Theorem cnmptkk 24002

Description: The composition of two curried functions is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem cnmptkk

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

Step	Hyp	Ref	Expression
1		cnmptkk.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
2	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
3		cnmptkk.l	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
4	3	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ (TopOn‘𝑍))
5		cnmptkk.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
6		topontop 23231	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
7	1, 6	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
8		cnmptkk.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ 𝑛-Locally Comp)
9		nllytop 23792	. . . . . . . . . 10 ⊢ (𝐿 ∈ 𝑛-Locally Comp → 𝐿 ∈ Top)
10	8, 9	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Top)
11		eqid 2761	. . . . . . . . . 10 ⊢ (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
12	11	xkotopon 23919	. . . . . . . . 9 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
13	7, 10, 12	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
14		cnmptkk.a	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
15		cnf2 23567	. . . . . . . 8 ⊢ ((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))
16	5, 13, 14, 15	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿))
17	16	fvmptelcdm 7113	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
18		cnf2 23567	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
19	2, 4, 17, 18	syl3anc 1398	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
20		eqid 2761	. . . . . 6 ⊢ (𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑦 ∈ 𝑌 ↦ 𝐴)
21	20	fmpt 7110	. . . . 5 ⊢ (∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 ↔ (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
22	19, 21	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍)
23		eqidd 2762	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑦 ∈ 𝑌 ↦ 𝐴))
24		eqidd 2762	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑍 ↦ 𝐵) = (𝑧 ∈ 𝑍 ↦ 𝐵))
25		cnmptkk.c	. . . 4 ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶)
26	22, 23, 24, 25	fmptcof 7131	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑦 ∈ 𝑌 ↦ 𝐶))
27	26	mpteq2dva 5198	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑦 ∈ 𝑌 ↦ 𝐴))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)))
28		cnmptkk.b	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑧 ∈ 𝑍 ↦ 𝐵)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐿)))
29		cnmptkk.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑊))
30		topontop 23231	. . . . 5 ⊢ (𝑀 ∈ (TopOn‘𝑊) → 𝑀 ∈ Top)
31	29, 30	syl 18	. . . 4 ⊢ (𝜑 → 𝑀 ∈ Top)
32		eqid 2761	. . . . 5 ⊢ (𝑀 ↑ko 𝐿) = (𝑀 ↑ko 𝐿)
33	32	xkotopon 23919	. . . 4 ⊢ ((𝐿 ∈ Top ∧ 𝑀 ∈ Top) → (𝑀 ↑ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
34	10, 31, 33	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑀 ↑ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
35		eqid 2761	. . . . 5 ⊢ (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓 ∘ 𝑔))
36	35	xkococn 23979	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ 𝑛-Locally Comp ∧ 𝑀 ∈ Top) → (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓 ∘ 𝑔)) ∈ (((𝑀 ↑ko 𝐿) ×t (𝐿 ↑ko 𝐾)) Cn (𝑀 ↑ko 𝐾)))
37	7, 8, 31, 36	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓 ∘ 𝑔)) ∈ (((𝑀 ↑ko 𝐿) ×t (𝐿 ↑ko 𝐾)) Cn (𝑀 ↑ko 𝐾)))
38		coeq1 5835	. . . 4 ⊢ (𝑓 = (𝑧 ∈ 𝑍 ↦ 𝐵) → (𝑓 ∘ 𝑔) = ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ 𝑔))
39		coeq2 5836	. . . 4 ⊢ (𝑔 = (𝑦 ∈ 𝑌 ↦ 𝐴) → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ 𝑔) = ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑦 ∈ 𝑌 ↦ 𝐴)))
40	38, 39	sylan9eq 2816	. . 3 ⊢ ((𝑓 = (𝑧 ∈ 𝑍 ↦ 𝐵) ∧ 𝑔 = (𝑦 ∈ 𝑌 ↦ 𝐴)) → (𝑓 ∘ 𝑔) = ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑦 ∈ 𝑌 ↦ 𝐴)))
41	5, 28, 14, 34, 13, 37, 40	cnmpt12 23986	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑦 ∈ 𝑌 ↦ 𝐴))) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾)))
42	27, 41	eqeltrrd 2862	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ↦ cmpt 5186   ∘ ccom 5655  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  Topctop 23211  TopOnctopon 23228   Cn ccn 23542  Compccmp 23704  𝑛-Locally cnlly 23784   ×_t ctx 23879   ↑_ko cxko 23880

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  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 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-1o 8476  df-2o 8477  df-map 8849  df-en 8974  df-dom 8975  df-fin 8977  df-fi 9403  df-rest 17593  df-topgen 17614  df-top 23212  df-topon 23229  df-bases 23264  df-ntr 23338  df-nei 23416  df-cn 23545  df-cmp 23705  df-nlly 23786  df-tx 23881  df-xko 23882

This theorem is used by: (None)