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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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