Theorem efmndtmd 23826

Description: The monoid of endofunctions on a set 𝐴 is a topological monoid. Formerly part of proof for symgtgp 23831. (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 18807	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ Mnd)
3		eqid 2731	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
4	1, 3	efmndtopn 18801	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
5		distopon 22721	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
6		eqid 2731	. . . . . . 7 ⊢ (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
7	6	pttoponconst 23322	. . . . . 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 18789	. . . . . . . . 9 ⊢ (Base‘𝑀) = (𝐴 ↑m 𝐴)
10	9	eleq2i 2824	. . . . . . . 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 3988	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝑀) ⊆ (𝐴 ↑m 𝐴))
14		resttopon 22886	. . . . 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 583	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)) ∈ (TopOn‘(Base‘𝑀)))
16	4, 15	eqeltrrd 2833	. . 3 ⊢ (𝐴 ∈ 𝑉 → (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
17		eqid 2731	. . . 4 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
18	3, 17	istps 22657	. . 3 ⊢ (𝑀 ∈ TopSp ↔ (TopOpen‘𝑀) ∈ (TopOn‘(Base‘𝑀)))
19	16, 18	sylibr 233	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ TopSp)
20		eqid 2731	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
21	1, 3, 20	efmndplusg 18798	. . . . . 6 ⊢ (+g‘𝑀) = (𝑥 ∈ (Base‘𝑀), 𝑦 ∈ (Base‘𝑀) ↦ (𝑥 ∘ 𝑦))
22		eqid 2731	. . . . . . 7 ⊢ ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)) = ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀))
23		distop 22719	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
24		eqid 2731	. . . . . . . . 9 ⊢ (𝒫 𝐴 ↑ko 𝒫 𝐴) = (𝒫 𝐴 ↑ko 𝒫 𝐴)
25	24	xkotopon 23325	. . . . . . . 8 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐴 ∈ Top) → (𝒫 𝐴 ↑ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
26	23, 23, 25	syl2anc 583	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ↑ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
27		cndis 23016	. . . . . . . . 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 4023	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝑀) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴))
30		disllycmp 23223	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Locally Comp)
31		llynlly 23202	. . . . . . . . 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 2731	. . . . . . . . 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 23385	. . . . . . . 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 23402	. . . . . 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 2836	. . . . 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 23380	. . . . . . . . . 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 7427	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝑀)))
41	40, 4	eqtrd 2771	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝑀)) = (TopOpen‘𝑀))
42	41, 41	oveq12d 7430	. . . . . 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 7427	. . . . 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 2834	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)))
45		vex 3477	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
46		vex 3477	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
47	45, 46	coex 7924	. . . . . . . . . 10 ⊢ (𝑥 ∘ 𝑦) ∈ V
48	21, 47	fnmpoi 8059	. . . . . . . . 9 ⊢ (+g‘𝑀) Fn ((Base‘𝑀) × (Base‘𝑀))
49		eqid 2731	. . . . . . . . . 10 ⊢ (+𝑓‘𝑀) = (+𝑓‘𝑀)
50	3, 20, 49	plusfeq 18574	. . . . . . . . 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 2740	. . . . . . 7 ⊢ (+g‘𝑀) = (+𝑓‘𝑀)
53	3, 52	mndplusf 18678	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (+g‘𝑀):((Base‘𝑀) × (Base‘𝑀))⟶(Base‘𝑀))
54		frn 6724	. . . . . 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 23011	. . . . 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 7428	. . 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 2834	. 2 ⊢ (𝐴 ∈ 𝑉 → (+g‘𝑀) ∈ (((TopOpen‘𝑀) ×t (TopOpen‘𝑀)) Cn (TopOpen‘𝑀)))
61	52, 17	istmd 23799	. 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 1540   ∈ wcel 2105   ⊆ wss 3948  𝒫 cpw 4602  {csn 4628   × cxp 5674  ran crn 5677   ∘ ccom 5680   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7412   ∈ cmpo 7414   ↑_m cmap 8823  Basecbs 17149  +_gcplusg 17202   ↾_t crest 17371  TopOpenctopn 17372  ∏_tcpt 17389  +_𝑓cplusf 18563  Mndcmnd 18660  EndoFMndcefmnd 18786  Topctop 22616  TopOnctopon 22633  TopSpctps 22655   Cn ccn 22949  Compccmp 23111  Locally clly 23189  𝑛-Locally cnlly 23190   ×_t ctx 23285   ↑_ko cxko 23286  TopMndctmd 23795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-er 8706  df-map 8825  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fi 9409  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12478  df-z 12564  df-uz 12828  df-fz 13490  df-struct 17085  df-slot 17120  df-ndx 17132  df-base 17150  df-plusg 17215  df-tset 17221  df-rest 17373  df-topn 17374  df-topgen 17394  df-pt 17395  df-plusf 18565  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-efmnd 18787  df-top 22617  df-topon 22634  df-topsp 22656  df-bases 22670  df-ntr 22745  df-nei 22823  df-cn 22952  df-cmp 23112  df-lly 23191  df-nlly 23192  df-tx 23287  df-xko 23288  df-tmd 23797

This theorem is referenced by:  symgtgp  23831