Theorem xkofvcn 23671

Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 23643.) (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
xkofvcn.1	⊢ 𝑋 = ∪ 𝑅
xkofvcn.2	⊢ 𝐹 = (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥 ∈ 𝑋 ↦ (𝑓‘𝑥))

Assertion

Ref	Expression
xkofvcn	⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝐹 ∈ (((𝑆 ↑ko 𝑅) ×t 𝑅) Cn 𝑆))

Distinct variable groups:   𝑥,𝑓,𝑅   𝑆,𝑓,𝑥   𝑓,𝑋,𝑥

Allowed substitution hints:   𝐹(𝑥,𝑓)

Proof of Theorem xkofvcn

Dummy variables 𝑔 ℎ 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xkofvcn.2	. 2 ⊢ 𝐹 = (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥 ∈ 𝑋 ↦ (𝑓‘𝑥))
2		nllytop 23460	. . . 4 ⊢ (𝑅 ∈ 𝑛-Locally Comp → 𝑅 ∈ Top)
3		eqid 2741	. . . . 5 ⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
4	3	xkotopon 23587	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
5	2, 4	sylan 587	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
6	2	adantr 482	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ Top)
7		xkofvcn.1	. . . . 5 ⊢ 𝑋 = ∪ 𝑅
8	7	toptopon 22904	. . . 4 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
9	6, 8	sylib 220	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ (TopOn‘𝑋))
10	5, 9	cnmpt1st 23655	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥 ∈ 𝑋 ↦ 𝑓) ∈ (((𝑆 ↑ko 𝑅) ×t 𝑅) Cn (𝑆 ↑ko 𝑅)))
11	5, 9	cnmpt2nd 23656	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (((𝑆 ↑ko 𝑅) ×t 𝑅) Cn 𝑅))
12		1on 8411	. . . . . . 7 ⊢ 1o ∈ On
13		distopon 22984	. . . . . . 7 ⊢ (1o ∈ On → 𝒫 1o ∈ (TopOn‘1o))
14	12, 13	mp1i 13	. . . . . 6 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1o ∈ (TopOn‘1o))
15		xkoccn 23606	. . . . . 6 ⊢ ((𝒫 1o ∈ (TopOn‘1o) ∧ 𝑅 ∈ (TopOn‘𝑋)) → (𝑦 ∈ 𝑋 ↦ (1o × {𝑦})) ∈ (𝑅 Cn (𝑅 ↑ko 𝒫 1o)))
16	14, 9, 15	syl2anc 591	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑦 ∈ 𝑋 ↦ (1o × {𝑦})) ∈ (𝑅 Cn (𝑅 ↑ko 𝒫 1o)))
17		sneq 4568	. . . . . 6 ⊢ (𝑦 = 𝑥 → {𝑦} = {𝑥})
18	17	xpeq2d 5651	. . . . 5 ⊢ (𝑦 = 𝑥 → (1o × {𝑦}) = (1o × {𝑥}))
19	5, 9, 11, 9, 16, 18	cnmpt21 23658	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥 ∈ 𝑋 ↦ (1o × {𝑥})) ∈ (((𝑆 ↑ko 𝑅) ×t 𝑅) Cn (𝑅 ↑ko 𝒫 1o)))
20		distop 22982	. . . . . 6 ⊢ (1o ∈ On → 𝒫 1o ∈ Top)
21	12, 20	mp1i 13	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1o ∈ Top)
22		eqid 2741	. . . . . 6 ⊢ (𝑅 ↑ko 𝒫 1o) = (𝑅 ↑ko 𝒫 1o)
23	22	xkotopon 23587	. . . . 5 ⊢ ((𝒫 1o ∈ Top ∧ 𝑅 ∈ Top) → (𝑅 ↑ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑅)))
24	21, 6, 23	syl2anc 591	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑅 ↑ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑅)))
25		simpl 484	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ 𝑛-Locally Comp)
26		simpr 486	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
27		eqid 2741	. . . . . 6 ⊢ (𝑔 ∈ (𝑅 Cn 𝑆), ℎ ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔 ∘ ℎ)) = (𝑔 ∈ (𝑅 Cn 𝑆), ℎ ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔 ∘ ℎ))
28	27	xkococn 23647	. . . . 5 ⊢ ((𝒫 1o ∈ Top ∧ 𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝑅 Cn 𝑆), ℎ ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔 ∘ ℎ)) ∈ (((𝑆 ↑ko 𝑅) ×t (𝑅 ↑ko 𝒫 1o)) Cn (𝑆 ↑ko 𝒫 1o)))
29	21, 25, 26, 28	syl3anc 1380	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝑅 Cn 𝑆), ℎ ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔 ∘ ℎ)) ∈ (((𝑆 ↑ko 𝑅) ×t (𝑅 ↑ko 𝒫 1o)) Cn (𝑆 ↑ko 𝒫 1o)))
30		coeq1 5802	. . . . 5 ⊢ (𝑔 = 𝑓 → (𝑔 ∘ ℎ) = (𝑓 ∘ ℎ))
31		coeq2 5803	. . . . 5 ⊢ (ℎ = (1o × {𝑥}) → (𝑓 ∘ ℎ) = (𝑓 ∘ (1o × {𝑥})))
32	30, 31	sylan9eq 2796	. . . 4 ⊢ ((𝑔 = 𝑓 ∧ ℎ = (1o × {𝑥})) → (𝑔 ∘ ℎ) = (𝑓 ∘ (1o × {𝑥})))
33	5, 9, 10, 19, 5, 24, 29, 32	cnmpt22 23661	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥 ∈ 𝑋 ↦ (𝑓 ∘ (1o × {𝑥}))) ∈ (((𝑆 ↑ko 𝑅) ×t 𝑅) Cn (𝑆 ↑ko 𝒫 1o)))
34		eqid 2741	. . . . 5 ⊢ (𝑆 ↑ko 𝒫 1o) = (𝑆 ↑ko 𝒫 1o)
35	34	xkotopon 23587	. . . 4 ⊢ ((𝒫 1o ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑆)))
36	21, 26, 35	syl2anc 591	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑆)))
37		0lt1o 8433	. . . . 5 ⊢ ∅ ∈ 1o
38	37	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → ∅ ∈ 1o)
39		unipw 5392	. . . . . 6 ⊢ ∪ 𝒫 1o = 1o
40	39	eqcomi 2750	. . . . 5 ⊢ 1o = ∪ 𝒫 1o
41	40	xkopjcn 23643	. . . 4 ⊢ ((𝒫 1o ∈ Top ∧ 𝑆 ∈ Top ∧ ∅ ∈ 1o) → (𝑔 ∈ (𝒫 1o Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆 ↑ko 𝒫 1o) Cn 𝑆))
42	21, 26, 38, 41	syl3anc 1380	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝒫 1o Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆 ↑ko 𝒫 1o) Cn 𝑆))
43		fveq1 6830	. . . 4 ⊢ (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = ((𝑓 ∘ (1o × {𝑥}))‘∅))
44		vex 3437	. . . . . . 7 ⊢ 𝑥 ∈ V
45	44	fconst 6717	. . . . . 6 ⊢ (1o × {𝑥}):1o⟶{𝑥}
46		fvco3 6931	. . . . . 6 ⊢ (((1o × {𝑥}):1o⟶{𝑥} ∧ ∅ ∈ 1o) → ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘((1o × {𝑥})‘∅)))
47	45, 37, 46	mp2an 699	. . . . 5 ⊢ ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘((1o × {𝑥})‘∅))
48	44	fvconst2 7152	. . . . . . 7 ⊢ (∅ ∈ 1o → ((1o × {𝑥})‘∅) = 𝑥)
49	37, 48	ax-mp 5	. . . . . 6 ⊢ ((1o × {𝑥})‘∅) = 𝑥
50	49	fveq2i 6834	. . . . 5 ⊢ (𝑓‘((1o × {𝑥})‘∅)) = (𝑓‘𝑥)
51	47, 50	eqtri 2764	. . . 4 ⊢ ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘𝑥)
52	43, 51	eqtrdi 2792	. . 3 ⊢ (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = (𝑓‘𝑥))
53	5, 9, 33, 36, 42, 52	cnmpt21 23658	. 2 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥 ∈ 𝑋 ↦ (𝑓‘𝑥)) ∈ (((𝑆 ↑ko 𝑅) ×t 𝑅) Cn 𝑆))
54	1, 53	eqeltrid 2845	1 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝐹 ∈ (((𝑆 ↑ko 𝑅) ×t 𝑅) Cn 𝑆))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∅c0 4264  𝒫 cpw 4532  {csn 4558  ∪ cuni 4841   ↦ cmpt 5156   × cxp 5619   ∘ ccom 5625  Oncon0 6314  ⟶wf 6485  ‘cfv 6489  (class class class)co 7360   ∈ cmpo 7362  1_oc1o 8392  Topctop 22880  TopOnctopon 22897   Cn ccn 23211  Compccmp 23373  𝑛-Locally cnlly 23452   ×_t ctx 23547   ↑_ko cxko 23548

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

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-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-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-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-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-1o 8399  df-2o 8400  df-map 8769  df-ixp 8840  df-en 8888  df-dom 8889  df-fin 8891  df-fi 9318  df-rest 17380  df-topgen 17401  df-pt 17402  df-top 22881  df-topon 22898  df-bases 22933  df-ntr 23007  df-nei 23085  df-cn 23214  df-cnp 23215  df-cmp 23374  df-nlly 23454  df-tx 23549  df-xko 23550

This theorem is referenced by:  cnmptk1p  23672  cnmptk2  23673