Theorem xkofvcn 23679

Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 23651.) (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 23468	. . . 4 ⊢ (𝑅 ∈ 𝑛-Locally Comp → 𝑅 ∈ Top)
3		eqid 2726	. . . . 5 ⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
4	3	xkotopon 23595	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
5	2, 4	sylan 578	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
6	2	adantr 479	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ Top)
7		xkofvcn.1	. . . . 5 ⊢ 𝑋 = ∪ 𝑅
8	7	toptopon 22910	. . . 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 23663	. . . 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 23664	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (((𝑆 ↑ko 𝑅) ×t 𝑅) Cn 𝑅))
12		1on 8508	. . . . . . 7 ⊢ 1o ∈ On
13		distopon 22991	. . . . . . 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 23614	. . . . . 6 ⊢ ((𝒫 1o ∈ (TopOn‘1o) ∧ 𝑅 ∈ (TopOn‘𝑋)) → (𝑦 ∈ 𝑋 ↦ (1o × {𝑦})) ∈ (𝑅 Cn (𝑅 ↑ko 𝒫 1o)))
16	14, 9, 15	syl2anc 582	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑦 ∈ 𝑋 ↦ (1o × {𝑦})) ∈ (𝑅 Cn (𝑅 ↑ko 𝒫 1o)))
17		sneq 4643	. . . . . 6 ⊢ (𝑦 = 𝑥 → {𝑦} = {𝑥})
18	17	xpeq2d 5712	. . . . 5 ⊢ (𝑦 = 𝑥 → (1o × {𝑦}) = (1o × {𝑥}))
19	5, 9, 11, 9, 16, 18	cnmpt21 23666	. . . 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 22989	. . . . . 6 ⊢ (1o ∈ On → 𝒫 1o ∈ Top)
21	12, 20	mp1i 13	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1o ∈ Top)
22		eqid 2726	. . . . . 6 ⊢ (𝑅 ↑ko 𝒫 1o) = (𝑅 ↑ko 𝒫 1o)
23	22	xkotopon 23595	. . . . 5 ⊢ ((𝒫 1o ∈ Top ∧ 𝑅 ∈ Top) → (𝑅 ↑ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑅)))
24	21, 6, 23	syl2anc 582	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑅 ↑ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑅)))
25		simpl 481	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ 𝑛-Locally Comp)
26		simpr 483	. . . . 5 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
27		eqid 2726	. . . . . 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 23655	. . . . 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 1368	. . . 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 5864	. . . . 5 ⊢ (𝑔 = 𝑓 → (𝑔 ∘ ℎ) = (𝑓 ∘ ℎ))
31		coeq2 5865	. . . . 5 ⊢ (ℎ = (1o × {𝑥}) → (𝑓 ∘ ℎ) = (𝑓 ∘ (1o × {𝑥})))
32	30, 31	sylan9eq 2786	. . . 4 ⊢ ((𝑔 = 𝑓 ∧ ℎ = (1o × {𝑥})) → (𝑔 ∘ ℎ) = (𝑓 ∘ (1o × {𝑥})))
33	5, 9, 10, 19, 5, 24, 29, 32	cnmpt22 23669	. . 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 2726	. . . . 5 ⊢ (𝑆 ↑ko 𝒫 1o) = (𝑆 ↑ko 𝒫 1o)
35	34	xkotopon 23595	. . . 4 ⊢ ((𝒫 1o ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑆)))
36	21, 26, 35	syl2anc 582	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑆)))
37		0lt1o 8534	. . . . 5 ⊢ ∅ ∈ 1o
38	37	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → ∅ ∈ 1o)
39		unipw 5456	. . . . . 6 ⊢ ∪ 𝒫 1o = 1o
40	39	eqcomi 2735	. . . . 5 ⊢ 1o = ∪ 𝒫 1o
41	40	xkopjcn 23651	. . . 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 1368	. . 3 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝒫 1o Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆 ↑ko 𝒫 1o) Cn 𝑆))
43		fveq1 6900	. . . 4 ⊢ (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = ((𝑓 ∘ (1o × {𝑥}))‘∅))
44		vex 3466	. . . . . . 7 ⊢ 𝑥 ∈ V
45	44	fconst 6788	. . . . . 6 ⊢ (1o × {𝑥}):1o⟶{𝑥}
46		fvco3 7001	. . . . . 6 ⊢ (((1o × {𝑥}):1o⟶{𝑥} ∧ ∅ ∈ 1o) → ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘((1o × {𝑥})‘∅)))
47	45, 37, 46	mp2an 690	. . . . 5 ⊢ ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘((1o × {𝑥})‘∅))
48	44	fvconst2 7221	. . . . . . 7 ⊢ (∅ ∈ 1o → ((1o × {𝑥})‘∅) = 𝑥)
49	37, 48	ax-mp 5	. . . . . 6 ⊢ ((1o × {𝑥})‘∅) = 𝑥
50	49	fveq2i 6904	. . . . 5 ⊢ (𝑓‘((1o × {𝑥})‘∅)) = (𝑓‘𝑥)
51	47, 50	eqtri 2754	. . . 4 ⊢ ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘𝑥)
52	43, 51	eqtrdi 2782	. . 3 ⊢ (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = (𝑓‘𝑥))
53	5, 9, 33, 36, 42, 52	cnmpt21 23666	. 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 2830	1 ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝐹 ∈ (((𝑆 ↑ko 𝑅) ×t 𝑅) Cn 𝑆))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ∅c0 4325  𝒫 cpw 4607  {csn 4633  ∪ cuni 4913   ↦ cmpt 5236   × cxp 5680   ∘ ccom 5686  Oncon0 6376  ⟶wf 6550  ‘cfv 6554  (class class class)co 7424   ∈ cmpo 7426  1_oc1o 8489  Topctop 22886  TopOnctopon 22903   Cn ccn 23219  Compccmp 23381  𝑛-Locally cnlly 23460   ×_t ctx 23555   ↑_ko cxko 23556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-1o 8496  df-2o 8497  df-map 8857  df-ixp 8927  df-en 8975  df-dom 8976  df-fin 8978  df-fi 9454  df-rest 17437  df-topgen 17458  df-pt 17459  df-top 22887  df-topon 22904  df-bases 22940  df-ntr 23015  df-nei 23093  df-cn 23222  df-cnp 23223  df-cmp 23382  df-nlly 23462  df-tx 23557  df-xko 23558

This theorem is referenced by:  cnmptk1p  23680  cnmptk2  23681