Theorem efmndtmd 24088

Description: The monoid of endofunctions on a set 𝐴 is a topological monoid. Formerly part of proof for symgtgp 24093. (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 18852	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ Mnd)
3		eqid 2741	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
4	1, 3	efmndtopn 18846	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
5		distopon 22984	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
6		eqid 2741	. . . . . . 7 ⊢ (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
7	6	pttoponconst 23584	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)))
8	5, 7	mpdan 694	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)))
9	1, 3	efmndbas 18834	. . . . . . . . 9 ⊢ (Base‘𝑀) = (𝐴 ↑m 𝐴)
10	9	eleq2i 2833	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝑀) ↔ 𝑥 ∈ (𝐴 ↑m 𝐴))
11	10	biimpi 218	. . . . . . 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 3923	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝑀) ⊆ (𝐴 ↑m 𝐴))
14		resttopon 23148	. . . . 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 591	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) ∈ (TopOn‘(Base‘𝑀)))
16	4, 15	eqeltrrd 2842	. . 3 ⊢ (𝐴 ∈ 𝑉 → (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
17		eqid 2741	. . . 4 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
18	3, 17	istps 22921	. . 3 ⊢ (𝑀 ∈ TopSp ↔ (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
19	16, 18	sylibr 236	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ TopSp)
20		eqid 2741	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
21	1, 3, 20	efmndplusg 18843	. . . . . 6 ⊢ (+g‘𝑀) = (𝑥 ∈ (Base‘𝑀), 𝑦 ∈ (Base‘𝑀) ↦ (𝑥 ∘ 𝑦))
22		eqid 2741	. . . . . . 7 ⊢ ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)) = ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀))
23		distop 22982	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
24		eqid 2741	. . . . . . . . 9 ⊢ (𝒫 𝐴 ↑ko 𝒫 𝐴) = (𝒫 𝐴 ↑ko 𝒫 𝐴)
25	24	xkotopon 23587	. . . . . . . 8 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐴 ∈ Top) → (𝒫 𝐴 ↑ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
26	23, 23, 25	syl2anc 591	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ↑ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
27		cndis 23278	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴 ↑m 𝐴))
28	5, 27	mpdan 694	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴 ↑m 𝐴))
29	13, 28	sseqtrrd 3954	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝑀) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴))
30		disllycmp 23485	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Locally Comp)
31		llynlly 23464	. . . . . . . . 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 2741	. . . . . . . . 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 23647	. . . . . . . 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 1380	. . . . . . 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 23664	. . . . . 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 2845	. . . . 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 23642	. . . . . . . . . 10 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝒫 𝐴 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
39	23, 38	mpancom 695	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
40	39	oveq1d 7375	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)))
41	40, 4	eqtrd 2776	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
42	41, 41	oveq12d 7378	. . . . . 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 7375	. . . . 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 2843	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)))
45		vex 3437	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
46		vex 3437	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
47	45, 46	coex 7874	. . . . . . . . . 10 ⊢ (𝑥 ∘ 𝑦) ∈ V
48	21, 47	fnmpoi 8016	. . . . . . . . 9 ⊢ (+g‘𝑀) Fn ((Base‘𝑀) × (Base‘𝑀))
49		eqid 2741	. . . . . . . . . 10 ⊢ (+𝑓‘𝑀) = (+𝑓‘𝑀)
50	3, 20, 49	plusfeq 18611	. . . . . . . . 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 2750	. . . . . . 7 ⊢ (+g‘𝑀) = (+𝑓‘𝑀)
53	3, 52	mndplusf 18715	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (+g‘𝑀):((Base‘𝑀) × (Base‘𝑀))⟶(Base‘𝑀))
54		frn 6666	. . . . . 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 23273	. . . . 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 1380	. . . 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 234	. . 3 ⊢ (𝐴 ∈ 𝑉 → (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀))))
59	41	oveq2d 7376	. . 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 2843	. 2 ⊢ (𝐴 ∈ 𝑉 → (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀)))
61	52, 17	istmd 24061	. 2 ⊢ (𝑀 ∈ TopMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ TopSp ∧ (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀))))
62	2, 19, 60, 61	syl3anbrc 1351	1 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ TopMnd)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1548   ∈ wcel 2121   ⊆ wss 3885  𝒫 cpw 4532  {csn 4558   × cxp 5619  ran crn 5622   ∘ ccom 5625   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489  (class class class)co 7360   ∈ cmpo 7362   ↑_m cmap 8767  Basecbs 17174  +_gcplusg 17215   ↾_t crest 17378  TopOpenctopn 17379  ∏_tcpt 17396  +_𝑓cplusf 18600  Mndcmnd 18697  EndoFMndcefmnd 18831  Topctop 22880  TopOnctopon 22897  TopSpctps 22919   Cn ccn 23211  Compccmp 23373  Locally clly 23451  𝑛-Locally cnlly 23452   ×_t ctx 23547   ↑_ko cxko 23548  TopMndctmd 24057

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-iin 4927  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fi 9318  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-uz 12784  df-fz 13457  df-struct 17112  df-slot 17147  df-ndx 17159  df-base 17175  df-plusg 17228  df-tset 17234  df-rest 17380  df-topn 17381  df-topgen 17401  df-pt 17402  df-plusf 18602  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-efmnd 18832  df-top 22881  df-topon 22898  df-topsp 22920  df-bases 22933  df-ntr 23007  df-nei 23085  df-cn 23214  df-cmp 23374  df-lly 23453  df-nlly 23454  df-tx 23549  df-xko 23550  df-tmd 24059

This theorem is referenced by:  symgtgp  24093