Theorem cnmptk1p 23694

Description: The evaluation of a curried function by a one-arg function is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
cnmptk1p.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmptk1p.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
cnmptk1p.l	⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
cnmptk1p.n	⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp)
cnmptk1p.a	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
cnmptk1p.b	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))
cnmptk1p.c	⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)

Assertion

Ref	Expression
cnmptk1p	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿))

Distinct variable groups:   𝑥,𝐽   𝑥,𝐾   𝑥,𝐿   𝑦,𝐵   𝑦,𝐶   𝑥,𝑦,𝑋   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦   𝑦,𝑍

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑥)   𝐽(𝑦)   𝐾(𝑦)   𝐿(𝑦)   𝑍(𝑥)

Proof of Theorem cnmptk1p

Dummy variables 𝑓 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . 4 ⊢ (𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑦 ∈ 𝑌 ↦ 𝐴)
2		cnmptk1p.c	. . . 4 ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)
3		cnmptk1p.j	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
4		cnmptk1p.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
5		cnmptk1p.b	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))
6		cnf2 23258	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑌)
7	3, 4, 5, 6	syl3anc 1372	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑌)
8	7	fvmptelcdm 7132	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑌)
9	2	eleq1d 2825	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∈ 𝑍 ↔ 𝐶 ∈ 𝑍))
10	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
11		cnmptk1p.l	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
12	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ (TopOn‘𝑍))
13		cnmptk1p.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp)
14		nllytop 23482	. . . . . . . . . . 11 ⊢ (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
15	13, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Top)
16		topontop 22920	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
17	11, 16	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ Top)
18		eqid 2736	. . . . . . . . . . 11 ⊢ (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
19	18	xkotopon 23609	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
20	15, 17, 19	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
21		cnmptk1p.a	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
22		cnf2 23258	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿))
23	3, 20, 21, 22	syl3anc 1372	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿))
24	23	fvmptelcdm 7132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
25		cnf2 23258	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
26	10, 12, 24, 25	syl3anc 1372	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
27	1	fmpt 7129	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 ↔ (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
28	26, 27	sylibr 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍)
29	9, 28, 8	rspcdva 3622	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ 𝑍)
30	1, 2, 8, 29	fvmptd3 7038	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵) = 𝐶)
31	30	mpteq2dva 5241	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
32		eqid 2736	. . . . 5 ⊢ (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
33		toponuni 22921	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
34	4, 33	syl 17	. . . . 5 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
35		mpoeq12 7507	. . . . 5 ⊢ (((𝐾 Cn 𝐿) = (𝐾 Cn 𝐿) ∧ 𝑌 = ∪ 𝐾) → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ 𝑌 ↦ (𝑓‘𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)))
36	32, 34, 35	sylancr 587	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ 𝑌 ↦ (𝑓‘𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)))
37		eqid 2736	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
38		eqid 2736	. . . . . 6 ⊢ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧))
39	37, 38	xkofvcn 23693	. . . . 5 ⊢ ((𝐾 ∈ 𝑛-Locally Comp ∧ 𝐿 ∈ Top) → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)) ∈ (((𝐿 ↑ko 𝐾) ×t 𝐾) Cn 𝐿))
40	13, 17, 39	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)) ∈ (((𝐿 ↑ko 𝐾) ×t 𝐾) Cn 𝐿))
41	36, 40	eqeltrd 2840	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ 𝑌 ↦ (𝑓‘𝑧)) ∈ (((𝐿 ↑ko 𝐾) ×t 𝐾) Cn 𝐿))
42		fveq1 6904	. . . 4 ⊢ (𝑓 = (𝑦 ∈ 𝑌 ↦ 𝐴) → (𝑓‘𝑧) = ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧))
43		fveq2 6905	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧) = ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵))
44	42, 43	sylan9eq 2796	. . 3 ⊢ ((𝑓 = (𝑦 ∈ 𝑌 ↦ 𝐴) ∧ 𝑧 = 𝐵) → (𝑓‘𝑧) = ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵))
45	3, 21, 5, 20, 4, 41, 44	cnmpt12 23676	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵)) ∈ (𝐽 Cn 𝐿))
46	31, 45	eqeltrrd 2841	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060  ∪ cuni 4906   ↦ cmpt 5224  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432   ∈ cmpo 7434  Topctop 22900  TopOnctopon 22917   Cn ccn 23233  Compccmp 23395  𝑛-Locally cnlly 23474   ×_t ctx 23569   ↑_ko cxko 23570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-1o 8507  df-2o 8508  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-fin 8990  df-fi 9452  df-rest 17468  df-topgen 17489  df-pt 17490  df-top 22901  df-topon 22918  df-bases 22954  df-ntr 23029  df-nei 23107  df-cn 23236  df-cnp 23237  df-cmp 23396  df-nlly 23476  df-tx 23571  df-xko 23572

This theorem is referenced by:  xkohmeo  23824