Theorem cnmpt12 23587

Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
cnmptid.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmpt11.a	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
cnmpt1t.b	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿))
cnmpt12.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
cnmpt12.l	⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
cnmpt12.c	⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀))
cnmpt12.d	⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → 𝐶 = 𝐷)

Assertion

Ref	Expression
cnmpt12	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐷) ∈ (𝐽 Cn 𝑀))

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

Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑦,𝑧)   𝐷(𝑥)   𝐽(𝑧)   𝐾(𝑧)   𝐿(𝑧)

Proof of Theorem cnmpt12

Step	Hyp	Ref	Expression
1		cnmptid.j	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		cnmpt12.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
3		cnmpt11.a	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
4		cnf2 23169	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
5	1, 2, 3, 4	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
6	5	fvmptelcdm 7067	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
7		cnmpt12.l	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
8		cnmpt1t.b	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿))
9		cnf2 23169	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍)
10	1, 7, 8, 9	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍)
11	10	fvmptelcdm 7067	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑍)
12	6, 11	jca 511	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍))
13		txtopon 23511	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
14	2, 7, 13	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
15		cnmpt12.c	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀))
16		cntop2 23161	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀) → 𝑀 ∈ Top)
17	15, 16	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ Top)
18		toptopon2 22838	. . . . . . . . . 10 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀))
19	17, 18	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀))
20		cnf2 23169	. . . . . . . . 9 ⊢ (((𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)) ∧ 𝑀 ∈ (TopOn‘∪ 𝑀) ∧ (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀)) → (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶):(𝑌 × 𝑍)⟶∪ 𝑀)
21	14, 19, 15, 20	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶):(𝑌 × 𝑍)⟶∪ 𝑀)
22		eqid 2729	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) = (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)
23	22	fmpo 8026	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑌 ∀𝑧 ∈ 𝑍 𝐶 ∈ ∪ 𝑀 ↔ (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶):(𝑌 × 𝑍)⟶∪ 𝑀)
24	21, 23	sylibr 234	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ 𝑍 𝐶 ∈ ∪ 𝑀)
25		r2al 3171	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑌 ∀𝑧 ∈ 𝑍 𝐶 ∈ ∪ 𝑀 ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀))
26	24, 25	sylib 218	. . . . . 6 ⊢ (𝜑 → ∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀))
27	26	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀))
28		eleq1 2816	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑌 ↔ 𝐴 ∈ 𝑌))
29		eleq1 2816	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ 𝑍 ↔ 𝐵 ∈ 𝑍))
30	28, 29	bi2anan9 638	. . . . . . 7 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ↔ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍)))
31		cnmpt12.d	. . . . . . . 8 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → 𝐶 = 𝐷)
32	31	eleq1d 2813	. . . . . . 7 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → (𝐶 ∈ ∪ 𝑀 ↔ 𝐷 ∈ ∪ 𝑀))
33	30, 32	imbi12d 344	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → (((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀) ↔ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍) → 𝐷 ∈ ∪ 𝑀)))
34	33	spc2gv 3563	. . . . 5 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍) → (∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀) → ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍) → 𝐷 ∈ ∪ 𝑀)))
35	12, 27, 12, 34	syl3c 66	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ ∪ 𝑀)
36	31, 22	ovmpoga 7523	. . . 4 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍 ∧ 𝐷 ∈ ∪ 𝑀) → (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵) = 𝐷)
37	6, 11, 35, 36	syl3anc 1373	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵) = 𝐷)
38	37	mpteq2dva 5195	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐷))
39	1, 3, 8, 15	cnmpt12f 23586	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵)) ∈ (𝐽 Cn 𝑀))
40	38, 39	eqeltrrd 2829	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐷) ∈ (𝐽 Cn 𝑀))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∪ cuni 4867   ↦ cmpt 5183   × cxp 5629  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  Topctop 22813  TopOnctopon 22830   Cn ccn 23144   ×_t ctx 23480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-map 8778  df-topgen 17382  df-top 22814  df-topon 22831  df-bases 22866  df-cn 23147  df-tx 23482

This theorem is referenced by:  cnmptkk  23603  cnmptk1p  23605  divccn  24797  iihalf1cn  24859  iihalf2cn  24862  icchmeo  24871  pcocn  24950  pcopt  24955  pcopt2  24956  pcoass  24957  mulcncf  25379  plycn  26199  psercn2  26365  resqrtcn  26692  sqrtcn  26693  efrlim  26912  rmulccn  33911  pl1cn  33938  cxpcncf1  34579  cvxpconn  35222  knoppcnlem10  36483  fprodcnlem  45590  cxpcncf2  45890