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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 21533 | . . . . . . 7 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝑅) | |
6 | 3, 5 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝑅) |
7 | 6, 4 | ssexd 5227 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ∈ V) |
8 | 7 | ralrimiva 3182 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑆 ∈ V) |
9 | iunexg 7663 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝑆 ∈ V) → ∪ 𝑘 ∈ 𝐴 𝑆 ∈ V) | |
10 | 2, 8, 9 | syl2anc 586 | . . 3 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝑆 ∈ V) |
11 | axac3 9885 | . . . 4 ⊢ CHOICE | |
12 | acacni 9565 | . . . 4 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → AC 𝐴 = V) | |
13 | 11, 2, 12 | sylancr 589 | . . 3 ⊢ (𝜑 → AC 𝐴 = V) |
14 | 10, 13 | eleqtrrd 2916 | . 2 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝑆 ∈ AC 𝐴) |
15 | 1, 2, 3, 4, 14 | ptclsg 22222 | 1 ⊢ (𝜑 → ((cls‘𝐽)‘X𝑘 ∈ 𝐴 𝑆) = X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ⊆ wss 3935 ∪ ciun 4918 ↦ cmpt 5145 ‘cfv 6354 Xcixp 8460 AC wacn 9366 CHOICEwac 9540 ∏tcpt 16711 TopOnctopon 21517 clsccl 21625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-ac2 9884 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-ixp 8461 df-en 8509 df-dom 8510 df-fin 8512 df-fi 8874 df-card 9367 df-acn 9370 df-ac 9541 df-topgen 16716 df-pt 16717 df-top 21501 df-topon 21518 df-bases 21553 df-cld 21626 df-ntr 21627 df-cls 21628 |
This theorem is referenced by: (None) |
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