Theorem ptcn 23656

Description: If every projection of a function is continuous, then the function itself is continuous into the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)

Hypotheses

Ref	Expression
ptcn.2	⊢ 𝐾 = (∏t‘𝐹)
ptcn.3	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
ptcn.4	⊢ (𝜑 → 𝐼 ∈ 𝑉)
ptcn.5	⊢ (𝜑 → 𝐹:𝐼⟶Top)
ptcn.6	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘)))

Assertion

Ref	Expression
ptcn	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ (𝐽 Cn 𝐾))

Distinct variable groups:   𝑥,𝑘,𝐹   𝑘,𝐼,𝑥   𝑘,𝐽   𝜑,𝑘,𝑥   𝑘,𝑋,𝑥   𝑥,𝐾   𝑘,𝑉,𝑥

Allowed substitution hints:   𝐴(𝑥,𝑘)   𝐽(𝑥)   𝐾(𝑘)

Proof of Theorem ptcn

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ptcn.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐽 ∈ (TopOn‘𝑋))
3		ptcn.5	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐼⟶Top)
4	3	ffvelcdmda 7118	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ Top)
5		toptopon2 22945	. . . . . . . . . 10 ⊢ ((𝐹‘𝑘) ∈ Top ↔ (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘)))
6	4, 5	sylib 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘)))
7		ptcn.6	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘)))
8		cnf2 23278	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘))) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘))
9	2, 6, 7, 8	syl3anc 1371	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘))
10	9	fvmptelcdm 7147	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ (𝐹‘𝑘))
11	10	an32s 651	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝐴 ∈ ∪ (𝐹‘𝑘))
12	11	ralrimiva 3152	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘))
13		ptcn.4	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
14	13	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐼 ∈ 𝑉)
15		mptelixpg 8993	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → ((𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) ↔ ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘)))
16	14, 15	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) ↔ ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘)))
17	12, 16	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘))
18		ptcn.2	. . . . . . 7 ⊢ 𝐾 = (∏t‘𝐹)
19	18	ptuni 23623	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶Top) → X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾)
20	13, 3, 19	syl2anc 583	. . . . 5 ⊢ (𝜑 → X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾)
21	20	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾)
22	17, 21	eleqtrd 2846	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ ∪ 𝐾)
23	22	fmpttd 7149	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)):𝑋⟶∪ 𝐾)
24	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
25	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐼 ∈ 𝑉)
26	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐹:𝐼⟶Top)
27		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
28	7	adantlr 714	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘)))
29		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑧 ∈ 𝑋)
30		toponuni 22941	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
31	1, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
32	31	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑋 = ∪ 𝐽)
33	29, 32	eleqtrd 2846	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑧 ∈ ∪ 𝐽)
34		eqid 2740	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
35	34	cncnpi 23307	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘)) ∧ 𝑧 ∈ ∪ 𝐽) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝑧))
36	28, 33, 35	syl2anc 583	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝑧))
37	18, 24, 25, 26, 27, 36	ptcnp 23651	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))
38	37	ralrimiva 3152	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))
39		pttop 23611	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶Top) → (∏t‘𝐹) ∈ Top)
40	13, 3, 39	syl2anc 583	. . . . 5 ⊢ (𝜑 → (∏t‘𝐹) ∈ Top)
41	18, 40	eqeltrid 2848	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
42		toptopon2 22945	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
43	41, 42	sylib 218	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
44		cncnp 23309	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ (𝐽 Cn 𝐾) ↔ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)):𝑋⟶∪ 𝐾 ∧ ∀𝑧 ∈ 𝑋 (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))))
45	1, 43, 44	syl2anc 583	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ (𝐽 Cn 𝐾) ↔ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)):𝑋⟶∪ 𝐾 ∧ ∀𝑧 ∈ 𝑋 (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))))
46	23, 38, 45	mpbir2and 712	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ (𝐽 Cn 𝐾))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∪ cuni 4931   ↦ cmpt 5249  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Xcixp 8955  ∏_tcpt 17498  Topctop 22920  TopOnctopon 22937   Cn ccn 23253   CnP ccnp 23254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-1o 8522  df-2o 8523  df-map 8886  df-ixp 8956  df-en 9004  df-dom 9005  df-fin 9007  df-fi 9480  df-topgen 17503  df-pt 17504  df-top 22921  df-topon 22938  df-bases 22974  df-cn 23256  df-cnp 23257

This theorem is referenced by:  pt1hmeo  23835  ptunhmeo  23837  symgtgp  24135  prdstmdd  24153  prdstgpd  24154  ptpconn  35201  broucube  37614