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Theorem ptcn 21340
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 = (∏t𝐹)
ptcn.3 (𝜑𝐽 ∈ (TopOn‘𝑋))
ptcn.4 (𝜑𝐼𝑉)
ptcn.5 (𝜑𝐹:𝐼⟶Top)
ptcn.6 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘)))
Assertion
Ref Expression
ptcn (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ (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.
StepHypRef Expression
1 ptcn.3 . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝑋))
21adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝐼) → 𝐽 ∈ (TopOn‘𝑋))
3 ptcn.5 . . . . . . . . . . . 12 (𝜑𝐹:𝐼⟶Top)
43ffvelrnda 6315 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ Top)
5 eqid 2621 . . . . . . . . . . . 12 (𝐹𝑘) = (𝐹𝑘)
65toptopon 20648 . . . . . . . . . . 11 ((𝐹𝑘) ∈ Top ↔ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
74, 6sylib 208 . . . . . . . . . 10 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
8 ptcn.6 . . . . . . . . . 10 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘)))
9 cnf2 20963 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘))) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
102, 7, 8, 9syl3anc 1323 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
11 eqid 2621 . . . . . . . . . 10 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
1211fmpt 6337 . . . . . . . . 9 (∀𝑥𝑋 𝐴 (𝐹𝑘) ↔ (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
1310, 12sylibr 224 . . . . . . . 8 ((𝜑𝑘𝐼) → ∀𝑥𝑋 𝐴 (𝐹𝑘))
1413r19.21bi 2927 . . . . . . 7 (((𝜑𝑘𝐼) ∧ 𝑥𝑋) → 𝐴 (𝐹𝑘))
1514an32s 845 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑘𝐼) → 𝐴 (𝐹𝑘))
1615ralrimiva 2960 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑘𝐼 𝐴 (𝐹𝑘))
17 ptcn.4 . . . . . . 7 (𝜑𝐼𝑉)
1817adantr 481 . . . . . 6 ((𝜑𝑥𝑋) → 𝐼𝑉)
19 mptelixpg 7889 . . . . . 6 (𝐼𝑉 → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘) ↔ ∀𝑘𝐼 𝐴 (𝐹𝑘)))
2018, 19syl 17 . . . . 5 ((𝜑𝑥𝑋) → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘) ↔ ∀𝑘𝐼 𝐴 (𝐹𝑘)))
2116, 20mpbird 247 . . . 4 ((𝜑𝑥𝑋) → (𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘))
22 ptcn.2 . . . . . . 7 𝐾 = (∏t𝐹)
2322ptuni 21307 . . . . . 6 ((𝐼𝑉𝐹:𝐼⟶Top) → X𝑘𝐼 (𝐹𝑘) = 𝐾)
2417, 3, 23syl2anc 692 . . . . 5 (𝜑X𝑘𝐼 (𝐹𝑘) = 𝐾)
2524adantr 481 . . . 4 ((𝜑𝑥𝑋) → X𝑘𝐼 (𝐹𝑘) = 𝐾)
2621, 25eleqtrd 2700 . . 3 ((𝜑𝑥𝑋) → (𝑘𝐼𝐴) ∈ 𝐾)
27 eqid 2621 . . 3 (𝑥𝑋 ↦ (𝑘𝐼𝐴)) = (𝑥𝑋 ↦ (𝑘𝐼𝐴))
2826, 27fmptd 6340 . 2 (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)):𝑋 𝐾)
291adantr 481 . . . 4 ((𝜑𝑧𝑋) → 𝐽 ∈ (TopOn‘𝑋))
3017adantr 481 . . . 4 ((𝜑𝑧𝑋) → 𝐼𝑉)
313adantr 481 . . . 4 ((𝜑𝑧𝑋) → 𝐹:𝐼⟶Top)
32 simpr 477 . . . 4 ((𝜑𝑧𝑋) → 𝑧𝑋)
338adantlr 750 . . . . 5 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → (𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘)))
34 simplr 791 . . . . . 6 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → 𝑧𝑋)
35 toponuni 20642 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
361, 35syl 17 . . . . . . 7 (𝜑𝑋 = 𝐽)
3736ad2antrr 761 . . . . . 6 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → 𝑋 = 𝐽)
3834, 37eleqtrd 2700 . . . . 5 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → 𝑧 𝐽)
39 eqid 2621 . . . . . 6 𝐽 = 𝐽
4039cncnpi 20992 . . . . 5 (((𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘)) ∧ 𝑧 𝐽) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝑧))
4133, 38, 40syl2anc 692 . . . 4 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝑧))
4222, 29, 30, 31, 32, 41ptcnp 21335 . . 3 ((𝜑𝑧𝑋) → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))
4342ralrimiva 2960 . 2 (𝜑 → ∀𝑧𝑋 (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))
44 pttop 21295 . . . . . 6 ((𝐼𝑉𝐹:𝐼⟶Top) → (∏t𝐹) ∈ Top)
4517, 3, 44syl2anc 692 . . . . 5 (𝜑 → (∏t𝐹) ∈ Top)
4622, 45syl5eqel 2702 . . . 4 (𝜑𝐾 ∈ Top)
47 eqid 2621 . . . . 5 𝐾 = 𝐾
4847toptopon 20648 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
4946, 48sylib 208 . . 3 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
50 cncnp 20994 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ 𝐾)) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ (𝐽 Cn 𝐾) ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)):𝑋 𝐾 ∧ ∀𝑧𝑋 (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))))
511, 49, 50syl2anc 692 . 2 (𝜑 → ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ (𝐽 Cn 𝐾) ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)):𝑋 𝐾 ∧ ∀𝑧𝑋 (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))))
5228, 43, 51mpbir2and 956 1 (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907   cuni 4402  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  Xcixp 7852  tcpt 16020  Topctop 20617  TopOnctopon 20618   Cn ccn 20938   CnP ccnp 20939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-fin 7903  df-fi 8261  df-topgen 16025  df-pt 16026  df-top 20621  df-bases 20622  df-topon 20623  df-cn 20941  df-cnp 20942
This theorem is referenced by:  pt1hmeo  21519  ptunhmeo  21521  symgtgp  21815  prdstmdd  21837  prdstgpd  21838  ptpconn  30923  broucube  33075
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