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Theorem ptcn 22211
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 483 . . . . . . . . 9 ((𝜑𝑘𝐼) → 𝐽 ∈ (TopOn‘𝑋))
3 ptcn.5 . . . . . . . . . . 11 (𝜑𝐹:𝐼⟶Top)
43ffvelrnda 6827 . . . . . . . . . 10 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ Top)
5 toptopon2 21502 . . . . . . . . . 10 ((𝐹𝑘) ∈ Top ↔ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
64, 5sylib 220 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
7 ptcn.6 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘)))
8 cnf2 21833 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘))) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
92, 6, 7, 8syl3anc 1367 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
109fvmptelrn 6853 . . . . . . 7 (((𝜑𝑘𝐼) ∧ 𝑥𝑋) → 𝐴 (𝐹𝑘))
1110an32s 650 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑘𝐼) → 𝐴 (𝐹𝑘))
1211ralrimiva 3169 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑘𝐼 𝐴 (𝐹𝑘))
13 ptcn.4 . . . . . . 7 (𝜑𝐼𝑉)
1413adantr 483 . . . . . 6 ((𝜑𝑥𝑋) → 𝐼𝑉)
15 mptelixpg 8477 . . . . . 6 (𝐼𝑉 → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘) ↔ ∀𝑘𝐼 𝐴 (𝐹𝑘)))
1614, 15syl 17 . . . . 5 ((𝜑𝑥𝑋) → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘) ↔ ∀𝑘𝐼 𝐴 (𝐹𝑘)))
1712, 16mpbird 259 . . . 4 ((𝜑𝑥𝑋) → (𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐹𝑘))
18 ptcn.2 . . . . . . 7 𝐾 = (∏t𝐹)
1918ptuni 22178 . . . . . 6 ((𝐼𝑉𝐹:𝐼⟶Top) → X𝑘𝐼 (𝐹𝑘) = 𝐾)
2013, 3, 19syl2anc 586 . . . . 5 (𝜑X𝑘𝐼 (𝐹𝑘) = 𝐾)
2120adantr 483 . . . 4 ((𝜑𝑥𝑋) → X𝑘𝐼 (𝐹𝑘) = 𝐾)
2217, 21eleqtrd 2913 . . 3 ((𝜑𝑥𝑋) → (𝑘𝐼𝐴) ∈ 𝐾)
2322fmpttd 6855 . 2 (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)):𝑋 𝐾)
241adantr 483 . . . 4 ((𝜑𝑧𝑋) → 𝐽 ∈ (TopOn‘𝑋))
2513adantr 483 . . . 4 ((𝜑𝑧𝑋) → 𝐼𝑉)
263adantr 483 . . . 4 ((𝜑𝑧𝑋) → 𝐹:𝐼⟶Top)
27 simpr 487 . . . 4 ((𝜑𝑧𝑋) → 𝑧𝑋)
287adantlr 713 . . . . 5 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → (𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘)))
29 simplr 767 . . . . . 6 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → 𝑧𝑋)
30 toponuni 21498 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
311, 30syl 17 . . . . . . 7 (𝜑𝑋 = 𝐽)
3231ad2antrr 724 . . . . . 6 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → 𝑋 = 𝐽)
3329, 32eleqtrd 2913 . . . . 5 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → 𝑧 𝐽)
34 eqid 2820 . . . . . 6 𝐽 = 𝐽
3534cncnpi 21862 . . . . 5 (((𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘)) ∧ 𝑧 𝐽) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝑧))
3628, 33, 35syl2anc 586 . . . 4 (((𝜑𝑧𝑋) ∧ 𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝑧))
3718, 24, 25, 26, 27, 36ptcnp 22206 . . 3 ((𝜑𝑧𝑋) → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))
3837ralrimiva 3169 . 2 (𝜑 → ∀𝑧𝑋 (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))
39 pttop 22166 . . . . . 6 ((𝐼𝑉𝐹:𝐼⟶Top) → (∏t𝐹) ∈ Top)
4013, 3, 39syl2anc 586 . . . . 5 (𝜑 → (∏t𝐹) ∈ Top)
4118, 40eqeltrid 2915 . . . 4 (𝜑𝐾 ∈ Top)
42 toptopon2 21502 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
4341, 42sylib 220 . . 3 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
44 cncnp 21864 . . 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 586 . 2 (𝜑 → ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ (𝐽 Cn 𝐾) ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)):𝑋 𝐾 ∧ ∀𝑧𝑋 (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧))))
4623, 38, 45mpbir2and 711 1 (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wcel 2114  wral 3125   cuni 4814  cmpt 5122  wf 6327  cfv 6331  (class class class)co 7133  Xcixp 8439  tcpt 16691  Topctop 21477  TopOnctopon 21494   Cn ccn 21808   CnP ccnp 21809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-iin 4898  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-oadd 8084  df-er 8267  df-map 8386  df-ixp 8440  df-en 8488  df-dom 8489  df-fin 8491  df-fi 8853  df-topgen 16696  df-pt 16697  df-top 21478  df-topon 21495  df-bases 21530  df-cn 21811  df-cnp 21812
This theorem is referenced by:  pt1hmeo  22390  ptunhmeo  22392  symgtgp  22690  prdstmdd  22708  prdstgpd  22709  ptpconn  32488  broucube  34967
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