Theorem cnmpt22 23669

Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
cnmpt21.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmpt21.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
cnmpt21.a	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
cnmpt2t.b	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
cnmpt22.l	⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
cnmpt22.m	⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑊))
cnmpt22.c	⊢ (𝜑 → (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
cnmpt22.d	⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝐶 = 𝐷)

Assertion

Ref	Expression
cnmpt22	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐷) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))

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

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

Proof of Theorem cnmpt22

Step	Hyp	Ref	Expression
1		df-ov 7427	. . . 4 ⊢ (𝐴(𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)𝐵) = ((𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)‘⟨𝐴, 𝐵⟩)
2		cnmpt21.j	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
3		cnmpt21.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
4		txtopon 23586	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5	2, 3, 4	syl2anc 582	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6		cnmpt22.l	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
7		cnmpt21.a	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
8		cnf2 23244	. . . . . . . . 9 ⊢ (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍)
9	5, 6, 7, 8	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍)
10		eqid 2726	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
11	10	fmpo 8082	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍)
12	9, 11	sylibr 233	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍)
13		rsp2 3265	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍))
14	12, 13	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍))
15	14	3impib 1113	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍)
16		cnmpt22.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑊))
17		cnmpt2t.b	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
18		cnf2 23244	. . . . . . . . 9 ⊢ (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑀 ∈ (TopOn‘𝑊) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶𝑊)
19	5, 16, 17, 18	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶𝑊)
20		eqid 2726	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)
21	20	fmpo 8082	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ 𝑊 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶𝑊)
22	19, 21	sylibr 233	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ 𝑊)
23		rsp2 3265	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ 𝑊 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ 𝑊))
24	22, 23	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ 𝑊))
25	24	3impib 1113	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ 𝑊)
26	15, 25	jca 510	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊))
27		txtopon 23586	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ (TopOn‘𝑍) ∧ 𝑀 ∈ (TopOn‘𝑊)) → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑍 × 𝑊)))
28	6, 16, 27	syl2anc 582	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑍 × 𝑊)))
29		cnmpt22.c	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
30		cntop2 23236	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁) → 𝑁 ∈ Top)
31	29, 30	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ Top)
32		toptopon2 22911	. . . . . . . . . . 11 ⊢ (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘∪ 𝑁))
33	31, 32	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘∪ 𝑁))
34		cnf2 23244	. . . . . . . . . 10 ⊢ (((𝐿 ×t 𝑀) ∈ (TopOn‘(𝑍 × 𝑊)) ∧ 𝑁 ∈ (TopOn‘∪ 𝑁) ∧ (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) → (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶):(𝑍 × 𝑊)⟶∪ 𝑁)
35	28, 33, 29, 34	syl3anc 1368	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶):(𝑍 × 𝑊)⟶∪ 𝑁)
36		eqid 2726	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶) = (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)
37	36	fmpo 8082	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑍 ∀𝑤 ∈ 𝑊 𝐶 ∈ ∪ 𝑁 ↔ (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶):(𝑍 × 𝑊)⟶∪ 𝑁)
38	35, 37	sylibr 233	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝑍 ∀𝑤 ∈ 𝑊 𝐶 ∈ ∪ 𝑁)
39		r2al 3185	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑍 ∀𝑤 ∈ 𝑊 𝐶 ∈ ∪ 𝑁 ↔ ∀𝑧∀𝑤((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) → 𝐶 ∈ ∪ 𝑁))
40	38, 39	sylib 217	. . . . . . 7 ⊢ (𝜑 → ∀𝑧∀𝑤((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) → 𝐶 ∈ ∪ 𝑁))
41	40	3ad2ant1 1130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ∀𝑧∀𝑤((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) → 𝐶 ∈ ∪ 𝑁))
42		eleq1 2814	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝑧 ∈ 𝑍 ↔ 𝐴 ∈ 𝑍))
43		eleq1 2814	. . . . . . . . 9 ⊢ (𝑤 = 𝐵 → (𝑤 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊))
44	42, 43	bi2anan9 636	. . . . . . . 8 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → ((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) ↔ (𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊)))
45		cnmpt22.d	. . . . . . . . 9 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝐶 = 𝐷)
46	45	eleq1d 2811	. . . . . . . 8 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → (𝐶 ∈ ∪ 𝑁 ↔ 𝐷 ∈ ∪ 𝑁))
47	44, 46	imbi12d 343	. . . . . . 7 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → (((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) → 𝐶 ∈ ∪ 𝑁) ↔ ((𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊) → 𝐷 ∈ ∪ 𝑁)))
48	47	spc2gv 3586	. . . . . 6 ⊢ ((𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊) → (∀𝑧∀𝑤((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) → 𝐶 ∈ ∪ 𝑁) → ((𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊) → 𝐷 ∈ ∪ 𝑁)))
49	26, 41, 26, 48	syl3c 66	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ ∪ 𝑁)
50	45, 36	ovmpoga 7580	. . . . 5 ⊢ ((𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊 ∧ 𝐷 ∈ ∪ 𝑁) → (𝐴(𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)𝐵) = 𝐷)
51	15, 25, 49, 50	syl3anc 1368	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (𝐴(𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)𝐵) = 𝐷)
52	1, 51	eqtr3id 2780	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ((𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)‘⟨𝐴, 𝐵⟩) = 𝐷)
53	52	mpoeq3dva 7502	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)‘⟨𝐴, 𝐵⟩)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐷))
54	2, 3, 7, 17	cnmpt2t 23668	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
55	2, 3, 54, 29	cnmpt21f 23667	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)‘⟨𝐴, 𝐵⟩)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
56	53, 55	eqeltrrd 2827	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐷) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084  ∀wal 1532   = wceq 1534   ∈ wcel 2099  ∀wral 3051  ⟨cop 4639  ∪ cuni 4913   × cxp 5680  ⟶wf 6550  ‘cfv 6554  (class class class)co 7424   ∈ cmpo 7426  Topctop 22886  TopOnctopon 22903   Cn ccn 23219   ×_t ctx 23555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-map 8857  df-topgen 17458  df-top 22887  df-topon 22904  df-bases 22940  df-cn 23222  df-tx 23557

This theorem is referenced by:  cnmpt22f  23670  xkofvcn  23679  cnmptk2  23681  divcn  24877  cnrehmeo  24969  reparphti  25014  pcorevlem  25044  cxpcn  26772  cvxsconn  35071