Theorem efmndtmd 23252

Description: The monoid of endofunctions on a set 𝐴 is a topological monoid. Formerly part of proof for symgtgp 23257. (Contributed by AV, 23-Feb-2024.)

Hypothesis

Ref	Expression
efmndtmd.g	⊢ 𝑀 = (EndoFMnd‘𝐴)

Assertion

Ref	Expression
efmndtmd	⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ TopMnd)

Proof of Theorem efmndtmd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efmndtmd.g	. . 3 ⊢ 𝑀 = (EndoFMnd‘𝐴)
2	1	efmndmnd 18528	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ Mnd)
3		eqid 2738	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
4	1, 3	efmndtopn 18522	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
5		distopon 22147	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
6		eqid 2738	. . . . . . 7 ⊢ (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
7	6	pttoponconst 22748	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)))
8	5, 7	mpdan 684	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)))
9	1, 3	efmndbas 18510	. . . . . . . . 9 ⊢ (Base‘𝑀) = (𝐴 ↑m 𝐴)
10	9	eleq2i 2830	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝑀) ↔ 𝑥 ∈ (𝐴 ↑m 𝐴))
11	10	biimpi 215	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ (𝐴 ↑m 𝐴))
12	11	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ (𝐴 ↑m 𝐴)))
13	12	ssrdv 3927	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝑀) ⊆ (𝐴 ↑m 𝐴))
14		resttopon 22312	. . . . 5 ⊢ (((∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)) ∧ (Base‘𝑀) ⊆ (𝐴 ↑m 𝐴)) → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) ∈ (TopOn‘(Base‘𝑀)))
15	8, 13, 14	syl2anc 584	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) ∈ (TopOn‘(Base‘𝑀)))
16	4, 15	eqeltrrd 2840	. . 3 ⊢ (𝐴 ∈ 𝑉 → (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
17		eqid 2738	. . . 4 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
18	3, 17	istps 22083	. . 3 ⊢ (𝑀 ∈ TopSp ↔ (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
19	16, 18	sylibr 233	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ TopSp)
20		eqid 2738	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
21	1, 3, 20	efmndplusg 18519	. . . . . 6 ⊢ (+g‘𝑀) = (𝑥 ∈ (Base‘𝑀), 𝑦 ∈ (Base‘𝑀) ↦ (𝑥 ∘ 𝑦))
22		eqid 2738	. . . . . . 7 ⊢ ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)) = ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀))
23		distop 22145	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
24		eqid 2738	. . . . . . . . 9 ⊢ (𝒫 𝐴 ↑ko 𝒫 𝐴) = (𝒫 𝐴 ↑ko 𝒫 𝐴)
25	24	xkotopon 22751	. . . . . . . 8 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐴 ∈ Top) → (𝒫 𝐴 ↑ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
26	23, 23, 25	syl2anc 584	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ↑ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
27		cndis 22442	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴 ↑m 𝐴))
28	5, 27	mpdan 684	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴 ↑m 𝐴))
29	13, 28	sseqtrrd 3962	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝑀) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴))
30		disllycmp 22649	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Locally Comp)
31		llynlly 22628	. . . . . . . . 9 ⊢ (𝒫 𝐴 ∈ Locally Comp → 𝒫 𝐴 ∈ 𝑛-Locally Comp)
32	30, 31	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ 𝑛-Locally Comp)
33		eqid 2738	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 Cn 𝒫 𝐴), 𝑦 ∈ (𝒫 𝐴 Cn 𝒫 𝐴) ↦ (𝑥 ∘ 𝑦)) = (𝑥 ∈ (𝒫 𝐴 Cn 𝒫 𝐴), 𝑦 ∈ (𝒫 𝐴 Cn 𝒫 𝐴) ↦ (𝑥 ∘ 𝑦))
34	33	xkococn 22811	. . . . . . . 8 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐴 ∈ 𝑛-Locally Comp ∧ 𝒫 𝐴 ∈ Top) → (𝑥 ∈ (𝒫 𝐴 Cn 𝒫 𝐴), 𝑦 ∈ (𝒫 𝐴 Cn 𝒫 𝐴) ↦ (𝑥 ∘ 𝑦)) ∈ (((𝒫 𝐴 ↑ko 𝒫 𝐴) ×t (𝒫 𝐴 ↑ko 𝒫 𝐴)) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)))
35	23, 32, 23, 34	syl3anc 1370	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (𝒫 𝐴 Cn 𝒫 𝐴), 𝑦 ∈ (𝒫 𝐴 Cn 𝒫 𝐴) ↦ (𝑥 ∘ 𝑦)) ∈ (((𝒫 𝐴 ↑ko 𝒫 𝐴) ×t (𝒫 𝐴 ↑ko 𝒫 𝐴)) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)))
36	22, 26, 29, 22, 26, 29, 35	cnmpt2res 22828	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝑀), 𝑦 ∈ (Base‘𝑀) ↦ (𝑥 ∘ 𝑦)) ∈ ((((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)) ×t ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀))) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)))
37	21, 36	eqeltrid 2843	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (+g‘𝑀) ∈ ((((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)) ×t ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀))) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)))
38		xkopt 22806	. . . . . . . . . 10 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝒫 𝐴 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
39	23, 38	mpancom 685	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
40	39	oveq1d 7290	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)))
41	40, 4	eqtrd 2778	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
42	41, 41	oveq12d 7293	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)) ×t ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀))) = ((TopOpen‘𝑀) ×t (TopOpen‘𝑀)))
43	42	oveq1d 7290	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)) ×t ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀))) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)) = (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)))
44	37, 43	eleqtrd 2841	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)))
45		vex 3436	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
46		vex 3436	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
47	45, 46	coex 7777	. . . . . . . . . 10 ⊢ (𝑥 ∘ 𝑦) ∈ V
48	21, 47	fnmpoi 7910	. . . . . . . . 9 ⊢ (+g‘𝑀) Fn ((Base‘𝑀) × (Base‘𝑀))
49		eqid 2738	. . . . . . . . . 10 ⊢ (+𝑓‘𝑀) = (+𝑓‘𝑀)
50	3, 20, 49	plusfeq 18334	. . . . . . . . 9 ⊢ ((+g‘𝑀) Fn ((Base‘𝑀) × (Base‘𝑀)) → (+𝑓‘𝑀) = (+g‘𝑀))
51	48, 50	ax-mp 5	. . . . . . . 8 ⊢ (+𝑓‘𝑀) = (+g‘𝑀)
52	51	eqcomi 2747	. . . . . . 7 ⊢ (+g‘𝑀) = (+𝑓‘𝑀)
53	3, 52	mndplusf 18403	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (+g‘𝑀):((Base‘𝑀) × (Base‘𝑀))⟶(Base‘𝑀))
54		frn 6607	. . . . . 6 ⊢ ((+g‘𝑀):((Base‘𝑀) × (Base‘𝑀))⟶(Base‘𝑀) → ran (+g‘𝑀) ⊆ (Base‘𝑀))
55	2, 53, 54	3syl 18	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ran (+g‘𝑀) ⊆ (Base‘𝑀))
56		cnrest2 22437	. . . . 5 ⊢ (((𝒫 𝐴 ↑ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)) ∧ ran (+g‘𝑀) ⊆ (Base‘𝑀) ∧ (Base‘𝑀) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴)) → ((+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)) ↔ (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)))))
57	26, 55, 29, 56	syl3anc 1370	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)) ↔ (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)))))
58	44, 57	mpbid 231	. . 3 ⊢ (𝐴 ∈ 𝑉 → (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀))))
59	41	oveq2d 7291	. . 3 ⊢ (𝐴 ∈ 𝑉 → (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀))) = (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀)))
60	58, 59	eleqtrd 2841	. 2 ⊢ (𝐴 ∈ 𝑉 → (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀)))
61	52, 17	istmd 23225	. 2 ⊢ (𝑀 ∈ TopMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ TopSp ∧ (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀))))
62	2, 19, 60, 61	syl3anbrc 1342	1 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ TopMnd)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887  𝒫 cpw 4533  {csn 4561   × cxp 5587  ran crn 5590   ∘ ccom 5593   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277   ↑_m cmap 8615  Basecbs 16912  +_gcplusg 16962   ↾_t crest 17131  TopOpenctopn 17132  ∏_tcpt 17149  +_𝑓cplusf 18323  Mndcmnd 18385  EndoFMndcefmnd 18507  Topctop 22042  TopOnctopon 22059  TopSpctps 22081   Cn ccn 22375  Compccmp 22537  Locally clly 22615  𝑛-Locally cnlly 22616   ×_t ctx 22711   ↑_ko cxko 22712  TopMndctmd 23221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fi 9170  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-struct 16848  df-slot 16883  df-ndx 16895  df-base 16913  df-plusg 16975  df-tset 16981  df-rest 17133  df-topn 17134  df-topgen 17154  df-pt 17155  df-plusf 18325  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-efmnd 18508  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-ntr 22171  df-nei 22249  df-cn 22378  df-cmp 22538  df-lly 22617  df-nlly 22618  df-tx 22713  df-xko 22714  df-tmd 23223

This theorem is referenced by:  symgtgp  23257