Theorem efmndtmd 24227

Description: The monoid of endofunctions on a set 𝐴 is a topological monoid. Formerly part of proof for symgtgp 24232. (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 18948	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ Mnd)
3		eqid 2769	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
4	1, 3	efmndtopn 18942	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
5		distopon 23123	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
6		eqid 2769	. . . . . . 7 ⊢ (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
7	6	pttoponconst 23723	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)))
8	5, 7	mpdan 699	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)))
9	1, 3	efmndbas 18930	. . . . . . . . 9 ⊢ (Base‘𝑀) = (𝐴 ↑m 𝐴)
10	9	eleq2i 2861	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝑀) ↔ 𝑥 ∈ (𝐴 ↑m 𝐴))
11	10	biimpi 219	. . . . . . 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 3949	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝑀) ⊆ (𝐴 ↑m 𝐴))
14		resttopon 23287	. . . . 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 595	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) ∈ (TopOn‘(Base‘𝑀)))
16	4, 15	eqeltrrd 2870	. . 3 ⊢ (𝐴 ∈ 𝑉 → (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
17		eqid 2769	. . . 4 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
18	3, 17	istps 23060	. . 3 ⊢ (𝑀 ∈ TopSp ↔ (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
19	16, 18	sylibr 237	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ TopSp)
20		eqid 2769	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
21	1, 3, 20	efmndplusg 18939	. . . . . 6 ⊢ (+g‘𝑀) = (𝑥 ∈ (Base‘𝑀), 𝑦 ∈ (Base‘𝑀) ↦ (𝑥 ∘ 𝑦))
22		eqid 2769	. . . . . . 7 ⊢ ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)) = ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀))
23		distop 23121	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
24		eqid 2769	. . . . . . . . 9 ⊢ (𝒫 𝐴 ↑ko 𝒫 𝐴) = (𝒫 𝐴 ↑ko 𝒫 𝐴)
25	24	xkotopon 23726	. . . . . . . 8 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐴 ∈ Top) → (𝒫 𝐴 ↑ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
26	23, 23, 25	syl2anc 595	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ↑ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
27		cndis 23417	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴 ↑m 𝐴))
28	5, 27	mpdan 699	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴 ↑m 𝐴))
29	13, 28	sseqtrrd 3980	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝑀) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴))
30		disllycmp 23624	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Locally Comp)
31		llynlly 23603	. . . . . . . . 9 ⊢ (𝒫 𝐴 ∈ Locally Comp → 𝒫 𝐴 ∈ 𝑛-Locally Comp)
32	30, 31	syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ 𝑛-Locally Comp)
33		eqid 2769	. . . . . . . . 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 23786	. . . . . . . 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 1396	. . . . . . 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 23803	. . . . . 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 2873	. . . . 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 23781	. . . . . . . . . 10 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝒫 𝐴 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
39	23, 38	mpancom 700	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
40	39	oveq1d 7426	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)))
41	40, 4	eqtrd 2804	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
42	41, 41	oveq12d 7429	. . . . . 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 7426	. . . . 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 2871	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)))
45		vex 3465	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
46		vex 3465	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
47	45, 46	coex 7927	. . . . . . . . . 10 ⊢ (𝑥 ∘ 𝑦) ∈ V
48	21, 47	fnmpoi 8067	. . . . . . . . 9 ⊢ (+g‘𝑀) Fn ((Base‘𝑀) × (Base‘𝑀))
49		eqid 2769	. . . . . . . . . 10 ⊢ (+𝑓‘𝑀) = (+𝑓‘𝑀)
50	3, 20, 49	plusfeq 18706	. . . . . . . . 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 2778	. . . . . . 7 ⊢ (+g‘𝑀) = (+𝑓‘𝑀)
53	3, 52	mndplusf 18810	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (+g‘𝑀):((Base‘𝑀) × (Base‘𝑀))⟶(Base‘𝑀))
54		frn 6714	. . . . . 6 ⊢ ((+g‘𝑀):((Base‘𝑀) × (Base‘𝑀))⟶(Base‘𝑀) → ran (+g‘𝑀) ⊆ (Base‘𝑀))
55	2, 53, 54	3syl 19	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ran (+g‘𝑀) ⊆ (Base‘𝑀))
56		cnrest2 23412	. . . . 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 1396	. . . 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 235	. . 3 ⊢ (𝐴 ∈ 𝑉 → (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀))))
59	41	oveq2d 7427	. . 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 2871	. 2 ⊢ (𝐴 ∈ 𝑉 → (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀)))
61	52, 17	istmd 24200	. 2 ⊢ (𝑀 ∈ TopMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ TopSp ∧ (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀))))
62	2, 19, 60, 61	syl3anbrc 1360	1 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ TopMnd)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149   ⊆ wss 3911  𝒫 cpw 4565  {csn 4592   × cxp 5660  ran crn 5663   ∘ ccom 5666   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413   ↑_m cmap 8824  Basecbs 17269  +_gcplusg 17310   ↾_t crest 17473  TopOpenctopn 17474  ∏_tcpt 17491  +_𝑓cplusf 18695  Mndcmnd 18792  EndoFMndcefmnd 18927  Topctop 23019  TopOnctopon 23036  TopSpctps 23058   Cn ccn 23350  Compccmp 23512  Locally clly 23590  𝑛-Locally cnlly 23591   ×_t ctx 23686   ↑_ko cxko 23687  TopMndctmd 24196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-er 8694  df-map 8826  df-ixp 8896  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fi 9371  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-z 12592  df-uz 12863  df-fz 13536  df-struct 17207  df-slot 17242  df-ndx 17254  df-base 17270  df-plusg 17323  df-tset 17329  df-rest 17475  df-topn 17476  df-topgen 17496  df-pt 17497  df-plusf 18697  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-efmnd 18928  df-top 23020  df-topon 23037  df-topsp 23059  df-bases 23072  df-ntr 23146  df-nei 23224  df-cn 23353  df-cmp 23513  df-lly 23592  df-nlly 23593  df-tx 23688  df-xko 23689  df-tmd 24198

This theorem is referenced by:  symgtgp  24232