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Theorem ptcn 23651
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 . . . . . . . . . 10 (𝜑𝐽 ∈ (TopOn‘𝑋))
21adantr 480 . . . . . . . . 9 ((𝜑𝑘𝐼) → 𝐽 ∈ (TopOn‘𝑋))
3 ptcn.5 . . . . . . . . . . 11 (𝜑𝐹:𝐼⟶Top)
43ffvelcdmda 7104 . . . . . . . . . 10 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ Top)
5 toptopon2 22940 . . . . . . . . . 10 ((𝐹𝑘) ∈ Top ↔ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
64, 5sylib 218 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
7 ptcn.6 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘)))
8 cnf2 23273 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘))) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
92, 6, 7, 8syl3anc 1370 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
109fvmptelcdm 7133 . . . . . . 7 (((𝜑𝑘𝐼) ∧ 𝑥𝑋) → 𝐴 (𝐹𝑘))
1110an32s 652 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑘𝐼) → 𝐴 (𝐹𝑘))
1211ralrimiva 3144 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑘𝐼 𝐴 (𝐹𝑘))
13 ptcn.4 . . . . . . 7 (𝜑𝐼𝑉)
1413adantr 480 . . . . . 6 ((𝜑𝑥𝑋) → 𝐼𝑉)
15 mptelixpg 8974 . . . . . 6 (𝐼𝑉 → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘) ↔ ∀𝑘𝐼 𝐴 (𝐹𝑘)))
1614, 15syl 17 . . . . 5 ((𝜑𝑥𝑋) → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘) ↔ ∀𝑘𝐼 𝐴 (𝐹𝑘)))
1712, 16mpbird 257 . . . 4 ((𝜑𝑥𝑋) → (𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘))
18 ptcn.2 . . . . . . 7 𝐾 = (∏t𝐹)
1918ptuni 23618 . . . . . 6 ((𝐼𝑉𝐹:𝐼⟶Top) → X𝑘𝐼 (𝐹𝑘) = 𝐾)
2013, 3, 19syl2anc 584 . . . . 5 (𝜑X𝑘𝐼 (𝐹𝑘) = 𝐾)
2120adantr 480 . . . 4 ((𝜑𝑥𝑋) → X𝑘𝐼 (𝐹𝑘) = 𝐾)
2217, 21eleqtrd 2841 . . 3 ((𝜑𝑥𝑋) → (𝑘𝐼𝐴) ∈ 𝐾)
2322fmpttd 7135 . 2 (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)):𝑋 𝐾)
241adantr 480 . . . 4 ((𝜑𝑧𝑋) → 𝐽 ∈ (TopOn‘𝑋))
2513adantr 480 . . . 4 ((𝜑𝑧𝑋) → 𝐼𝑉)
263adantr 480 . . . 4 ((𝜑𝑧𝑋) → 𝐹:𝐼⟶Top)
27 simpr 484 . . . 4 ((𝜑𝑧𝑋) → 𝑧𝑋)
287adantlr 715 . . . . 5 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → (𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘)))
29 simplr 769 . . . . . 6 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → 𝑧𝑋)
30 toponuni 22936 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
311, 30syl 17 . . . . . . 7 (𝜑𝑋 = 𝐽)
3231ad2antrr 726 . . . . . 6 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → 𝑋 = 𝐽)
3329, 32eleqtrd 2841 . . . . 5 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → 𝑧 𝐽)
34 eqid 2735 . . . . . 6 𝐽 = 𝐽
3534cncnpi 23302 . . . . 5 (((𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘)) ∧ 𝑧 𝐽) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝑧))
3628, 33, 35syl2anc 584 . . . 4 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝑧))
3718, 24, 25, 26, 27, 36ptcnp 23646 . . 3 ((𝜑𝑧𝑋) → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))
3837ralrimiva 3144 . 2 (𝜑 → ∀𝑧𝑋 (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))
39 pttop 23606 . . . . . 6 ((𝐼𝑉𝐹:𝐼⟶Top) → (∏t𝐹) ∈ Top)
4013, 3, 39syl2anc 584 . . . . 5 (𝜑 → (∏t𝐹) ∈ Top)
4118, 40eqeltrid 2843 . . . 4 (𝜑𝐾 ∈ Top)
42 toptopon2 22940 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
4341, 42sylib 218 . . 3 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
44 cncnp 23304 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ 𝐾)) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ (𝐽 Cn 𝐾) ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)):𝑋 𝐾 ∧ ∀𝑧𝑋 (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))))
451, 43, 44syl2anc 584 . 2 (𝜑 → ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ (𝐽 Cn 𝐾) ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)):𝑋 𝐾 ∧ ∀𝑧𝑋 (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))))
4623, 38, 45mpbir2and 713 1 (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059   cuni 4912  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  Xcixp 8936  tcpt 17485  Topctop 22915  TopOnctopon 22932   Cn ccn 23248   CnP ccnp 23249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-topgen 17490  df-pt 17491  df-top 22916  df-topon 22933  df-bases 22969  df-cn 23251  df-cnp 23252
This theorem is referenced by:  pt1hmeo  23830  ptunhmeo  23832  symgtgp  24130  prdstmdd  24148  prdstgpd  24149  ptpconn  35218  broucube  37641
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