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Mirrors > Home > MPE Home > Th. List > prdstps | Structured version Visualization version GIF version |
Description: A structure product of topologies 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 6267 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ TopSp) |
4 | eqid 2610 | . . . . . . 7 ⊢ (Base‘(𝑅‘𝑥)) = (Base‘(𝑅‘𝑥)) | |
5 | eqid 2610 | . . . . . . 7 ⊢ (TopOpen‘(𝑅‘𝑥)) = (TopOpen‘(𝑅‘𝑥)) | |
6 | 4, 5 | istps 20551 | . . . . . 6 ⊢ ((𝑅‘𝑥) ∈ TopSp ↔ (TopOpen‘(𝑅‘𝑥)) ∈ (TopOn‘(Base‘(𝑅‘𝑥)))) |
7 | 3, 6 | sylib 207 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (TopOpen‘(𝑅‘𝑥)) ∈ (TopOn‘(Base‘(𝑅‘𝑥)))) |
8 | 7 | ralrimiva 2949 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (TopOpen‘(𝑅‘𝑥)) ∈ (TopOn‘(Base‘(𝑅‘𝑥)))) |
9 | eqid 2610 | . . . . 5 ⊢ (∏t‘(𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥)))) = (∏t‘(𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥)))) | |
10 | 9 | pttopon 21209 | . . . 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 691 | . . 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 6394 | . . . . . 6 ⊢ ((𝑅:𝐼⟶TopSp ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ V) | |
15 | 2, 1, 14 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ V) |
16 | eqid 2610 | . . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
17 | fdm 5964 | . . . . . 6 ⊢ (𝑅:𝐼⟶TopSp → dom 𝑅 = 𝐼) | |
18 | 2, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝑅 = 𝐼) |
19 | eqid 2610 | . . . . 5 ⊢ (TopSet‘𝑌) = (TopSet‘𝑌) | |
20 | 12, 13, 15, 16, 18, 19 | prdstset 15949 | . . . 4 ⊢ (𝜑 → (TopSet‘𝑌) = (∏t‘(TopOpen ∘ 𝑅))) |
21 | topnfn 15909 | . . . . . . 7 ⊢ TopOpen Fn V | |
22 | dffn2 5960 | . . . . . . 7 ⊢ (TopOpen Fn V ↔ TopOpen:V⟶V) | |
23 | 21, 22 | mpbi 219 | . . . . . 6 ⊢ TopOpen:V⟶V |
24 | ssv 3588 | . . . . . . 7 ⊢ TopSp ⊆ V | |
25 | fss 5969 | . . . . . . 7 ⊢ ((𝑅:𝐼⟶TopSp ∧ TopSp ⊆ V) → 𝑅:𝐼⟶V) | |
26 | 2, 24, 25 | sylancl 693 | . . . . . 6 ⊢ (𝜑 → 𝑅:𝐼⟶V) |
27 | fcompt 6306 | . . . . . 6 ⊢ ((TopOpen:V⟶V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) = (𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥)))) | |
28 | 23, 26, 27 | sylancr 694 | . . . . 5 ⊢ (𝜑 → (TopOpen ∘ 𝑅) = (𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥)))) |
29 | 28 | fveq2d 6107 | . . . 4 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥))))) |
30 | 20, 29 | eqtrd 2644 | . . 3 ⊢ (𝜑 → (TopSet‘𝑌) = (∏t‘(𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥))))) |
31 | 12, 13, 15, 16, 18 | prdsbas 15940 | . . . 4 ⊢ (𝜑 → (Base‘𝑌) = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) |
32 | 31 | fveq2d 6107 | . . 3 ⊢ (𝜑 → (TopOn‘(Base‘𝑌)) = (TopOn‘X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))) |
33 | 11, 30, 32 | 3eltr4d 2703 | . 2 ⊢ (𝜑 → (TopSet‘𝑌) ∈ (TopOn‘(Base‘𝑌))) |
34 | 16, 19 | tsettps 20558 | . 2 ⊢ ((TopSet‘𝑌) ∈ (TopOn‘(Base‘𝑌)) → 𝑌 ∈ TopSp) |
35 | 33, 34 | syl 17 | 1 ⊢ (𝜑 → 𝑌 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 ↦ cmpt 4643 dom cdm 5038 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Xcixp 7794 Basecbs 15695 TopSetcts 15774 TopOpenctopn 15905 ∏tcpt 15922 Xscprds 15929 TopOnctopon 20518 TopSpctps 20519 |
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-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
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-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-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-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-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 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-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-topgen 15927 df-pt 15928 df-prds 15931 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 |
This theorem is referenced by: pwstps 21243 xpstps 21423 prdstmdd 21737 |
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