Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2740 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2740 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
4 | simpl 483 | . . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑅 ∈ TopSp) | |
5 | simpr 485 | . . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑆 ∈ TopSp) | |
6 | eqid 2740 | . . 3 ⊢ (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) | |
7 | eqid 2740 | . . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
8 | eqid 2740 | . . 3 ⊢ ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) = ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | xpsval 17279 | . 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 17280 | . 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 17278 | . . . 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 6726 | . . 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 6721 | . . 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 6786 | . . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (Scalar‘𝑅) ∈ V) | |
17 | 2on 8302 | . . . 4 ⊢ 2o ∈ On | |
18 | 17 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 2o ∈ On) |
19 | xpscf 17274 | . . . 4 ⊢ ({〈∅, 𝑅〉, 〈1o, 𝑆〉}:2o⟶TopSp ↔ (𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp)) | |
20 | 19 | biimpri 227 | . . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → {〈∅, 𝑅〉, 〈1o, 𝑆〉}:2o⟶TopSp) |
21 | 8, 16, 18, 20 | prdstps 22778 | . 2 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) ∈ TopSp) |
22 | 9, 10, 15, 21 | imastps 22870 | 1 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑇 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ∅c0 4262 {cpr 4569 〈cop 4573 × cxp 5588 ◡ccnv 5589 ran crn 5591 Oncon0 6265 ⟶wf 6428 –onto→wfo 6430 –1-1-onto→wf1o 6431 ‘cfv 6432 (class class class)co 7271 ∈ cmpo 7273 1oc1o 8281 2oc2o 8282 Basecbs 16910 Scalarcsca 16963 Xscprds 17154 ×s cxps 17215 TopSpctps 22079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-er 8481 df-map 8600 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fi 9148 df-sup 9179 df-inf 9180 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-fz 13239 df-struct 16846 df-slot 16881 df-ndx 16893 df-base 16911 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-topgen 17152 df-pt 17153 df-prds 17156 df-qtop 17216 df-imas 17217 df-xps 17219 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |