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Theorem ptcmpg 23445
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 23446). (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 2902 . . . 4 𝑘(𝐹𝑎)
3 nfcv 2902 . . . 4 𝑎(𝐹𝑘)
4 nfcv 2902 . . . 4 𝑘((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏)
5 nfcv 2902 . . . 4 𝑢((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏)
6 nfcv 2902 . . . 4 𝑎((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)
7 nfcv 2902 . . . 4 𝑏((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)
8 fveq2 6847 . . . 4 (𝑎 = 𝑘 → (𝐹𝑎) = (𝐹𝑘))
9 fveq2 6847 . . . . . . . 8 (𝑎 = 𝑘 → (𝑤𝑎) = (𝑤𝑘))
109mpteq2dv 5212 . . . . . . 7 (𝑎 = 𝑘 → (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) = (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)))
1110cnveqd 5836 . . . . . 6 (𝑎 = 𝑘(𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) = (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)))
1211imaeq1d 6017 . . . . 5 (𝑎 = 𝑘 → ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏) = ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑏))
13 imaeq2 6014 . . . . 5 (𝑏 = 𝑢 → ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑏) = ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢))
1412, 13sylan9eq 2791 . . . 4 ((𝑎 = 𝑘𝑏 = 𝑢) → ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏) = ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢))
152, 3, 4, 5, 6, 7, 8, 14cbvmpox 7455 . . 3 (𝑎𝐴, 𝑏 ∈ (𝐹𝑎) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑎)) “ 𝑏)) = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢))
16 fveq2 6847 . . . . 5 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
1716unieqd 4884 . . . 4 (𝑛 = 𝑚 (𝐹𝑛) = (𝐹𝑚))
1817cbvixpv 8860 . . 3 X𝑛𝐴 (𝐹𝑛) = X𝑚𝐴 (𝐹𝑚)
19 simp1 1136 . . 3 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐴𝑉)
20 simp2 1137 . . 3 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐹:𝐴⟶Comp)
21 cmptop 22783 . . . . . . . 8 (𝑘 ∈ Comp → 𝑘 ∈ Top)
2221ssriv 3951 . . . . . . 7 Comp ⊆ Top
23 fss 6690 . . . . . . 7 ((𝐹:𝐴⟶Comp ∧ Comp ⊆ Top) → 𝐹:𝐴⟶Top)
2420, 22, 23sylancl 586 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐹:𝐴⟶Top)
251ptuni 22982 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = 𝐽)
2619, 24, 25syl2anc 584 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛𝐴 (𝐹𝑛) = 𝐽)
27 ptcmpg.2 . . . . 5 𝑋 = 𝐽
2826, 27eqtr4di 2789 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛𝐴 (𝐹𝑛) = 𝑋)
29 simp3 1138 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝑋 ∈ (UFL ∩ dom card))
3028, 29eqeltrd 2832 . . 3 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛𝐴 (𝐹𝑛) ∈ (UFL ∩ dom card))
3115, 18, 19, 20, 30ptcmplem5 23444 . 2 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → (∏t𝐹) ∈ Comp)
321, 31eqeltrid 2836 1 ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  cin 3912  wss 3913   cuni 4870  cmpt 5193  ccnv 5637  dom cdm 5638  cima 5641  wf 6497  cfv 6501  cmpo 7364  Xcixp 8842  cardccrd 9880  tcpt 17334  Topctop 22279  Compccmp 22774  UFLcufl 23288
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-omul 8422  df-er 8655  df-map 8774  df-ixp 8843  df-en 8891  df-dom 8892  df-fin 8894  df-fi 9356  df-wdom 9510  df-card 9884  df-acn 9887  df-topgen 17339  df-pt 17340  df-fbas 20830  df-fg 20831  df-top 22280  df-topon 22297  df-bases 22333  df-cld 22407  df-ntr 22408  df-cls 22409  df-nei 22486  df-cmp 22775  df-fil 23234  df-ufil 23289  df-ufl 23290  df-flim 23327  df-fcls 23329
This theorem is referenced by:  ptcmp  23446  dfac21  41451
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