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Theorem ptcmpg 22672
 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 22673). (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 2955 . . . 4 𝑘(𝐹𝑎)
3 nfcv 2955 . . . 4 𝑎(𝐹𝑘)
4 nfcv 2955 . . . 4 𝑘((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏)
5 nfcv 2955 . . . 4 𝑢((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏)
6 nfcv 2955 . . . 4 𝑎((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)
7 nfcv 2955 . . . 4 𝑏((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)
8 fveq2 6646 . . . 4 (𝑎 = 𝑘 → (𝐹𝑎) = (𝐹𝑘))
9 fveq2 6646 . . . . . . . 8 (𝑎 = 𝑘 → (𝑤𝑎) = (𝑤𝑘))
109mpteq2dv 5127 . . . . . . 7 (𝑎 = 𝑘 → (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) = (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)))
1110cnveqd 5711 . . . . . 6 (𝑎 = 𝑘(𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) = (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)))
1211imaeq1d 5896 . . . . 5 (𝑎 = 𝑘 → ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏) = ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑏))
13 imaeq2 5893 . . . . 5 (𝑏 = 𝑢 → ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑏) = ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢))
1412, 13sylan9eq 2853 . . . 4 ((𝑎 = 𝑘𝑏 = 𝑢) → ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏) = ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢))
152, 3, 4, 5, 6, 7, 8, 14cbvmpox 7227 . . 3 (𝑎𝐴, 𝑏 ∈ (𝐹𝑎) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏)) = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢))
16 fveq2 6646 . . . . 5 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
1716unieqd 4815 . . . 4 (𝑛 = 𝑚 (𝐹𝑛) = (𝐹𝑚))
1817cbvixpv 8465 . . 3 X𝑛𝐴 (𝐹𝑛) = X𝑚𝐴 (𝐹𝑚)
19 simp1 1133 . . 3 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐴𝑉)
20 simp2 1134 . . 3 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐹:𝐴⟶Comp)
21 cmptop 22010 . . . . . . . 8 (𝑘 ∈ Comp → 𝑘 ∈ Top)
2221ssriv 3919 . . . . . . 7 Comp ⊆ Top
23 fss 6502 . . . . . . 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 22209 . . . . . 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 2851 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛𝐴 (𝐹𝑛) = 𝑋)
29 simp3 1135 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝑋 ∈ (UFL ∩ dom card))
3028, 29eqeltrd 2890 . . 3 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛𝐴 (𝐹𝑛) ∈ (UFL ∩ dom card))
3115, 18, 19, 20, 30ptcmplem5 22671 . 2 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → (∏t𝐹) ∈ Comp)
321, 31eqeltrid 2894 1 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐽 ∈ Comp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ∩ cin 3880   ⊆ wss 3881  ∪ cuni 4801   ↦ cmpt 5111  ◡ccnv 5519  dom cdm 5520   “ cima 5523  ⟶wf 6321  ‘cfv 6325   ∈ cmpo 7138  Xcixp 8447  cardccrd 9351  ∏tcpt 16707  Topctop 21508  Compccmp 22001  UFLcufl 22515 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-iin 4885  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-se 5480  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-isom 6334  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-2o 8089  df-oadd 8092  df-omul 8093  df-er 8275  df-map 8394  df-ixp 8448  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-fi 8862  df-wdom 9016  df-card 9355  df-acn 9358  df-topgen 16712  df-pt 16713  df-fbas 20092  df-fg 20093  df-top 21509  df-topon 21526  df-bases 21561  df-cld 21634  df-ntr 21635  df-cls 21636  df-nei 21713  df-cmp 22002  df-fil 22461  df-ufil 22516  df-ufl 22517  df-flim 22554  df-fcls 22556 This theorem is referenced by:  ptcmp  22673  dfac21  40053
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