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Mirrors > Home > MPE Home > Th. List > xpstps | Structured version Visualization version GIF version |
Description: A binary product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.) |
Ref | Expression |
---|---|
xpstps.t | ⊢ 𝑇 = (𝑅 ×s 𝑆) |
Ref | Expression |
---|---|
xpstps | ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑇 ∈ TopSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpstps.t | . . 3 ⊢ 𝑇 = (𝑅 ×s 𝑆) | |
2 | eqid 2798 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2798 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
4 | simpl 486 | . . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑅 ∈ TopSp) | |
5 | simpr 488 | . . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑆 ∈ TopSp) | |
6 | eqid 2798 | . . 3 ⊢ (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) | |
7 | eqid 2798 | . . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
8 | eqid 2798 | . . 3 ⊢ ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) = ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | xpsval 16835 | . 2 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑇 = (◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) “s ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}))) |
10 | 1, 2, 3, 4, 5, 6, 7, 8 | xpsrnbas 16836 | . 2 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = (Base‘((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}))) |
11 | 6 | xpsff1o2 16834 | . . . 4 ⊢ (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):((Base‘𝑅) × (Base‘𝑆))–1-1-onto→ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) |
12 | 11 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):((Base‘𝑅) × (Base‘𝑆))–1-1-onto→ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})) |
13 | f1ocnv 6602 | . . 3 ⊢ ((𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):((Base‘𝑅) × (Base‘𝑆))–1-1-onto→ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) → ◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–1-1-onto→((Base‘𝑅) × (Base‘𝑆))) | |
14 | f1ofo 6597 | . . 3 ⊢ (◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–1-1-onto→((Base‘𝑅) × (Base‘𝑆)) → ◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–onto→((Base‘𝑅) × (Base‘𝑆))) | |
15 | 12, 13, 14 | 3syl 18 | . 2 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–onto→((Base‘𝑅) × (Base‘𝑆))) |
16 | fvexd 6660 | . . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (Scalar‘𝑅) ∈ V) | |
17 | 2on 8094 | . . . 4 ⊢ 2o ∈ On | |
18 | 17 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 2o ∈ On) |
19 | xpscf 16830 | . . . 4 ⊢ ({〈∅, 𝑅〉, 〈1o, 𝑆〉}:2o⟶TopSp ↔ (𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp)) | |
20 | 19 | biimpri 231 | . . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → {〈∅, 𝑅〉, 〈1o, 𝑆〉}:2o⟶TopSp) |
21 | 8, 16, 18, 20 | prdstps 22234 | . 2 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) ∈ TopSp) |
22 | 9, 10, 15, 21 | imastps 22326 | 1 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑇 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 {cpr 4527 〈cop 4531 × cxp 5517 ◡ccnv 5518 ran crn 5520 Oncon0 6159 ⟶wf 6320 –onto→wfo 6322 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 1oc1o 8078 2oc2o 8079 Basecbs 16475 Scalarcsca 16560 Xscprds 16711 ×s cxps 16771 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-2o 8086 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-inf 8891 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-qtop 16772 df-imas 16773 df-xps 16775 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 |
This theorem is referenced by: (None) |
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