Theorem cnmptk1p 23747

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 2764	. . . 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 23311	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑌)
7	3, 4, 5, 6	syl3anc 1392	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑌)
8	7	fvmptelcdm 7096	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑌)
9	2	eleq1d 2849	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∈ 𝑍 ↔ 𝐶 ∈ 𝑍))
10	4	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
11		cnmptk1p.l	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
12	11	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ (TopOn‘𝑍))
13		cnmptk1p.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp)
14		nllytop 23535	. . . . . . . . . . 11 ⊢ (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
15	13, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Top)
16		topontop 22975	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
17	11, 16	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ Top)
18		eqid 2764	. . . . . . . . . . 11 ⊢ (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
19	18	xkotopon 23662	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
20	15, 17, 19	syl2anc 593	. . . . . . . . 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 23311	. . . . . . . . 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 1392	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿))
24	23	fvmptelcdm 7096	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
25		cnf2 23311	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
26	10, 12, 24, 25	syl3anc 1392	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
27	1	fmpt 7093	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 ↔ (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
28	26, 27	sylibr 236	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍)
29	9, 28, 8	rspcdva 3584	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ 𝑍)
30	1, 2, 8, 29	fvmptd3 7001	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵) = 𝐶)
31	30	mpteq2dva 5195	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
32		eqid 2764	. . . . 5 ⊢ (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
33		toponuni 22976	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
34	4, 33	syl 17	. . . . 5 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
35		mpoeq12 7471	. . . . 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 596	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ 𝑌 ↦ (𝑓‘𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)))
37		eqid 2764	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
38		eqid 2764	. . . . . 6 ⊢ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧))
39	37, 38	xkofvcn 23746	. . . . 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 593	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)) ∈ (((𝐿 ↑ko 𝐾) ×t 𝐾) Cn 𝐿))
41	36, 40	eqeltrd 2864	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ 𝑌 ↦ (𝑓‘𝑧)) ∈ (((𝐿 ↑ko 𝐾) ×t 𝐾) Cn 𝐿))
42		fveq1 6868	. . . 4 ⊢ (𝑓 = (𝑦 ∈ 𝑌 ↦ 𝐴) → (𝑓‘𝑧) = ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧))
43		fveq2 6869	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧) = ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵))
44	42, 43	sylan9eq 2819	. . 3 ⊢ ((𝑓 = (𝑦 ∈ 𝑌 ↦ 𝐴) ∧ 𝑧 = 𝐵) → (𝑓‘𝑧) = ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵))
45	3, 21, 5, 20, 4, 41, 44	cnmpt12 23729	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵)) ∈ (𝐽 Cn 𝐿))
46	31, 45	eqeltrrd 2865	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∪ cuni 4867   ↦ cmpt 5183  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398   ∈ cmpo 7400  Topctop 22955  TopOnctopon 22972   Cn ccn 23286  Compccmp 23448  𝑛-Locally cnlly 23527   ×_t ctx 23622   ↑_ko cxko 23623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-1o 8439  df-2o 8440  df-map 8812  df-ixp 8882  df-en 8930  df-dom 8931  df-fin 8933  df-fi 9359  df-rest 17453  df-topgen 17474  df-pt 17475  df-top 22956  df-topon 22973  df-bases 23008  df-ntr 23082  df-nei 23160  df-cn 23289  df-cnp 23290  df-cmp 23449  df-nlly 23529  df-tx 23624  df-xko 23625

This theorem is referenced by:  xkohmeo  23877