Theorem cnmpt1k 23711

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 23278	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑍)
5	1, 2, 3, 4	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑍)
6		eqid 2740	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
7	6	fmpt 7144	. . . . . 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 2741	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴))
11		eqidd 2741	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑧 ∈ 𝑍 ↦ 𝐵) = (𝑧 ∈ 𝑍 ↦ 𝐵))
12		cnmpt1k.c	. . . 4 ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶)
13	9, 10, 11, 12	fmptcof 7164	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
14	13	mpteq2dva 5266	. 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 22940	. . . . 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 22940	. . . . 5 ⊢ (𝑀 ∈ (TopOn‘𝑊) → 𝑀 ∈ Top)
21	19, 20	syl 17	. . . 4 ⊢ (𝜑 → 𝑀 ∈ Top)
22		eqid 2740	. . . . 5 ⊢ (𝑀 ↑ko 𝐿) = (𝑀 ↑ko 𝐿)
23	22	xkotopon 23629	. . . 4 ⊢ ((𝐿 ∈ Top ∧ 𝑀 ∈ Top) → (𝑀 ↑ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
24	18, 21, 23	syl2anc 583	. . 3 ⊢ (𝜑 → (𝑀 ↑ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
25	21, 3	xkoco1cn 23686	. . 3 ⊢ (𝜑 → (𝑤 ∈ (𝐿 Cn 𝑀) ↦ (𝑤 ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) ∈ ((𝑀 ↑ko 𝐿) Cn (𝑀 ↑ko 𝐽)))
26		coeq1 5882	. . 3 ⊢ (𝑤 = (𝑧 ∈ 𝑍 ↦ 𝐵) → (𝑤 ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)))
27	15, 16, 24, 25, 26	cnmpt11 23692	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) ∈ (𝐾 Cn (𝑀 ↑ko 𝐽)))
28	14, 27	eqeltrrd 2845	1 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑥 ∈ 𝑋 ↦ 𝐶)) ∈ (𝐾 Cn (𝑀 ↑ko 𝐽)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ↦ cmpt 5249   ∘ ccom 5704  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Topctop 22920  TopOnctopon 22937   Cn ccn 23253   ↑_ko cxko 23590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-1o 8522  df-2o 8523  df-map 8886  df-en 9004  df-dom 9005  df-fin 9007  df-fi 9480  df-rest 17482  df-topgen 17503  df-top 22921  df-topon 22938  df-bases 22974  df-cn 23256  df-cmp 23416  df-xko 23592

This theorem is referenced by: (None)