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Mirrors > Home > MPE Home > Th. List > prdstps | Structured version Visualization version GIF version |
Description: A structure product of topological spaces is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.) |
Ref | Expression |
---|---|
prdstopn.y | ⊢ 𝑌 = (𝑆Xs𝑅) |
prdstopn.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
prdstopn.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
prdstps.r | ⊢ (𝜑 → 𝑅:𝐼⟶TopSp) |
Ref | Expression |
---|---|
prdstps | ⊢ (𝜑 → 𝑌 ∈ TopSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdstopn.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
2 | prdstps.r | . . . . . . 7 ⊢ (𝜑 → 𝑅:𝐼⟶TopSp) | |
3 | 2 | ffvelrnda 6828 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ TopSp) |
4 | eqid 2798 | . . . . . . 7 ⊢ (Base‘(𝑅‘𝑥)) = (Base‘(𝑅‘𝑥)) | |
5 | eqid 2798 | . . . . . . 7 ⊢ (TopOpen‘(𝑅‘𝑥)) = (TopOpen‘(𝑅‘𝑥)) | |
6 | 4, 5 | istps 21539 | . . . . . 6 ⊢ ((𝑅‘𝑥) ∈ TopSp ↔ (TopOpen‘(𝑅‘𝑥)) ∈ (TopOn‘(Base‘(𝑅‘𝑥)))) |
7 | 3, 6 | sylib 221 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (TopOpen‘(𝑅‘𝑥)) ∈ (TopOn‘(Base‘(𝑅‘𝑥)))) |
8 | 7 | ralrimiva 3149 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (TopOpen‘(𝑅‘𝑥)) ∈ (TopOn‘(Base‘(𝑅‘𝑥)))) |
9 | eqid 2798 | . . . . 5 ⊢ (∏t‘(𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥)))) = (∏t‘(𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥)))) | |
10 | 9 | pttopon 22201 | . . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ ∀𝑥 ∈ 𝐼 (TopOpen‘(𝑅‘𝑥)) ∈ (TopOn‘(Base‘(𝑅‘𝑥)))) → (∏t‘(𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥)))) ∈ (TopOn‘X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))) |
11 | 1, 8, 10 | syl2anc 587 | . . 3 ⊢ (𝜑 → (∏t‘(𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥)))) ∈ (TopOn‘X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))) |
12 | prdstopn.y | . . . . 5 ⊢ 𝑌 = (𝑆Xs𝑅) | |
13 | prdstopn.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
14 | fex 6966 | . . . . . 6 ⊢ ((𝑅:𝐼⟶TopSp ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ V) | |
15 | 2, 1, 14 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ V) |
16 | eqid 2798 | . . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
17 | 2 | fdmd 6497 | . . . . 5 ⊢ (𝜑 → dom 𝑅 = 𝐼) |
18 | eqid 2798 | . . . . 5 ⊢ (TopSet‘𝑌) = (TopSet‘𝑌) | |
19 | 12, 13, 15, 16, 17, 18 | prdstset 16731 | . . . 4 ⊢ (𝜑 → (TopSet‘𝑌) = (∏t‘(TopOpen ∘ 𝑅))) |
20 | topnfn 16691 | . . . . . . 7 ⊢ TopOpen Fn V | |
21 | dffn2 6489 | . . . . . . 7 ⊢ (TopOpen Fn V ↔ TopOpen:V⟶V) | |
22 | 20, 21 | mpbi 233 | . . . . . 6 ⊢ TopOpen:V⟶V |
23 | ssv 3939 | . . . . . . 7 ⊢ TopSp ⊆ V | |
24 | fss 6501 | . . . . . . 7 ⊢ ((𝑅:𝐼⟶TopSp ∧ TopSp ⊆ V) → 𝑅:𝐼⟶V) | |
25 | 2, 23, 24 | sylancl 589 | . . . . . 6 ⊢ (𝜑 → 𝑅:𝐼⟶V) |
26 | fcompt 6872 | . . . . . 6 ⊢ ((TopOpen:V⟶V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) = (𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥)))) | |
27 | 22, 25, 26 | sylancr 590 | . . . . 5 ⊢ (𝜑 → (TopOpen ∘ 𝑅) = (𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥)))) |
28 | 27 | fveq2d 6649 | . . . 4 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥))))) |
29 | 19, 28 | eqtrd 2833 | . . 3 ⊢ (𝜑 → (TopSet‘𝑌) = (∏t‘(𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥))))) |
30 | 12, 13, 15, 16, 17 | prdsbas 16722 | . . . 4 ⊢ (𝜑 → (Base‘𝑌) = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) |
31 | 30 | fveq2d 6649 | . . 3 ⊢ (𝜑 → (TopOn‘(Base‘𝑌)) = (TopOn‘X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))) |
32 | 11, 29, 31 | 3eltr4d 2905 | . 2 ⊢ (𝜑 → (TopSet‘𝑌) ∈ (TopOn‘(Base‘𝑌))) |
33 | 16, 18 | tsettps 21546 | . 2 ⊢ ((TopSet‘𝑌) ∈ (TopOn‘(Base‘𝑌)) → 𝑌 ∈ TopSp) |
34 | 32, 33 | syl 17 | 1 ⊢ (𝜑 → 𝑌 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ⊆ wss 3881 ↦ cmpt 5110 ∘ ccom 5523 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Xcixp 8444 Basecbs 16475 TopSetcts 16563 TopOpenctopn 16687 ∏tcpt 16704 Xscprds 16711 TopOnctopon 21515 TopSpctps 21537 |
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-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
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-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-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-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-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-sdom 8495 df-fin 8496 df-fi 8859 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-topgen 16709 df-pt 16710 df-prds 16713 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 |
This theorem is referenced by: pwstps 22235 xpstps 22415 prdstmdd 22729 |
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