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Mirrors > Home > MPE Home > Th. List > ptcls | Structured version Visualization version GIF version |
Description: The closure of a box in the product topology is the box formed from the closures of the factors. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
ptcls.2 | ⊢ 𝐽 = (∏t‘(𝑘 ∈ 𝐴 ↦ 𝑅)) |
ptcls.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ptcls.j | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑅 ∈ (TopOn‘𝑋)) |
ptcls.c | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ⊆ 𝑋) |
Ref | Expression |
---|---|
ptcls | ⊢ (𝜑 → ((cls‘𝐽)‘X𝑘 ∈ 𝐴 𝑆) = X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ptcls.2 | . 2 ⊢ 𝐽 = (∏t‘(𝑘 ∈ 𝐴 ↦ 𝑅)) | |
2 | ptcls.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | ptcls.j | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑅 ∈ (TopOn‘𝑋)) | |
4 | ptcls.c | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ⊆ 𝑋) | |
5 | toponmax 21531 | . . . . . . 7 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝑅) | |
6 | 3, 5 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝑅) |
7 | 6, 4 | ssexd 5192 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ∈ V) |
8 | 7 | ralrimiva 3149 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑆 ∈ V) |
9 | iunexg 7646 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝑆 ∈ V) → ∪ 𝑘 ∈ 𝐴 𝑆 ∈ V) | |
10 | 2, 8, 9 | syl2anc 587 | . . 3 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝑆 ∈ V) |
11 | axac3 9875 | . . . 4 ⊢ CHOICE | |
12 | acacni 9551 | . . . 4 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → AC 𝐴 = V) | |
13 | 11, 2, 12 | sylancr 590 | . . 3 ⊢ (𝜑 → AC 𝐴 = V) |
14 | 10, 13 | eleqtrrd 2893 | . 2 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝑆 ∈ AC 𝐴) |
15 | 1, 2, 3, 4, 14 | ptclsg 22220 | 1 ⊢ (𝜑 → ((cls‘𝐽)‘X𝑘 ∈ 𝐴 𝑆) = X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ⊆ wss 3881 ∪ ciun 4881 ↦ cmpt 5110 ‘cfv 6324 Xcixp 8444 AC wacn 9351 CHOICEwac 9526 ∏tcpt 16704 TopOnctopon 21515 clsccl 21623 |
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-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-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-fin 8496 df-fi 8859 df-card 9352 df-acn 9355 df-ac 9527 df-topgen 16709 df-pt 16710 df-top 21499 df-topon 21516 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 |
This theorem is referenced by: (None) |
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