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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 6113 | . . . . 5 ⊢ (∏t‘𝐹) ∈ V | |
2 | 1 | uniex 6851 | . . . 4 ⊢ ∪ (∏t‘𝐹) ∈ V |
3 | axac3 9169 | . . . . 5 ⊢ CHOICE | |
4 | acufl 21531 | . . . . 5 ⊢ (CHOICE → UFL = V) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ UFL = V |
6 | 2, 5 | eleqtrri 2687 | . . 3 ⊢ ∪ (∏t‘𝐹) ∈ UFL |
7 | cardeqv 9174 | . . . 4 ⊢ dom card = V | |
8 | 2, 7 | eleqtrri 2687 | . . 3 ⊢ ∪ (∏t‘𝐹) ∈ dom card |
9 | elin 3758 | . . 3 ⊢ (∪ (∏t‘𝐹) ∈ (UFL ∩ dom card) ↔ (∪ (∏t‘𝐹) ∈ UFL ∧ ∪ (∏t‘𝐹) ∈ dom card)) | |
10 | 6, 8, 9 | mpbir2an 957 | . 2 ⊢ ∪ (∏t‘𝐹) ∈ (UFL ∩ dom card) |
11 | eqid 2610 | . . 3 ⊢ (∏t‘𝐹) = (∏t‘𝐹) | |
12 | eqid 2610 | . . 3 ⊢ ∪ (∏t‘𝐹) = ∪ (∏t‘𝐹) | |
13 | 11, 12 | ptcmpg 21671 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ ∪ (∏t‘𝐹) ∈ (UFL ∩ dom card)) → (∏t‘𝐹) ∈ Comp) |
14 | 10, 13 | mp3an3 1405 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp) → (∏t‘𝐹) ∈ Comp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ∪ cuni 4372 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 cardccrd 8644 CHOICEwac 8821 ∏tcpt 15922 Compccmp 20999 UFLcufl 21514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-ac2 9168 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-rpss 6835 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-wdom 8347 df-card 8648 df-acn 8651 df-ac 8822 df-cda 8873 df-topgen 15927 df-pt 15928 df-fbas 19564 df-fg 19565 df-top 20521 df-bases 20522 df-topon 20523 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-cmp 21000 df-fil 21460 df-ufil 21515 df-ufl 21516 df-flim 21553 df-fcls 21555 |
This theorem is referenced by: (None) |
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