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Theorem ptcmpg 21771
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 21772). (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 2761 . . . 4 𝑘(𝐹𝑎)
3 nfcv 2761 . . . 4 𝑎(𝐹𝑘)
4 nfcv 2761 . . . 4 𝑘((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏)
5 nfcv 2761 . . . 4 𝑢((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏)
6 nfcv 2761 . . . 4 𝑎((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)
7 nfcv 2761 . . . 4 𝑏((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)
8 fveq2 6148 . . . 4 (𝑎 = 𝑘 → (𝐹𝑎) = (𝐹𝑘))
9 fveq2 6148 . . . . . . . 8 (𝑎 = 𝑘 → (𝑤𝑎) = (𝑤𝑘))
109mpteq2dv 4705 . . . . . . 7 (𝑎 = 𝑘 → (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) = (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)))
1110cnveqd 5258 . . . . . 6 (𝑎 = 𝑘(𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) = (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)))
1211imaeq1d 5424 . . . . 5 (𝑎 = 𝑘 → ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏) = ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑏))
13 imaeq2 5421 . . . . 5 (𝑏 = 𝑢 → ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑏) = ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢))
1412, 13sylan9eq 2675 . . . 4 ((𝑎 = 𝑘𝑏 = 𝑢) → ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏) = ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢))
152, 3, 4, 5, 6, 7, 8, 14cbvmpt2x 6686 . . 3 (𝑎𝐴, 𝑏 ∈ (𝐹𝑎) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏)) = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢))
16 fveq2 6148 . . . . 5 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
1716unieqd 4412 . . . 4 (𝑛 = 𝑚 (𝐹𝑛) = (𝐹𝑚))
1817cbvixpv 7870 . . 3 X𝑛𝐴 (𝐹𝑛) = X𝑚𝐴 (𝐹𝑚)
19 simp1 1059 . . 3 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐴𝑉)
20 simp2 1060 . . 3 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐹:𝐴⟶Comp)
21 cmptop 21108 . . . . . . . 8 (𝑘 ∈ Comp → 𝑘 ∈ Top)
2221ssriv 3587 . . . . . . 7 Comp ⊆ Top
23 fss 6013 . . . . . . 7 ((𝐹:𝐴⟶Comp ∧ Comp ⊆ Top) → 𝐹:𝐴⟶Top)
2420, 22, 23sylancl 693 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐹:𝐴⟶Top)
251ptuni 21307 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = 𝐽)
2619, 24, 25syl2anc 692 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛𝐴 (𝐹𝑛) = 𝐽)
27 ptcmpg.2 . . . . 5 𝑋 = 𝐽
2826, 27syl6eqr 2673 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛𝐴 (𝐹𝑛) = 𝑋)
29 simp3 1061 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝑋 ∈ (UFL ∩ dom card))
3028, 29eqeltrd 2698 . . 3 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛𝐴 (𝐹𝑛) ∈ (UFL ∩ dom card))
3115, 18, 19, 20, 30ptcmplem5 21770 . 2 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → (∏t𝐹) ∈ Comp)
321, 31syl5eqel 2702 1 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  cin 3554  wss 3555   cuni 4402  cmpt 4673  ccnv 5073  dom cdm 5074  cima 5077  wf 5843  cfv 5847  cmpt2 6606  Xcixp 7852  cardccrd 8705  tcpt 16020  Topctop 20617  Compccmp 21099  UFLcufl 21614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fi 8261  df-wdom 8408  df-card 8709  df-acn 8712  df-topgen 16025  df-pt 16026  df-fbas 19662  df-fg 19663  df-top 20621  df-bases 20622  df-topon 20623  df-cld 20733  df-ntr 20734  df-cls 20735  df-nei 20812  df-cmp 21100  df-fil 21560  df-ufil 21615  df-ufl 21616  df-flim 21653  df-fcls 21655
This theorem is referenced by:  ptcmp  21772  dfac21  37116
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