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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 6359 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ TopSp) |
4 | eqid 2622 | . . . . . . 7 ⊢ (Base‘(𝑅‘𝑥)) = (Base‘(𝑅‘𝑥)) | |
5 | eqid 2622 | . . . . . . 7 ⊢ (TopOpen‘(𝑅‘𝑥)) = (TopOpen‘(𝑅‘𝑥)) | |
6 | 4, 5 | istps 20738 | . . . . . 6 ⊢ ((𝑅‘𝑥) ∈ TopSp ↔ (TopOpen‘(𝑅‘𝑥)) ∈ (TopOn‘(Base‘(𝑅‘𝑥)))) |
7 | 3, 6 | sylib 208 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (TopOpen‘(𝑅‘𝑥)) ∈ (TopOn‘(Base‘(𝑅‘𝑥)))) |
8 | 7 | ralrimiva 2966 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (TopOpen‘(𝑅‘𝑥)) ∈ (TopOn‘(Base‘(𝑅‘𝑥)))) |
9 | eqid 2622 | . . . . 5 ⊢ (∏t‘(𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥)))) = (∏t‘(𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥)))) | |
10 | 9 | pttopon 21399 | . . . 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 693 | . . 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 6490 | . . . . . 6 ⊢ ((𝑅:𝐼⟶TopSp ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ V) | |
15 | 2, 1, 14 | syl2anc 693 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ V) |
16 | eqid 2622 | . . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
17 | fdm 6051 | . . . . . 6 ⊢ (𝑅:𝐼⟶TopSp → dom 𝑅 = 𝐼) | |
18 | 2, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝑅 = 𝐼) |
19 | eqid 2622 | . . . . 5 ⊢ (TopSet‘𝑌) = (TopSet‘𝑌) | |
20 | 12, 13, 15, 16, 18, 19 | prdstset 16126 | . . . 4 ⊢ (𝜑 → (TopSet‘𝑌) = (∏t‘(TopOpen ∘ 𝑅))) |
21 | topnfn 16086 | . . . . . . 7 ⊢ TopOpen Fn V | |
22 | dffn2 6047 | . . . . . . 7 ⊢ (TopOpen Fn V ↔ TopOpen:V⟶V) | |
23 | 21, 22 | mpbi 220 | . . . . . 6 ⊢ TopOpen:V⟶V |
24 | ssv 3625 | . . . . . . 7 ⊢ TopSp ⊆ V | |
25 | fss 6056 | . . . . . . 7 ⊢ ((𝑅:𝐼⟶TopSp ∧ TopSp ⊆ V) → 𝑅:𝐼⟶V) | |
26 | 2, 24, 25 | sylancl 694 | . . . . . 6 ⊢ (𝜑 → 𝑅:𝐼⟶V) |
27 | fcompt 6400 | . . . . . 6 ⊢ ((TopOpen:V⟶V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) = (𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥)))) | |
28 | 23, 26, 27 | sylancr 695 | . . . . 5 ⊢ (𝜑 → (TopOpen ∘ 𝑅) = (𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥)))) |
29 | 28 | fveq2d 6195 | . . . 4 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥))))) |
30 | 20, 29 | eqtrd 2656 | . . 3 ⊢ (𝜑 → (TopSet‘𝑌) = (∏t‘(𝑥 ∈ 𝐼 ↦ (TopOpen‘(𝑅‘𝑥))))) |
31 | 12, 13, 15, 16, 18 | prdsbas 16117 | . . . 4 ⊢ (𝜑 → (Base‘𝑌) = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) |
32 | 31 | fveq2d 6195 | . . 3 ⊢ (𝜑 → (TopOn‘(Base‘𝑌)) = (TopOn‘X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))) |
33 | 11, 30, 32 | 3eltr4d 2716 | . 2 ⊢ (𝜑 → (TopSet‘𝑌) ∈ (TopOn‘(Base‘𝑌))) |
34 | 16, 19 | tsettps 20745 | . 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 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⊆ wss 3574 ↦ cmpt 4729 dom cdm 5114 ∘ ccom 5118 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Xcixp 7908 Basecbs 15857 TopSetcts 15947 TopOpenctopn 16082 ∏tcpt 16099 Xscprds 16106 TopOnctopon 20715 TopSpctps 20736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-topgen 16104 df-pt 16105 df-prds 16108 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 |
This theorem is referenced by: pwstps 21433 xpstps 21613 prdstmdd 21927 |
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