Theorem efmndtmd 23825

Description: The monoid of endofunctions on a set 𝐴 is a topological monoid. Formerly part of proof for symgtgp 23830. (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 18806	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ Mnd)
3		eqid 2730	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
4	1, 3	efmndtopn 18800	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
5		distopon 22720	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
6		eqid 2730	. . . . . . 7 ⊢ (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
7	6	pttoponconst 23321	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)))
8	5, 7	mpdan 683	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)))
9	1, 3	efmndbas 18788	. . . . . . . . 9 ⊢ (Base‘𝑀) = (𝐴 ↑m 𝐴)
10	9	eleq2i 2823	. . . . . . . 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 3987	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝑀) ⊆ (𝐴 ↑m 𝐴))
14		resttopon 22885	. . . . 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 582	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) ∈ (TopOn‘(Base‘𝑀)))
16	4, 15	eqeltrrd 2832	. . 3 ⊢ (𝐴 ∈ 𝑉 → (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
17		eqid 2730	. . . 4 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
18	3, 17	istps 22656	. . 3 ⊢ (𝑀 ∈ TopSp ↔ (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
19	16, 18	sylibr 233	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ TopSp)
20		eqid 2730	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
21	1, 3, 20	efmndplusg 18797	. . . . . 6 ⊢ (+g‘𝑀) = (𝑥 ∈ (Base‘𝑀), 𝑦 ∈ (Base‘𝑀) ↦ (𝑥 ∘ 𝑦))
22		eqid 2730	. . . . . . 7 ⊢ ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)) = ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀))
23		distop 22718	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
24		eqid 2730	. . . . . . . . 9 ⊢ (𝒫 𝐴 ↑ko 𝒫 𝐴) = (𝒫 𝐴 ↑ko 𝒫 𝐴)
25	24	xkotopon 23324	. . . . . . . 8 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐴 ∈ Top) → (𝒫 𝐴 ↑ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
26	23, 23, 25	syl2anc 582	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ↑ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
27		cndis 23015	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴 ↑m 𝐴))
28	5, 27	mpdan 683	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴 ↑m 𝐴))
29	13, 28	sseqtrrd 4022	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝑀) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴))
30		disllycmp 23222	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Locally Comp)
31		llynlly 23201	. . . . . . . . 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 2730	. . . . . . . . 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 23384	. . . . . . . 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 1369	. . . . . . 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 23401	. . . . . 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 2835	. . . . 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 23379	. . . . . . . . . 10 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝒫 𝐴 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
39	23, 38	mpancom 684	. . . . . . . . 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 2770	. . . . . . 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 2833	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)))
45		vex 3476	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
46		vex 3476	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
47	45, 46	coex 7923	. . . . . . . . . 10 ⊢ (𝑥 ∘ 𝑦) ∈ V
48	21, 47	fnmpoi 8058	. . . . . . . . 9 ⊢ (+g‘𝑀) Fn ((Base‘𝑀) × (Base‘𝑀))
49		eqid 2730	. . . . . . . . . 10 ⊢ (+𝑓‘𝑀) = (+𝑓‘𝑀)
50	3, 20, 49	plusfeq 18573	. . . . . . . . 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 2739	. . . . . . 7 ⊢ (+g‘𝑀) = (+𝑓‘𝑀)
53	3, 52	mndplusf 18677	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (+g‘𝑀):((Base‘𝑀) × (Base‘𝑀))⟶(Base‘𝑀))
54		frn 6723	. . . . . 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 23010	. . . . 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 1369	. . . 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 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 2833	. 2 ⊢ (𝐴 ∈ 𝑉 → (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀)))
61	52, 17	istmd 23798	. 2 ⊢ (𝑀 ∈ TopMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ TopSp ∧ (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀))))
62	2, 19, 60, 61	syl3anbrc 1341	1 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ TopMnd)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2104   ⊆ wss 3947  𝒫 cpw 4601  {csn 4627   × cxp 5673  ran crn 5676   ∘ ccom 5679   Fn wfn 6537  ⟶wf 6538  ‘cfv 6542  (class class class)co 7411   ∈ cmpo 7413   ↑_m cmap 8822  Basecbs 17148  +_gcplusg 17201   ↾_t crest 17370  TopOpenctopn 17371  ∏_tcpt 17388  +_𝑓cplusf 18562  Mndcmnd 18659  EndoFMndcefmnd 18785  Topctop 22615  TopOnctopon 22632  TopSpctps 22654   Cn ccn 22948  Compccmp 23110  Locally clly 23188  𝑛-Locally cnlly 23189   ×_t ctx 23284   ↑_ko cxko 23285  TopMndctmd 23794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fi 9408  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-struct 17084  df-slot 17119  df-ndx 17131  df-base 17149  df-plusg 17214  df-tset 17220  df-rest 17372  df-topn 17373  df-topgen 17393  df-pt 17394  df-plusf 18564  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-efmnd 18786  df-top 22616  df-topon 22633  df-topsp 22655  df-bases 22669  df-ntr 22744  df-nei 22822  df-cn 22951  df-cmp 23111  df-lly 23190  df-nlly 23191  df-tx 23286  df-xko 23287  df-tmd 23796

This theorem is referenced by:  symgtgp  23830