Theorem cnmptkk 23511

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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
3		cnmptkk.l	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
4	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ (TopOn‘𝑍))
5		cnmptkk.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
6		topontop 22739	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
7	1, 6	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
8		cnmptkk.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ 𝑛-Locally Comp)
9		nllytop 23301	. . . . . . . . . 10 ⊢ (𝐿 ∈ 𝑛-Locally Comp → 𝐿 ∈ Top)
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Top)
11		eqid 2724	. . . . . . . . . 10 ⊢ (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
12	11	xkotopon 23428	. . . . . . . . 9 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
13	7, 10, 12	syl2anc 583	. . . . . . . 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 23077	. . . . . . . 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 1368	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿))
17	16	fvmptelcdm 7105	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
18		cnf2 23077	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
19	2, 4, 17, 18	syl3anc 1368	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
20		eqid 2724	. . . . . 6 ⊢ (𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑦 ∈ 𝑌 ↦ 𝐴)
21	20	fmpt 7102	. . . . 5 ⊢ (∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 ↔ (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
22	19, 21	sylibr 233	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍)
23		eqidd 2725	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑦 ∈ 𝑌 ↦ 𝐴))
24		eqidd 2725	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑍 ↦ 𝐵) = (𝑧 ∈ 𝑍 ↦ 𝐵))
25		cnmptkk.c	. . . 4 ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶)
26	22, 23, 24, 25	fmptcof 7121	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑦 ∈ 𝑌 ↦ 𝐶))
27	26	mpteq2dva 5239	. 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 22739	. . . . 5 ⊢ (𝑀 ∈ (TopOn‘𝑊) → 𝑀 ∈ Top)
31	29, 30	syl 17	. . . 4 ⊢ (𝜑 → 𝑀 ∈ Top)
32		eqid 2724	. . . . 5 ⊢ (𝑀 ↑ko 𝐿) = (𝑀 ↑ko 𝐿)
33	32	xkotopon 23428	. . . 4 ⊢ ((𝐿 ∈ Top ∧ 𝑀 ∈ Top) → (𝑀 ↑ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
34	10, 31, 33	syl2anc 583	. . 3 ⊢ (𝜑 → (𝑀 ↑ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
35		eqid 2724	. . . . 5 ⊢ (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓 ∘ 𝑔))
36	35	xkococn 23488	. . . 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 1368	. . 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 5848	. . . 4 ⊢ (𝑓 = (𝑧 ∈ 𝑍 ↦ 𝐵) → (𝑓 ∘ 𝑔) = ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ 𝑔))
39		coeq2 5849	. . . 4 ⊢ (𝑔 = (𝑦 ∈ 𝑌 ↦ 𝐴) → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ 𝑔) = ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑦 ∈ 𝑌 ↦ 𝐴)))
40	38, 39	sylan9eq 2784	. . 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 23495	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑦 ∈ 𝑌 ↦ 𝐴))) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾)))
42	27, 41	eqeltrrd 2826	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3053   ↦ cmpt 5222   ∘ ccom 5671  ⟶wf 6530  ‘cfv 6534  (class class class)co 7402   ∈ cmpo 7404  Topctop 22719  TopOnctopon 22736   Cn ccn 23052  Compccmp 23214  𝑛-Locally cnlly 23293   ×_t ctx 23388   ↑_ko cxko 23389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-iin 4991  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-1o 8462  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-fin 8940  df-fi 9403  df-rest 17369  df-topgen 17390  df-top 22720  df-topon 22737  df-bases 22773  df-ntr 22848  df-nei 22926  df-cn 23055  df-cmp 23215  df-nlly 23295  df-tx 23390  df-xko 23391

This theorem is referenced by: (None)