Theorem xkofvcn 24003

Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 23975.) (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 23792	. . . 4 ⊢ (𝑅 ∈ 𝑛-Locally Comp → 𝑅 ∈ Top)
3		eqid 2760	. . . . 5 ⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
4	3	xkotopon 23919	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
5	2, 4	sylan 592	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
6	2	adantr 486	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ Top)
7		xkofvcn.1	. . . . 5 ⊢ 𝑋 = ∪ 𝑅
8	7	toptopon 23235	. . . 4 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
9	6, 8	sylib 221	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ (TopOn‘𝑋))
10	5, 9	cnmpt1st 23987	. . . 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 23988	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (((𝑆 ↑ko 𝑅) ×t 𝑅) Cn 𝑅))
12		1on 8489	. . . . . . 7 ⊢ 1o ∈ On
13		distopon 23315	. . . . . . 7 ⊢ (1o ∈ On → 𝒫 1o ∈ (TopOn‘1o))
14	12, 13	mp1i 14	. . . . . 6 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1o ∈ (TopOn‘1o))
15		xkoccn 23938	. . . . . 6 ⊢ ((𝒫 1o ∈ (TopOn‘1o) ∧ 𝑅 ∈ (TopOn‘𝑋)) → (𝑦 ∈ 𝑋 ↦ (1o × {𝑦})) ∈ (𝑅 Cn (𝑅 ↑ko 𝒫 1o)))
16	14, 9, 15	syl2anc 596	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑦 ∈ 𝑋 ↦ (1o × {𝑦})) ∈ (𝑅 Cn (𝑅 ↑ko 𝒫 1o)))
17		sneq 4593	. . . . . 6 ⊢ (𝑦 = 𝑥 → {𝑦} = {𝑥})
18	17	xpeq2d 5680	. . . . 5 ⊢ (𝑦 = 𝑥 → (1o × {𝑦}) = (1o × {𝑥}))
19	5, 9, 11, 9, 16, 18	cnmpt21 23990	. . . 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 23313	. . . . . 6 ⊢ (1o ∈ On → 𝒫 1o ∈ Top)
21	12, 20	mp1i 14	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1o ∈ Top)
22		eqid 2760	. . . . . 6 ⊢ (𝑅 ↑ko 𝒫 1o) = (𝑅 ↑ko 𝒫 1o)
23	22	xkotopon 23919	. . . . 5 ⊢ ((𝒫 1o ∈ Top ∧ 𝑅 ∈ Top) → (𝑅 ↑ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑅)))
24	21, 6, 23	syl2anc 596	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑅 ↑ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑅)))
25		simpl 488	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ 𝑛-Locally Comp)
26		simpr 490	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
27		eqid 2760	. . . . . 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 23979	. . . . 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 1398	. . . 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 5834	. . . . 5 ⊢ (𝑔 = 𝑓 → (𝑔 ∘ ℎ) = (𝑓 ∘ ℎ))
31		coeq2 5835	. . . . 5 ⊢ (ℎ = (1o × {𝑥}) → (𝑓 ∘ ℎ) = (𝑓 ∘ (1o × {𝑥})))
32	30, 31	sylan9eq 2815	. . . 4 ⊢ ((𝑔 = 𝑓 ∧ ℎ = (1o × {𝑥})) → (𝑔 ∘ ℎ) = (𝑓 ∘ (1o × {𝑥})))
33	5, 9, 10, 19, 5, 24, 29, 32	cnmpt22 23993	. . 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 2760	. . . . 5 ⊢ (𝑆 ↑ko 𝒫 1o) = (𝑆 ↑ko 𝒫 1o)
35	34	xkotopon 23919	. . . 4 ⊢ ((𝒫 1o ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑆)))
36	21, 26, 35	syl2anc 596	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑆)))
37		0lt1o 8512	. . . . 5 ⊢ ∅ ∈ 1o
38	37	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → ∅ ∈ 1o)
39		unipw 5417	. . . . . 6 ⊢ ∪ 𝒫 1o = 1o
40	39	eqcomi 2769	. . . . 5 ⊢ 1o = ∪ 𝒫 1o
41	40	xkopjcn 23975	. . . 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 1398	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝒫 1o Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆 ↑ko 𝒫 1o) Cn 𝑆))
43		fveq1 6884	. . . 4 ⊢ (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = ((𝑓 ∘ (1o × {𝑥}))‘∅))
44		vex 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
45	44	fconst 6768	. . . . . 6 ⊢ (1o × {𝑥}):1o⟶{𝑥}
46		fvco3 6985	. . . . . 6 ⊢ (((1o × {𝑥}):1o⟶{𝑥} ∧ ∅ ∈ 1o) → ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘((1o × {𝑥})‘∅)))
47	45, 37, 46	mp2an 705	. . . . 5 ⊢ ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘((1o × {𝑥})‘∅))
48	44	fvconst2 7210	. . . . . . 7 ⊢ (∅ ∈ 1o → ((1o × {𝑥})‘∅) = 𝑥)
49	37, 48	ax-mp 5	. . . . . 6 ⊢ ((1o × {𝑥})‘∅) = 𝑥
50	49	fveq2i 6888	. . . . 5 ⊢ (𝑓‘((1o × {𝑥})‘∅)) = (𝑓‘𝑥)
51	47, 50	eqtri 2783	. . . 4 ⊢ ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘𝑥)
52	43, 51	eqtrdi 2811	. . 3 ⊢ (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = (𝑓‘𝑥))
53	5, 9, 33, 36, 42, 52	cnmpt21 23990	. 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 2864	1 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝐹 ∈ (((𝑆 ↑ko 𝑅) ×t 𝑅) Cn 𝑆))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∅c0 4278  𝒫 cpw 4556  {csn 4583  ∪ cuni 4866   ↦ cmpt 5185   × cxp 5648   ∘ ccom 5654  Oncon0 6362  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1_oc1o 8469  Topctop 23211  TopOnctopon 23228   Cn ccn 23542  Compccmp 23704  𝑛-Locally cnlly 23784   ×_t ctx 23879   ↑_ko cxko 23880

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-1o 8476  df-2o 8477  df-map 8849  df-ixp 8926  df-en 8974  df-dom 8975  df-fin 8977  df-fi 9403  df-rest 17593  df-topgen 17614  df-pt 17615  df-top 23212  df-topon 23229  df-bases 23264  df-ntr 23338  df-nei 23416  df-cn 23545  df-cnp 23546  df-cmp 23705  df-nlly 23786  df-tx 23881  df-xko 23882

This theorem is used by:  cnmptk1p  24004  cnmptk2  24005