Theorem xkofvcn 21813

Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 21785.) (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 21602	. . . 4 ⊢ (𝑅 ∈ 𝑛-Locally Comp → 𝑅 ∈ Top)
3		eqid 2797	. . . . 5 ⊢ (𝑆 ^ko 𝑅) = (𝑆 ^ko 𝑅)
4	3	xkotopon 21729	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
5	2, 4	sylan 576	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
6	2	adantr 473	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ Top)
7		xkofvcn.1	. . . . 5 ⊢ 𝑋 = ∪ 𝑅
8	7	toptopon 21047	. . . 4 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
9	6, 8	sylib 210	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ (TopOn‘𝑋))
10	5, 9	cnmpt1st 21797	. . . 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 21798	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn 𝑅))
12		1on 7804	. . . . . . 7 ⊢ 1𝑜 ∈ On
13		distopon 21127	. . . . . . 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 21748	. . . . . 6 ⊢ ((𝒫 1𝑜 ∈ (TopOn‘1𝑜) ∧ 𝑅 ∈ (TopOn‘𝑋)) → (𝑦 ∈ 𝑋 ↦ (1𝑜 × {𝑦})) ∈ (𝑅 Cn (𝑅 ^ko 𝒫 1𝑜)))
16	14, 9, 15	syl2anc 580	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑦 ∈ 𝑋 ↦ (1𝑜 × {𝑦})) ∈ (𝑅 Cn (𝑅 ^ko 𝒫 1𝑜)))
17		sneq 4376	. . . . . 6 ⊢ (𝑦 = 𝑥 → {𝑦} = {𝑥})
18	17	xpeq2d 5340	. . . . 5 ⊢ (𝑦 = 𝑥 → (1𝑜 × {𝑦}) = (1𝑜 × {𝑥}))
19	5, 9, 11, 9, 16, 18	cnmpt21 21800	. . . 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 21125	. . . . . 6 ⊢ (1𝑜 ∈ On → 𝒫 1𝑜 ∈ Top)
21	12, 20	mp1i 13	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1𝑜 ∈ Top)
22		eqid 2797	. . . . . 6 ⊢ (𝑅 ^ko 𝒫 1𝑜) = (𝑅 ^ko 𝒫 1𝑜)
23	22	xkotopon 21729	. . . . 5 ⊢ ((𝒫 1𝑜 ∈ Top ∧ 𝑅 ∈ Top) → (𝑅 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑅)))
24	21, 6, 23	syl2anc 580	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑅 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑅)))
25		simpl 475	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ 𝑛-Locally Comp)
26		simpr 478	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
27		eqid 2797	. . . . . 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 21789	. . . . 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 1491	. . . 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 5481	. . . . 5 ⊢ (𝑔 = 𝑓 → (𝑔 ∘ ℎ) = (𝑓 ∘ ℎ))
31		coeq2 5482	. . . . 5 ⊢ (ℎ = (1𝑜 × {𝑥}) → (𝑓 ∘ ℎ) = (𝑓 ∘ (1𝑜 × {𝑥})))
32	30, 31	sylan9eq 2851	. . . 4 ⊢ ((𝑔 = 𝑓 ∧ ℎ = (1𝑜 × {𝑥})) → (𝑔 ∘ ℎ) = (𝑓 ∘ (1𝑜 × {𝑥})))
33	5, 9, 10, 19, 5, 24, 29, 32	cnmpt22 21803	. . 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 2797	. . . . 5 ⊢ (𝑆 ^ko 𝒫 1𝑜) = (𝑆 ^ko 𝒫 1𝑜)
35	34	xkotopon 21729	. . . 4 ⊢ ((𝒫 1𝑜 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑆)))
36	21, 26, 35	syl2anc 580	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑆)))
37		0lt1o 7822	. . . . 5 ⊢ ∅ ∈ 1𝑜
38	37	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → ∅ ∈ 1𝑜)
39		unipw 5107	. . . . . 6 ⊢ ∪ 𝒫 1𝑜 = 1𝑜
40	39	eqcomi 2806	. . . . 5 ⊢ 1𝑜 = ∪ 𝒫 1𝑜
41	40	xkopjcn 21785	. . . 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 1491	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝒫 1𝑜 Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆 ^ko 𝒫 1𝑜) Cn 𝑆))
43		fveq1 6408	. . . 4 ⊢ (𝑔 = (𝑓 ∘ (1𝑜 × {𝑥})) → (𝑔‘∅) = ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅))
44		vex 3386	. . . . . . 7 ⊢ 𝑥 ∈ V
45	44	fconst 6304	. . . . . 6 ⊢ (1𝑜 × {𝑥}):1𝑜⟶{𝑥}
46		fvco3 6498	. . . . . 6 ⊢ (((1𝑜 × {𝑥}):1𝑜⟶{𝑥} ∧ ∅ ∈ 1𝑜) → ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅) = (𝑓‘((1𝑜 × {𝑥})‘∅)))
47	45, 37, 46	mp2an 684	. . . . 5 ⊢ ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅) = (𝑓‘((1𝑜 × {𝑥})‘∅))
48	44	fvconst2 6696	. . . . . . 7 ⊢ (∅ ∈ 1𝑜 → ((1𝑜 × {𝑥})‘∅) = 𝑥)
49	37, 48	ax-mp 5	. . . . . 6 ⊢ ((1𝑜 × {𝑥})‘∅) = 𝑥
50	49	fveq2i 6412	. . . . 5 ⊢ (𝑓‘((1𝑜 × {𝑥})‘∅)) = (𝑓‘𝑥)
51	47, 50	eqtri 2819	. . . 4 ⊢ ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅) = (𝑓‘𝑥)
52	43, 51	syl6eq 2847	. . 3 ⊢ (𝑔 = (𝑓 ∘ (1𝑜 × {𝑥})) → (𝑔‘∅) = (𝑓‘𝑥))
53	5, 9, 33, 36, 42, 52	cnmpt21 21800	. 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	syl5eqel 2880	1 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝐹 ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn 𝑆))

Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ∅c0 4113  𝒫 cpw 4347  {csn 4366  ∪ cuni 4626   ↦ cmpt 4920   × cxp 5308   ∘ ccom 5314  Oncon0 5939  ⟶wf 6095  ‘cfv 6099  (class class class)co 6876   ↦ cmpt2 6878  1_𝑜c1o 7790  Topctop 21023  TopOnctopon 21040   Cn ccn 21354  Compccmp 21515  𝑛-Locally cnlly 21594   ×_t ctx 21689   ^_ko cxko 21690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-iin 4711  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-1st 7399  df-2nd 7400  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-1o 7797  df-2o 7798  df-oadd 7801  df-er 7980  df-map 8095  df-ixp 8147  df-en 8194  df-dom 8195  df-sdom 8196  df-fin 8197  df-fi 8557  df-rest 16395  df-topgen 16416  df-pt 16417  df-top 21024  df-topon 21041  df-bases 21076  df-ntr 21150  df-nei 21228  df-cn 21357  df-cnp 21358  df-cmp 21516  df-nlly 21596  df-tx 21691  df-xko 21692

This theorem is referenced by:  cnmptk1p  21814  cnmptk2  21815