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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 2824 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2824 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
4 | simpl 485 | . . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑅 ∈ TopSp) | |
5 | simpr 487 | . . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑆 ∈ TopSp) | |
6 | eqid 2824 | . . 3 ⊢ (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) | |
7 | eqid 2824 | . . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
8 | eqid 2824 | . . 3 ⊢ ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) = ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | xpsval 16846 | . 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 16847 | . 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 16845 | . . . 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 6630 | . . 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 6625 | . . 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 6688 | . . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (Scalar‘𝑅) ∈ V) | |
17 | 2on 8114 | . . . 4 ⊢ 2o ∈ On | |
18 | 17 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 2o ∈ On) |
19 | xpscf 16841 | . . . 4 ⊢ ({〈∅, 𝑅〉, 〈1o, 𝑆〉}:2o⟶TopSp ↔ (𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp)) | |
20 | 19 | biimpri 230 | . . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → {〈∅, 𝑅〉, 〈1o, 𝑆〉}:2o⟶TopSp) |
21 | 8, 16, 18, 20 | prdstps 22240 | . 2 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) ∈ TopSp) |
22 | 9, 10, 15, 21 | imastps 22332 | 1 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑇 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∅c0 4294 {cpr 4572 〈cop 4576 × cxp 5556 ◡ccnv 5557 ran crn 5559 Oncon0 6194 ⟶wf 6354 –onto→wfo 6356 –1-1-onto→wf1o 6357 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 1oc1o 8098 2oc2o 8099 Basecbs 16486 Scalarcsca 16571 Xscprds 16722 ×s cxps 16782 TopSpctps 21543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fi 8878 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-topgen 16720 df-pt 16721 df-prds 16724 df-qtop 16783 df-imas 16784 df-xps 16786 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 |
This theorem is referenced by: (None) |
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