Theorem ptcn 23122

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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐽 ∈ (TopOn‘𝑋))
3		ptcn.5	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐼⟶Top)
4	3	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ Top)
5		toptopon2 22411	. . . . . . . . . 10 ⊢ ((𝐹‘𝑘) ∈ Top ↔ (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘)))
6	4, 5	sylib 217	. . . . . . . . 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 22744	. . . . . . . . 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 7109	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ (𝐹‘𝑘))
11	10	an32s 650	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝐴 ∈ ∪ (𝐹‘𝑘))
12	11	ralrimiva 3146	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘))
13		ptcn.4	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
14	13	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐼 ∈ 𝑉)
15		mptelixpg 8925	. . . . . 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 256	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘))
18		ptcn.2	. . . . . . 7 ⊢ 𝐾 = (∏t‘𝐹)
19	18	ptuni 23089	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶Top) → X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾)
20	13, 3, 19	syl2anc 584	. . . . 5 ⊢ (𝜑 → X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾)
21	20	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾)
22	17, 21	eleqtrd 2835	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ ∪ 𝐾)
23	22	fmpttd 7111	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)):𝑋⟶∪ 𝐾)
24	1	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
25	13	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐼 ∈ 𝑉)
26	3	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐹:𝐼⟶Top)
27		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
28	7	adantlr 713	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘)))
29		simplr 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑧 ∈ 𝑋)
30		toponuni 22407	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
31	1, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
32	31	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑋 = ∪ 𝐽)
33	29, 32	eleqtrd 2835	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑧 ∈ ∪ 𝐽)
34		eqid 2732	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
35	34	cncnpi 22773	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘)) ∧ 𝑧 ∈ ∪ 𝐽) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝑧))
36	28, 33, 35	syl2anc 584	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝑧))
37	18, 24, 25, 26, 27, 36	ptcnp 23117	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))
38	37	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))
39		pttop 23077	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶Top) → (∏t‘𝐹) ∈ Top)
40	13, 3, 39	syl2anc 584	. . . . 5 ⊢ (𝜑 → (∏t‘𝐹) ∈ Top)
41	18, 40	eqeltrid 2837	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
42		toptopon2 22411	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
43	41, 42	sylib 217	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
44		cncnp 22775	. . 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 584	. 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 711	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ (𝐽 Cn 𝐾))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∪ cuni 4907   ↦ cmpt 5230  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405  Xcixp 8887  ∏_tcpt 17380  Topctop 22386  TopOnctopon 22403   Cn ccn 22719   CnP ccnp 22720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-1o 8462  df-er 8699  df-map 8818  df-ixp 8888  df-en 8936  df-dom 8937  df-fin 8939  df-fi 9402  df-topgen 17385  df-pt 17386  df-top 22387  df-topon 22404  df-bases 22440  df-cn 22722  df-cnp 22723

This theorem is referenced by:  pt1hmeo  23301  ptunhmeo  23303  symgtgp  23601  prdstmdd  23619  prdstgpd  23620  ptpconn  34212  broucube  36510