![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ptcmpg | Structured version Visualization version GIF version |
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 23393). (Contributed by Mario Carneiro, 27-Aug-2015.) |
Ref | Expression |
---|---|
ptcmpg.1 | ⊢ 𝐽 = (∏t‘𝐹) |
ptcmpg.2 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
ptcmpg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐽 ∈ Comp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ptcmpg.1 | . 2 ⊢ 𝐽 = (∏t‘𝐹) | |
2 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑎) | |
3 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑎(𝐹‘𝑘) | |
4 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑘(◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) “ 𝑏) | |
5 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑢(◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) “ 𝑏) | |
6 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑎(◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) | |
7 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑏(◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) | |
8 | fveq2 6839 | . . . 4 ⊢ (𝑎 = 𝑘 → (𝐹‘𝑎) = (𝐹‘𝑘)) | |
9 | fveq2 6839 | . . . . . . . 8 ⊢ (𝑎 = 𝑘 → (𝑤‘𝑎) = (𝑤‘𝑘)) | |
10 | 9 | mpteq2dv 5205 | . . . . . . 7 ⊢ (𝑎 = 𝑘 → (𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) = (𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘))) |
11 | 10 | cnveqd 5829 | . . . . . 6 ⊢ (𝑎 = 𝑘 → ◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) = ◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘))) |
12 | 11 | imaeq1d 6010 | . . . . 5 ⊢ (𝑎 = 𝑘 → (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) “ 𝑏) = (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑏)) |
13 | imaeq2 6007 | . . . . 5 ⊢ (𝑏 = 𝑢 → (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑏) = (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)) | |
14 | 12, 13 | sylan9eq 2796 | . . . 4 ⊢ ((𝑎 = 𝑘 ∧ 𝑏 = 𝑢) → (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) “ 𝑏) = (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)) |
15 | 2, 3, 4, 5, 6, 7, 8, 14 | cbvmpox 7446 | . . 3 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ (𝐹‘𝑎) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) “ 𝑏)) = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)) |
16 | fveq2 6839 | . . . . 5 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚)) | |
17 | 16 | unieqd 4877 | . . . 4 ⊢ (𝑛 = 𝑚 → ∪ (𝐹‘𝑛) = ∪ (𝐹‘𝑚)) |
18 | 17 | cbvixpv 8849 | . . 3 ⊢ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = X𝑚 ∈ 𝐴 ∪ (𝐹‘𝑚) |
19 | simp1 1136 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐴 ∈ 𝑉) | |
20 | simp2 1137 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐹:𝐴⟶Comp) | |
21 | cmptop 22730 | . . . . . . . 8 ⊢ (𝑘 ∈ Comp → 𝑘 ∈ Top) | |
22 | 21 | ssriv 3946 | . . . . . . 7 ⊢ Comp ⊆ Top |
23 | fss 6682 | . . . . . . 7 ⊢ ((𝐹:𝐴⟶Comp ∧ Comp ⊆ Top) → 𝐹:𝐴⟶Top) | |
24 | 20, 22, 23 | sylancl 586 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐹:𝐴⟶Top) |
25 | 1 | ptuni 22929 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ 𝐽) |
26 | 19, 24, 25 | syl2anc 584 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ 𝐽) |
27 | ptcmpg.2 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
28 | 26, 27 | eqtr4di 2794 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = 𝑋) |
29 | simp3 1138 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝑋 ∈ (UFL ∩ dom card)) | |
30 | 28, 29 | eqeltrd 2838 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ (UFL ∩ dom card)) |
31 | 15, 18, 19, 20, 30 | ptcmplem5 23391 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → (∏t‘𝐹) ∈ Comp) |
32 | 1, 31 | eqeltrid 2842 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹: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 3907 ⊆ wss 3908 ∪ cuni 4863 ↦ cmpt 5186 ◡ccnv 5630 dom cdm 5631 “ cima 5634 ⟶wf 6489 ‘cfv 6493 ∈ cmpo 7355 Xcixp 8831 cardccrd 9867 ∏tcpt 17312 Topctop 22226 Compccmp 22721 UFLcufl 23235 |
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 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-oadd 8412 df-omul 8413 df-er 8644 df-map 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-fin 8883 df-fi 9343 df-wdom 9497 df-card 9871 df-acn 9874 df-topgen 17317 df-pt 17318 df-fbas 20778 df-fg 20779 df-top 22227 df-topon 22244 df-bases 22280 df-cld 22354 df-ntr 22355 df-cls 22356 df-nei 22433 df-cmp 22722 df-fil 23181 df-ufil 23236 df-ufl 23237 df-flim 23274 df-fcls 23276 |
This theorem is referenced by: ptcmp 23393 dfac21 41331 |
Copyright terms: Public domain | W3C validator |