Theorem xkofvcn 23188

Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 23160.) (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 22977	. . . 4 ⊢ (𝑅 ∈ 𝑛-Locally Comp → 𝑅 ∈ Top)
3		eqid 2733	. . . . 5 ⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
4	3	xkotopon 23104	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
5	2, 4	sylan 581	. . 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 22419	. . . 4 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
9	6, 8	sylib 217	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ (TopOn‘𝑋))
10	5, 9	cnmpt1st 23172	. . . 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 23173	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (((𝑆 ↑ko 𝑅) ×t 𝑅) Cn 𝑅))
12		1on 8478	. . . . . . 7 ⊢ 1o ∈ On
13		distopon 22500	. . . . . . 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 23123	. . . . . 6 ⊢ ((𝒫 1o ∈ (TopOn‘1o) ∧ 𝑅 ∈ (TopOn‘𝑋)) → (𝑦 ∈ 𝑋 ↦ (1o × {𝑦})) ∈ (𝑅 Cn (𝑅 ↑ko 𝒫 1o)))
16	14, 9, 15	syl2anc 585	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑦 ∈ 𝑋 ↦ (1o × {𝑦})) ∈ (𝑅 Cn (𝑅 ↑ko 𝒫 1o)))
17		sneq 4639	. . . . . 6 ⊢ (𝑦 = 𝑥 → {𝑦} = {𝑥})
18	17	xpeq2d 5707	. . . . 5 ⊢ (𝑦 = 𝑥 → (1o × {𝑦}) = (1o × {𝑥}))
19	5, 9, 11, 9, 16, 18	cnmpt21 23175	. . . 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 22498	. . . . . 6 ⊢ (1o ∈ On → 𝒫 1o ∈ Top)
21	12, 20	mp1i 13	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1o ∈ Top)
22		eqid 2733	. . . . . 6 ⊢ (𝑅 ↑ko 𝒫 1o) = (𝑅 ↑ko 𝒫 1o)
23	22	xkotopon 23104	. . . . 5 ⊢ ((𝒫 1o ∈ Top ∧ 𝑅 ∈ Top) → (𝑅 ↑ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑅)))
24	21, 6, 23	syl2anc 585	. . . 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 2733	. . . . . 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 23164	. . . . 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 1372	. . . 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 5858	. . . . 5 ⊢ (𝑔 = 𝑓 → (𝑔 ∘ ℎ) = (𝑓 ∘ ℎ))
31		coeq2 5859	. . . . 5 ⊢ (ℎ = (1o × {𝑥}) → (𝑓 ∘ ℎ) = (𝑓 ∘ (1o × {𝑥})))
32	30, 31	sylan9eq 2793	. . . 4 ⊢ ((𝑔 = 𝑓 ∧ ℎ = (1o × {𝑥})) → (𝑔 ∘ ℎ) = (𝑓 ∘ (1o × {𝑥})))
33	5, 9, 10, 19, 5, 24, 29, 32	cnmpt22 23178	. . 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 2733	. . . . 5 ⊢ (𝑆 ↑ko 𝒫 1o) = (𝑆 ↑ko 𝒫 1o)
35	34	xkotopon 23104	. . . 4 ⊢ ((𝒫 1o ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑆)))
36	21, 26, 35	syl2anc 585	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑆)))
37		0lt1o 8504	. . . . 5 ⊢ ∅ ∈ 1o
38	37	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → ∅ ∈ 1o)
39		unipw 5451	. . . . . 6 ⊢ ∪ 𝒫 1o = 1o
40	39	eqcomi 2742	. . . . 5 ⊢ 1o = ∪ 𝒫 1o
41	40	xkopjcn 23160	. . . 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 1372	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝒫 1o Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆 ↑ko 𝒫 1o) Cn 𝑆))
43		fveq1 6891	. . . 4 ⊢ (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = ((𝑓 ∘ (1o × {𝑥}))‘∅))
44		vex 3479	. . . . . . 7 ⊢ 𝑥 ∈ V
45	44	fconst 6778	. . . . . 6 ⊢ (1o × {𝑥}):1o⟶{𝑥}
46		fvco3 6991	. . . . . 6 ⊢ (((1o × {𝑥}):1o⟶{𝑥} ∧ ∅ ∈ 1o) → ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘((1o × {𝑥})‘∅)))
47	45, 37, 46	mp2an 691	. . . . 5 ⊢ ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘((1o × {𝑥})‘∅))
48	44	fvconst2 7205	. . . . . . 7 ⊢ (∅ ∈ 1o → ((1o × {𝑥})‘∅) = 𝑥)
49	37, 48	ax-mp 5	. . . . . 6 ⊢ ((1o × {𝑥})‘∅) = 𝑥
50	49	fveq2i 6895	. . . . 5 ⊢ (𝑓‘((1o × {𝑥})‘∅)) = (𝑓‘𝑥)
51	47, 50	eqtri 2761	. . . 4 ⊢ ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘𝑥)
52	43, 51	eqtrdi 2789	. . 3 ⊢ (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = (𝑓‘𝑥))
53	5, 9, 33, 36, 42, 52	cnmpt21 23175	. 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 2838	1 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝐹 ∈ (((𝑆 ↑ko 𝑅) ×t 𝑅) Cn 𝑆))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∅c0 4323  𝒫 cpw 4603  {csn 4629  ∪ cuni 4909   ↦ cmpt 5232   × cxp 5675   ∘ ccom 5681  Oncon0 6365  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1_oc1o 8459  Topctop 22395  TopOnctopon 22412   Cn ccn 22728  Compccmp 22890  𝑛-Locally cnlly 22969   ×_t ctx 23064   ↑_ko cxko 23065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-1o 8466  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-fin 8943  df-fi 9406  df-rest 17368  df-topgen 17389  df-pt 17390  df-top 22396  df-topon 22413  df-bases 22449  df-ntr 22524  df-nei 22602  df-cn 22731  df-cnp 22732  df-cmp 22891  df-nlly 22971  df-tx 23066  df-xko 23067

This theorem is referenced by:  cnmptk1p  23189  cnmptk2  23190