Theorem xkofvcn 21708

Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 21680.) (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 21497	. . . 4 ⊢ (𝑅 ∈ 𝑛-Locally Comp → 𝑅 ∈ Top)
3		eqid 2771	. . . . 5 ⊢ (𝑆 ^ko 𝑅) = (𝑆 ^ko 𝑅)
4	3	xkotopon 21624	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
5	2, 4	sylan 569	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
6	2	adantr 466	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ Top)
7		xkofvcn.1	. . . . 5 ⊢ 𝑋 = ∪ 𝑅
8	7	toptopon 20942	. . . 4 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
9	6, 8	sylib 208	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ (TopOn‘𝑋))
10	5, 9	cnmpt1st 21692	. . . 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 21693	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn 𝑅))
12		1on 7720	. . . . . . 7 ⊢ 1𝑜 ∈ On
13		distopon 21022	. . . . . . 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 21643	. . . . . 6 ⊢ ((𝒫 1𝑜 ∈ (TopOn‘1𝑜) ∧ 𝑅 ∈ (TopOn‘𝑋)) → (𝑦 ∈ 𝑋 ↦ (1𝑜 × {𝑦})) ∈ (𝑅 Cn (𝑅 ^ko 𝒫 1𝑜)))
16	14, 9, 15	syl2anc 573	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑦 ∈ 𝑋 ↦ (1𝑜 × {𝑦})) ∈ (𝑅 Cn (𝑅 ^ko 𝒫 1𝑜)))
17		sneq 4326	. . . . . 6 ⊢ (𝑦 = 𝑥 → {𝑦} = {𝑥})
18	17	xpeq2d 5279	. . . . 5 ⊢ (𝑦 = 𝑥 → (1𝑜 × {𝑦}) = (1𝑜 × {𝑥}))
19	5, 9, 11, 9, 16, 18	cnmpt21 21695	. . . 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 21020	. . . . . 6 ⊢ (1𝑜 ∈ On → 𝒫 1𝑜 ∈ Top)
21	12, 20	mp1i 13	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1𝑜 ∈ Top)
22		eqid 2771	. . . . . 6 ⊢ (𝑅 ^ko 𝒫 1𝑜) = (𝑅 ^ko 𝒫 1𝑜)
23	22	xkotopon 21624	. . . . 5 ⊢ ((𝒫 1𝑜 ∈ Top ∧ 𝑅 ∈ Top) → (𝑅 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑅)))
24	21, 6, 23	syl2anc 573	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑅 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑅)))
25		simpl 468	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ 𝑛-Locally Comp)
26		simpr 471	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
27		eqid 2771	. . . . . 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 21684	. . . . 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 1476	. . . 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 5418	. . . . 5 ⊢ (𝑔 = 𝑓 → (𝑔 ∘ ℎ) = (𝑓 ∘ ℎ))
31		coeq2 5419	. . . . 5 ⊢ (ℎ = (1𝑜 × {𝑥}) → (𝑓 ∘ ℎ) = (𝑓 ∘ (1𝑜 × {𝑥})))
32	30, 31	sylan9eq 2825	. . . 4 ⊢ ((𝑔 = 𝑓 ∧ ℎ = (1𝑜 × {𝑥})) → (𝑔 ∘ ℎ) = (𝑓 ∘ (1𝑜 × {𝑥})))
33	5, 9, 10, 19, 5, 24, 29, 32	cnmpt22 21698	. . 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 2771	. . . . 5 ⊢ (𝑆 ^ko 𝒫 1𝑜) = (𝑆 ^ko 𝒫 1𝑜)
35	34	xkotopon 21624	. . . 4 ⊢ ((𝒫 1𝑜 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑆)))
36	21, 26, 35	syl2anc 573	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑆)))
37		0lt1o 7738	. . . . 5 ⊢ ∅ ∈ 1𝑜
38	37	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → ∅ ∈ 1𝑜)
39		unipw 5046	. . . . . 6 ⊢ ∪ 𝒫 1𝑜 = 1𝑜
40	39	eqcomi 2780	. . . . 5 ⊢ 1𝑜 = ∪ 𝒫 1𝑜
41	40	xkopjcn 21680	. . . 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 1476	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝒫 1𝑜 Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆 ^ko 𝒫 1𝑜) Cn 𝑆))
43		fveq1 6331	. . . 4 ⊢ (𝑔 = (𝑓 ∘ (1𝑜 × {𝑥})) → (𝑔‘∅) = ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅))
44		vex 3354	. . . . . . 7 ⊢ 𝑥 ∈ V
45	44	fconst 6231	. . . . . 6 ⊢ (1𝑜 × {𝑥}):1𝑜⟶{𝑥}
46		fvco3 6417	. . . . . 6 ⊢ (((1𝑜 × {𝑥}):1𝑜⟶{𝑥} ∧ ∅ ∈ 1𝑜) → ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅) = (𝑓‘((1𝑜 × {𝑥})‘∅)))
47	45, 37, 46	mp2an 672	. . . . 5 ⊢ ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅) = (𝑓‘((1𝑜 × {𝑥})‘∅))
48	44	fvconst2 6613	. . . . . . 7 ⊢ (∅ ∈ 1𝑜 → ((1𝑜 × {𝑥})‘∅) = 𝑥)
49	37, 48	ax-mp 5	. . . . . 6 ⊢ ((1𝑜 × {𝑥})‘∅) = 𝑥
50	49	fveq2i 6335	. . . . 5 ⊢ (𝑓‘((1𝑜 × {𝑥})‘∅)) = (𝑓‘𝑥)
51	47, 50	eqtri 2793	. . . 4 ⊢ ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅) = (𝑓‘𝑥)
52	43, 51	syl6eq 2821	. . 3 ⊢ (𝑔 = (𝑓 ∘ (1𝑜 × {𝑥})) → (𝑔‘∅) = (𝑓‘𝑥))
53	5, 9, 33, 36, 42, 52	cnmpt21 21695	. 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 2854	1 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝐹 ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn 𝑆))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∅c0 4063  𝒫 cpw 4297  {csn 4316  ∪ cuni 4574   ↦ cmpt 4863   × cxp 5247   ∘ ccom 5253  Oncon0 5866  ⟶wf 6027  ‘cfv 6031  (class class class)co 6793   ↦ cmpt2 6795  1_𝑜c1o 7706  Topctop 20918  TopOnctopon 20935   Cn ccn 21249  Compccmp 21410  𝑛-Locally cnlly 21489   ×_t ctx 21584   ^_ko cxko 21585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fi 8473  df-rest 16291  df-topgen 16312  df-pt 16313  df-top 20919  df-topon 20936  df-bases 20971  df-ntr 21045  df-nei 21123  df-cn 21252  df-cnp 21253  df-cmp 21411  df-nlly 21491  df-tx 21586  df-xko 21587

This theorem is referenced by:  cnmptk1p  21709  cnmptk2  21710