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Theorem ptcn 21801
 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 474 . . . . . . . . . 10 ((𝜑𝑘𝐼) → 𝐽 ∈ (TopOn‘𝑋))
3 ptcn.5 . . . . . . . . . . . 12 (𝜑𝐹:𝐼⟶Top)
43ffvelrnda 6608 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ Top)
5 eqid 2825 . . . . . . . . . . . 12 (𝐹𝑘) = (𝐹𝑘)
65toptopon 21092 . . . . . . . . . . 11 ((𝐹𝑘) ∈ Top ↔ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
74, 6sylib 210 . . . . . . . . . 10 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
8 ptcn.6 . . . . . . . . . 10 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘)))
9 cnf2 21424 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘))) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
102, 7, 8, 9syl3anc 1496 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
11 eqid 2825 . . . . . . . . . 10 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
1211fmpt 6629 . . . . . . . . 9 (∀𝑥𝑋 𝐴 (𝐹𝑘) ↔ (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
1310, 12sylibr 226 . . . . . . . 8 ((𝜑𝑘𝐼) → ∀𝑥𝑋 𝐴 (𝐹𝑘))
1413r19.21bi 3141 . . . . . . 7 (((𝜑𝑘𝐼) ∧ 𝑥𝑋) → 𝐴 (𝐹𝑘))
1514an32s 644 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑘𝐼) → 𝐴 (𝐹𝑘))
1615ralrimiva 3175 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑘𝐼 𝐴 (𝐹𝑘))
17 ptcn.4 . . . . . . 7 (𝜑𝐼𝑉)
1817adantr 474 . . . . . 6 ((𝜑𝑥𝑋) → 𝐼𝑉)
19 mptelixpg 8212 . . . . . 6 (𝐼𝑉 → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘) ↔ ∀𝑘𝐼 𝐴 (𝐹𝑘)))
2018, 19syl 17 . . . . 5 ((𝜑𝑥𝑋) → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘) ↔ ∀𝑘𝐼 𝐴 (𝐹𝑘)))
2116, 20mpbird 249 . . . 4 ((𝜑𝑥𝑋) → (𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘))
22 ptcn.2 . . . . . . 7 𝐾 = (∏t𝐹)
2322ptuni 21768 . . . . . 6 ((𝐼𝑉𝐹:𝐼⟶Top) → X𝑘𝐼 (𝐹𝑘) = 𝐾)
2417, 3, 23syl2anc 581 . . . . 5 (𝜑X𝑘𝐼 (𝐹𝑘) = 𝐾)
2524adantr 474 . . . 4 ((𝜑𝑥𝑋) → X𝑘𝐼 (𝐹𝑘) = 𝐾)
2621, 25eleqtrd 2908 . . 3 ((𝜑𝑥𝑋) → (𝑘𝐼𝐴) ∈ 𝐾)
2726fmpttd 6634 . 2 (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)):𝑋 𝐾)
281adantr 474 . . . 4 ((𝜑𝑧𝑋) → 𝐽 ∈ (TopOn‘𝑋))
2917adantr 474 . . . 4 ((𝜑𝑧𝑋) → 𝐼𝑉)
303adantr 474 . . . 4 ((𝜑𝑧𝑋) → 𝐹:𝐼⟶Top)
31 simpr 479 . . . 4 ((𝜑𝑧𝑋) → 𝑧𝑋)
328adantlr 708 . . . . 5 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → (𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘)))
33 simplr 787 . . . . . 6 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → 𝑧𝑋)
34 toponuni 21089 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
351, 34syl 17 . . . . . . 7 (𝜑𝑋 = 𝐽)
3635ad2antrr 719 . . . . . 6 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → 𝑋 = 𝐽)
3733, 36eleqtrd 2908 . . . . 5 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → 𝑧 𝐽)
38 eqid 2825 . . . . . 6 𝐽 = 𝐽
3938cncnpi 21453 . . . . 5 (((𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘)) ∧ 𝑧 𝐽) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝑧))
4032, 37, 39syl2anc 581 . . . 4 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝑧))
4122, 28, 29, 30, 31, 40ptcnp 21796 . . 3 ((𝜑𝑧𝑋) → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))
4241ralrimiva 3175 . 2 (𝜑 → ∀𝑧𝑋 (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))
43 pttop 21756 . . . . . 6 ((𝐼𝑉𝐹:𝐼⟶Top) → (∏t𝐹) ∈ Top)
4417, 3, 43syl2anc 581 . . . . 5 (𝜑 → (∏t𝐹) ∈ Top)
4522, 44syl5eqel 2910 . . . 4 (𝜑𝐾 ∈ Top)
46 eqid 2825 . . . . 5 𝐾 = 𝐾
4746toptopon 21092 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
4845, 47sylib 210 . . 3 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
49 cncnp 21455 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ 𝐾)) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ (𝐽 Cn 𝐾) ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)):𝑋 𝐾 ∧ ∀𝑧𝑋 (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))))
501, 48, 49syl2anc 581 . 2 (𝜑 → ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ (𝐽 Cn 𝐾) ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)):𝑋 𝐾 ∧ ∀𝑧𝑋 (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))))
5127, 42, 50mpbir2and 706 1 (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ (𝐽 Cn 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∀wral 3117  ∪ cuni 4658   ↦ cmpt 4952  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905  Xcixp 8175  ∏tcpt 16452  Topctop 21068  TopOnctopon 21085   Cn ccn 21399   CnP ccnp 21400 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-iin 4743  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-map 8124  df-ixp 8176  df-en 8223  df-dom 8224  df-fin 8226  df-fi 8586  df-topgen 16457  df-pt 16458  df-top 21069  df-topon 21086  df-bases 21121  df-cn 21402  df-cnp 21403 This theorem is referenced by:  pt1hmeo  21980  ptunhmeo  21982  symgtgp  22275  prdstmdd  22297  prdstgpd  22298  ptpconn  31761  broucube  33987
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