Theorem xkofvcn 23645

Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 23617.) (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 23434	. . . 4 ⊢ (𝑅 ∈ 𝑛-Locally Comp → 𝑅 ∈ Top)
3		eqid 2737	. . . . 5 ⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
4	3	xkotopon 23561	. . . 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 480	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ Top)
7		xkofvcn.1	. . . . 5 ⊢ 𝑋 = ∪ 𝑅
8	7	toptopon 22878	. . . 4 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
9	6, 8	sylib 218	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ (TopOn‘𝑋))
10	5, 9	cnmpt1st 23629	. . . 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 23630	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (((𝑆 ↑ko 𝑅) ×t 𝑅) Cn 𝑅))
12		1on 8421	. . . . . . 7 ⊢ 1o ∈ On
13		distopon 22958	. . . . . . 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 23580	. . . . . 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 4592	. . . . . 6 ⊢ (𝑦 = 𝑥 → {𝑦} = {𝑥})
18	17	xpeq2d 5664	. . . . 5 ⊢ (𝑦 = 𝑥 → (1o × {𝑦}) = (1o × {𝑥}))
19	5, 9, 11, 9, 16, 18	cnmpt21 23632	. . . 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 22956	. . . . . 6 ⊢ (1o ∈ On → 𝒫 1o ∈ Top)
21	12, 20	mp1i 13	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1o ∈ Top)
22		eqid 2737	. . . . . 6 ⊢ (𝑅 ↑ko 𝒫 1o) = (𝑅 ↑ko 𝒫 1o)
23	22	xkotopon 23561	. . . . 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 482	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ 𝑛-Locally Comp)
26		simpr 484	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
27		eqid 2737	. . . . . 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 23621	. . . . 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 1374	. . . 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 5816	. . . . 5 ⊢ (𝑔 = 𝑓 → (𝑔 ∘ ℎ) = (𝑓 ∘ ℎ))
31		coeq2 5817	. . . . 5 ⊢ (ℎ = (1o × {𝑥}) → (𝑓 ∘ ℎ) = (𝑓 ∘ (1o × {𝑥})))
32	30, 31	sylan9eq 2792	. . . 4 ⊢ ((𝑔 = 𝑓 ∧ ℎ = (1o × {𝑥})) → (𝑔 ∘ ℎ) = (𝑓 ∘ (1o × {𝑥})))
33	5, 9, 10, 19, 5, 24, 29, 32	cnmpt22 23635	. . 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 2737	. . . . 5 ⊢ (𝑆 ↑ko 𝒫 1o) = (𝑆 ↑ko 𝒫 1o)
35	34	xkotopon 23561	. . . 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 8443	. . . . 5 ⊢ ∅ ∈ 1o
38	37	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → ∅ ∈ 1o)
39		unipw 5407	. . . . . 6 ⊢ ∪ 𝒫 1o = 1o
40	39	eqcomi 2746	. . . . 5 ⊢ 1o = ∪ 𝒫 1o
41	40	xkopjcn 23617	. . . 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 1374	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝒫 1o Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆 ↑ko 𝒫 1o) Cn 𝑆))
43		fveq1 6843	. . . 4 ⊢ (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = ((𝑓 ∘ (1o × {𝑥}))‘∅))
44		vex 3446	. . . . . . 7 ⊢ 𝑥 ∈ V
45	44	fconst 6730	. . . . . 6 ⊢ (1o × {𝑥}):1o⟶{𝑥}
46		fvco3 6943	. . . . . 6 ⊢ (((1o × {𝑥}):1o⟶{𝑥} ∧ ∅ ∈ 1o) → ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘((1o × {𝑥})‘∅)))
47	45, 37, 46	mp2an 693	. . . . 5 ⊢ ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘((1o × {𝑥})‘∅))
48	44	fvconst2 7162	. . . . . . 7 ⊢ (∅ ∈ 1o → ((1o × {𝑥})‘∅) = 𝑥)
49	37, 48	ax-mp 5	. . . . . 6 ⊢ ((1o × {𝑥})‘∅) = 𝑥
50	49	fveq2i 6847	. . . . 5 ⊢ (𝑓‘((1o × {𝑥})‘∅)) = (𝑓‘𝑥)
51	47, 50	eqtri 2760	. . . 4 ⊢ ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘𝑥)
52	43, 51	eqtrdi 2788	. . 3 ⊢ (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = (𝑓‘𝑥))
53	5, 9, 33, 36, 42, 52	cnmpt21 23632	. 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 2841	1 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝐹 ∈ (((𝑆 ↑ko 𝑅) ×t 𝑅) Cn 𝑆))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∅c0 4287  𝒫 cpw 4556  {csn 4582  ∪ cuni 4865   ↦ cmpt 5181   × cxp 5632   ∘ ccom 5638  Oncon0 6327  ⟶wf 6498  ‘cfv 6502  (class class class)co 7370   ∈ cmpo 7372  1_oc1o 8402  Topctop 22854  TopOnctopon 22871   Cn ccn 23185  Compccmp 23347  𝑛-Locally cnlly 23426   ×_t ctx 23521   ↑_ko cxko 23522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-1o 8409  df-2o 8410  df-map 8779  df-ixp 8850  df-en 8898  df-dom 8899  df-fin 8901  df-fi 9328  df-rest 17356  df-topgen 17377  df-pt 17378  df-top 22855  df-topon 22872  df-bases 22907  df-ntr 22981  df-nei 23059  df-cn 23188  df-cnp 23189  df-cmp 23348  df-nlly 23428  df-tx 23523  df-xko 23524

This theorem is referenced by:  cnmptk1p  23646  cnmptk2  23647