Theorem cnmpt1k 21485

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 21053	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑍)
5	1, 2, 3, 4	syl3anc 1326	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑍)
6		eqid 2622	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
7	6	fmpt 6381	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑍 ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑍)
8	5, 7	sylibr 224	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑍)
9	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑍)
10		eqidd 2623	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴))
11		eqidd 2623	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑧 ∈ 𝑍 ↦ 𝐵) = (𝑧 ∈ 𝑍 ↦ 𝐵))
12		cnmpt1k.c	. . . 4 ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶)
13	9, 10, 11, 12	fmptcof 6397	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
14	13	mpteq2dva 4744	. 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 20718	. . . . 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 20718	. . . . 5 ⊢ (𝑀 ∈ (TopOn‘𝑊) → 𝑀 ∈ Top)
21	19, 20	syl 17	. . . 4 ⊢ (𝜑 → 𝑀 ∈ Top)
22		eqid 2622	. . . . 5 ⊢ (𝑀 ^ko 𝐿) = (𝑀 ^ko 𝐿)
23	22	xkotopon 21403	. . . 4 ⊢ ((𝐿 ∈ Top ∧ 𝑀 ∈ Top) → (𝑀 ^ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
24	18, 21, 23	syl2anc 693	. . 3 ⊢ (𝜑 → (𝑀 ^ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
25	21, 3	xkoco1cn 21460	. . 3 ⊢ (𝜑 → (𝑤 ∈ (𝐿 Cn 𝑀) ↦ (𝑤 ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) ∈ ((𝑀 ^ko 𝐿) Cn (𝑀 ^ko 𝐽)))
26		coeq1 5279	. . 3 ⊢ (𝑤 = (𝑧 ∈ 𝑍 ↦ 𝐵) → (𝑤 ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)))
27	15, 16, 24, 25, 26	cnmpt11 21466	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) ∈ (𝐾 Cn (𝑀 ^ko 𝐽)))
28	14, 27	eqeltrrd 2702	1 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑥 ∈ 𝑋 ↦ 𝐶)) ∈ (𝐾 Cn (𝑀 ^ko 𝐽)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ↦ cmpt 4729   ∘ ccom 5118  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Topctop 20698  TopOnctopon 20715   Cn ccn 21028   ^_ko cxko 21364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-cmp 21190  df-xko 21366

This theorem is referenced by: (None)