)
503, 20, 49plusfeq 18674 . . . . . . . . 9 ((+g𝑀) Fn ((Base‘𝑀) × (Base‘𝑀)) → (+𝑓𝑀) = (+g𝑀))
5148, 50ax-mp 5 . . . . . . . 8 (+𝑓𝑀) = (+g𝑀)
5251eqcomi 2744 . . . . . . 7 (+g𝑀) = (+𝑓𝑀)
533, 52mndplusf 18778 . . . . . 6 (𝑀 ∈ Mnd → (+g𝑀):((Base‘𝑀) × (Base‘𝑀))⟶(Base‘𝑀))
54 frn 6744 . . . . . 6 ((+g𝑀):((Base‘𝑀) × (Base‘𝑀))⟶(Base‘𝑀) → ran (+g𝑀) ⊆ (Base‘𝑀))
552, 53, 543syl 18 . . . . 5 (𝐴𝑉 → ran (+g𝑀) ⊆ (Base‘𝑀))
56 cnrest2 23310 . . . . 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 232 . . 3 (𝐴𝑉 → (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn ((𝒫 𝐴ko 𝒫 𝐴) ↾t (Base‘𝑀))))
5941oveq2d 7447 . . 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 2841 . 2 (𝐴𝑉 → (+g𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀)))
6152, 17istmd 24098 . 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 206   = wceq 1537  wcel 2106  wss 3963  𝒫 cpw 4605  {csn 4631   × cxp 5687  ran crn 5690  ccom 5693   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  m cmap 8865  Basecbs 17245  +gcplusg 17298  t crest 17467  TopOpenctopn 17468  tcpt 17485  +𝑓cplusf 18663  Mndcmnd 18760  EndoFMndcefmnd 18894  Topctop 22915  TopOnctopon 22932  TopSpctps 22954   Cn ccn 23248  Compccmp 23410  Locally clly 23488  𝑛-Locally cnlly 23489   ×t ctx 23584  ko cxko 23585  TopMndctmd 24094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fi 9449  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-tset 17317  df-rest 17469  df-topn 17470  df-topgen 17490  df-pt 17491  df-plusf 18665  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-efmnd 18895  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-ntr 23044  df-nei 23122  df-cn 23251  df-cmp 23411  df-lly 23490  df-nlly 23491  df-tx 23586  df-xko 23587  df-tmd 24096
This theorem is referenced by:  symgtgp  24130
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