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Mirrors > Home > MPE Home > Th. List > ptcmp | Structured version Visualization version GIF version |
Description: Tychonoff's theorem: The product of compact spaces is compact. The proof uses the Axiom of Choice. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
ptcmp | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp) → (∏t‘𝐹) ∈ Comp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6658 | . . . . 5 ⊢ (∏t‘𝐹) ∈ V | |
2 | 1 | uniex 7447 | . . . 4 ⊢ ∪ (∏t‘𝐹) ∈ V |
3 | axac3 9875 | . . . . 5 ⊢ CHOICE | |
4 | acufl 22522 | . . . . 5 ⊢ (CHOICE → UFL = V) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ UFL = V |
6 | 2, 5 | eleqtrri 2889 | . . 3 ⊢ ∪ (∏t‘𝐹) ∈ UFL |
7 | cardeqv 9880 | . . . 4 ⊢ dom card = V | |
8 | 2, 7 | eleqtrri 2889 | . . 3 ⊢ ∪ (∏t‘𝐹) ∈ dom card |
9 | 6, 8 | elini 4120 | . 2 ⊢ ∪ (∏t‘𝐹) ∈ (UFL ∩ dom card) |
10 | eqid 2798 | . . 3 ⊢ (∏t‘𝐹) = (∏t‘𝐹) | |
11 | eqid 2798 | . . 3 ⊢ ∪ (∏t‘𝐹) = ∪ (∏t‘𝐹) | |
12 | 10, 11 | ptcmpg 22662 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ ∪ (∏t‘𝐹) ∈ (UFL ∩ dom card)) → (∏t‘𝐹) ∈ Comp) |
13 | 9, 12 | mp3an3 1447 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp) → (∏t‘𝐹) ∈ Comp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 ∪ cuni 4800 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 cardccrd 9348 CHOICEwac 9526 ∏tcpt 16704 Compccmp 21991 UFLcufl 22505 |
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 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-ac2 9874 |
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 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-rpss 7429 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fi 8859 df-wdom 9013 df-dju 9314 df-card 9352 df-acn 9355 df-ac 9527 df-topgen 16709 df-pt 16710 df-fbas 20088 df-fg 20089 df-top 21499 df-topon 21516 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-cmp 21992 df-fil 22451 df-ufil 22506 df-ufl 22507 df-flim 22544 df-fcls 22546 |
This theorem is referenced by: (None) |
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