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Theorem ptcmpg 22954
Description: Tychonoff's theorem: The product of compact spaces is compact. The choice principles needed are encoded in the last hypothesis: the base set of the product must be well-orderable and satisfy the ultrafilter lemma. Both these assumptions are satisfied if 𝒫 𝒫 𝑋 is well-orderable, so if we assume the Axiom of Choice we can eliminate them (see ptcmp 22955). (Contributed by Mario Carneiro, 27-Aug-2015.)
Hypotheses
Ref Expression
ptcmpg.1 𝐽 = (∏t𝐹)
ptcmpg.2 𝑋 = 𝐽
Assertion
Ref Expression
ptcmpg ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐽 ∈ Comp)

Proof of Theorem ptcmpg
Dummy variables 𝑎 𝑏 𝑘 𝑚 𝑛 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptcmpg.1 . 2 𝐽 = (∏t𝐹)
2 nfcv 2904 . . . 4 𝑘(𝐹𝑎)
3 nfcv 2904 . . . 4 𝑎(𝐹𝑘)
4 nfcv 2904 . . . 4 𝑘((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏)
5 nfcv 2904 . . . 4 𝑢((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏)
6 nfcv 2904 . . . 4 𝑎((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)
7 nfcv 2904 . . . 4 𝑏((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)
8 fveq2 6717 . . . 4 (𝑎 = 𝑘 → (𝐹𝑎) = (𝐹𝑘))
9 fveq2 6717 . . . . . . . 8 (𝑎 = 𝑘 → (𝑤𝑎) = (𝑤𝑘))
109mpteq2dv 5151 . . . . . . 7 (𝑎 = 𝑘 → (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) = (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)))
1110cnveqd 5744 . . . . . 6 (𝑎 = 𝑘(𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) = (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)))
1211imaeq1d 5928 . . . . 5 (𝑎 = 𝑘 → ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏) = ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑏))
13 imaeq2 5925 . . . . 5 (𝑏 = 𝑢 → ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑏) = ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢))
1412, 13sylan9eq 2798 . . . 4 ((𝑎 = 𝑘𝑏 = 𝑢) → ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏) = ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢))
152, 3, 4, 5, 6, 7, 8, 14cbvmpox 7304 . . 3 (𝑎𝐴, 𝑏 ∈ (𝐹𝑎) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏)) = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢))
16 fveq2 6717 . . . . 5 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
1716unieqd 4833 . . . 4 (𝑛 = 𝑚 (𝐹𝑛) = (𝐹𝑚))
1817cbvixpv 8596 . . 3 X𝑛𝐴 (𝐹𝑛) = X𝑚𝐴 (𝐹𝑚)
19 simp1 1138 . . 3 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐴𝑉)
20 simp2 1139 . . 3 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐹:𝐴⟶Comp)
21 cmptop 22292 . . . . . . . 8 (𝑘 ∈ Comp → 𝑘 ∈ Top)
2221ssriv 3905 . . . . . . 7 Comp ⊆ Top
23 fss 6562 . . . . . . 7 ((𝐹:𝐴⟶Comp ∧ Comp ⊆ Top) → 𝐹:𝐴⟶Top)
2420, 22, 23sylancl 589 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐹:𝐴⟶Top)
251ptuni 22491 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = 𝐽)
2619, 24, 25syl2anc 587 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛𝐴 (𝐹𝑛) = 𝐽)
27 ptcmpg.2 . . . . 5 𝑋 = 𝐽
2826, 27eqtr4di 2796 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛𝐴 (𝐹𝑛) = 𝑋)
29 simp3 1140 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝑋 ∈ (UFL ∩ dom card))
3028, 29eqeltrd 2838 . . 3 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛𝐴 (𝐹𝑛) ∈ (UFL ∩ dom card))
3115, 18, 19, 20, 30ptcmplem5 22953 . 2 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → (∏t𝐹) ∈ Comp)
321, 31eqeltrid 2842 1 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1089   = wceq 1543  wcel 2110  cin 3865  wss 3866   cuni 4819  cmpt 5135  ccnv 5550  dom cdm 5551  cima 5554  wf 6376  cfv 6380  cmpo 7215  Xcixp 8578  cardccrd 9551  tcpt 16943  Topctop 21790  Compccmp 22283  UFLcufl 22797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-omul 8207  df-er 8391  df-map 8510  df-ixp 8579  df-en 8627  df-dom 8628  df-fin 8630  df-fi 9027  df-wdom 9181  df-card 9555  df-acn 9558  df-topgen 16948  df-pt 16949  df-fbas 20360  df-fg 20361  df-top 21791  df-topon 21808  df-bases 21843  df-cld 21916  df-ntr 21917  df-cls 21918  df-nei 21995  df-cmp 22284  df-fil 22743  df-ufil 22798  df-ufl 22799  df-flim 22836  df-fcls 22838
This theorem is referenced by:  ptcmp  22955  dfac21  40594
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