Theorem ptcn 22232

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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐽 ∈ (TopOn‘𝑋))
3		ptcn.5	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐼⟶Top)
4	3	ffvelrnda 6828	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ Top)
5		toptopon2 21523	. . . . . . . . . 10 ⊢ ((𝐹‘𝑘) ∈ Top ↔ (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘)))
6	4, 5	sylib 221	. . . . . . . . 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 21854	. . . . . . . . 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 1368	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘))
10	9	fvmptelrn 6854	. . . . . . 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 3149	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘))
13		ptcn.4	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
14	13	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐼 ∈ 𝑉)
15		mptelixpg 8482	. . . . . 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 260	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘))
18		ptcn.2	. . . . . . 7 ⊢ 𝐾 = (∏t‘𝐹)
19	18	ptuni 22199	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶Top) → X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾)
20	13, 3, 19	syl2anc 587	. . . . 5 ⊢ (𝜑 → X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾)
21	20	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾)
22	17, 21	eleqtrd 2892	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ ∪ 𝐾)
23	22	fmpttd 6856	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)):𝑋⟶∪ 𝐾)
24	1	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
25	13	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐼 ∈ 𝑉)
26	3	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐹:𝐼⟶Top)
27		simpr 488	. . . 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 21519	. . . . . . . 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 2892	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑧 ∈ ∪ 𝐽)
34		eqid 2798	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
35	34	cncnpi 21883	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘)) ∧ 𝑧 ∈ ∪ 𝐽) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝑧))
36	28, 33, 35	syl2anc 587	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝑧))
37	18, 24, 25, 26, 27, 36	ptcnp 22227	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))
38	37	ralrimiva 3149	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))
39		pttop 22187	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶Top) → (∏t‘𝐹) ∈ Top)
40	13, 3, 39	syl2anc 587	. . . . 5 ⊢ (𝜑 → (∏t‘𝐹) ∈ Top)
41	18, 40	eqeltrid 2894	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
42		toptopon2 21523	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
43	41, 42	sylib 221	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
44		cncnp 21885	. . 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 587	. 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∪ cuni 4800   ↦ cmpt 5110  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  Xcixp 8444  ∏_tcpt 16704  Topctop 21498  TopOnctopon 21515   Cn ccn 21829   CnP ccnp 21830

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-fin 8496  df-fi 8859  df-topgen 16709  df-pt 16710  df-top 21499  df-topon 21516  df-bases 21551  df-cn 21832  df-cnp 21833

This theorem is referenced by:  pt1hmeo  22411  ptunhmeo  22413  symgtgp  22711  prdstmdd  22729  prdstgpd  22730  ptpconn  32593  broucube  35091