Theorem cnmpt1k 23597

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

Hypotheses

Ref	Expression
cnmptk1.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmptk1.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
cnmptk1.l	⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
cnmpt1k.m	⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑊))
cnmpt1k.a	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿))
cnmpt1k.b	⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑧 ∈ 𝑍 ↦ 𝐵)) ∈ (𝐾 Cn (𝑀 ↑ko 𝐿)))
cnmpt1k.c	⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
cnmpt1k	⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑥 ∈ 𝑋 ↦ 𝐶)) ∈ (𝐾 Cn (𝑀 ↑ko 𝐽)))

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

Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑥)   𝐵(𝑦,𝑧)   𝐶(𝑥,𝑦)   𝐽(𝑧)   𝐾(𝑧)   𝐿(𝑧)   𝑀(𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑧)   𝑌(𝑧)

Proof of Theorem cnmpt1k

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnmptk1.j	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		cnmptk1.l	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
3		cnmpt1k.a	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿))
4		cnf2 23164	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑍)
5	1, 2, 3, 4	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑍)
6		eqid 2731	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
7	6	fmpt 7043	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑍 ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑍)
8	5, 7	sylibr 234	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑍)
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑍)
10		eqidd 2732	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴))
11		eqidd 2732	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑧 ∈ 𝑍 ↦ 𝐵) = (𝑧 ∈ 𝑍 ↦ 𝐵))
12		cnmpt1k.c	. . . 4 ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶)
13	9, 10, 11, 12	fmptcof 7063	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
14	13	mpteq2dva 5182	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) = (𝑦 ∈ 𝑌 ↦ (𝑥 ∈ 𝑋 ↦ 𝐶)))
15		cnmptk1.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
16		cnmpt1k.b	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑧 ∈ 𝑍 ↦ 𝐵)) ∈ (𝐾 Cn (𝑀 ↑ko 𝐿)))
17		topontop 22828	. . . . 5 ⊢ (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
18	2, 17	syl 17	. . . 4 ⊢ (𝜑 → 𝐿 ∈ Top)
19		cnmpt1k.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑊))
20		topontop 22828	. . . . 5 ⊢ (𝑀 ∈ (TopOn‘𝑊) → 𝑀 ∈ Top)
21	19, 20	syl 17	. . . 4 ⊢ (𝜑 → 𝑀 ∈ Top)
22		eqid 2731	. . . . 5 ⊢ (𝑀 ↑ko 𝐿) = (𝑀 ↑ko 𝐿)
23	22	xkotopon 23515	. . . 4 ⊢ ((𝐿 ∈ Top ∧ 𝑀 ∈ Top) → (𝑀 ↑ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
24	18, 21, 23	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑀 ↑ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
25	21, 3	xkoco1cn 23572	. . 3 ⊢ (𝜑 → (𝑤 ∈ (𝐿 Cn 𝑀) ↦ (𝑤 ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) ∈ ((𝑀 ↑ko 𝐿) Cn (𝑀 ↑ko 𝐽)))
26		coeq1 5796	. . 3 ⊢ (𝑤 = (𝑧 ∈ 𝑍 ↦ 𝐵) → (𝑤 ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)))
27	15, 16, 24, 25, 26	cnmpt11 23578	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) ∈ (𝐾 Cn (𝑀 ↑ko 𝐽)))
28	14, 27	eqeltrrd 2832	1 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑥 ∈ 𝑋 ↦ 𝐶)) ∈ (𝐾 Cn (𝑀 ↑ko 𝐽)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ↦ cmpt 5170   ∘ ccom 5618  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Topctop 22808  TopOnctopon 22825   Cn ccn 23139   ↑_ko cxko 23476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-1o 8385  df-2o 8386  df-map 8752  df-en 8870  df-dom 8871  df-fin 8873  df-fi 9295  df-rest 17326  df-topgen 17347  df-top 22809  df-topon 22826  df-bases 22861  df-cn 23142  df-cmp 23302  df-xko 23478

This theorem is referenced by: (None)