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Mirrors > Home > MPE Home > Th. List > txtopon | Structured version Visualization version GIF version |
Description: The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 22-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
txtopon | ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 20939 | . . 3 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top) | |
2 | topontop 20939 | . . 3 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top) | |
3 | txtop 21594 | . . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top) | |
4 | 1, 2, 3 | syl2an 577 | . 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ Top) |
5 | eqid 2771 | . . . . 5 ⊢ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) | |
6 | eqid 2771 | . . . . 5 ⊢ ∪ 𝑅 = ∪ 𝑅 | |
7 | eqid 2771 | . . . . 5 ⊢ ∪ 𝑆 = ∪ 𝑆 | |
8 | 5, 6, 7 | txuni2 21590 | . . . 4 ⊢ (∪ 𝑅 × ∪ 𝑆) = ∪ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) |
9 | toponuni 20940 | . . . . 5 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑅) | |
10 | toponuni 20940 | . . . . 5 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝑆) | |
11 | xpeq12 5275 | . . . . 5 ⊢ ((𝑋 = ∪ 𝑅 ∧ 𝑌 = ∪ 𝑆) → (𝑋 × 𝑌) = (∪ 𝑅 × ∪ 𝑆)) | |
12 | 9, 10, 11 | syl2an 577 | . . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = (∪ 𝑅 × ∪ 𝑆)) |
13 | 5 | txbasex 21591 | . . . . 5 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V) |
14 | unitg 20993 | . . . . 5 ⊢ (ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V → ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) = ∪ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) = ∪ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) |
16 | 8, 12, 15 | 3eqtr4a 2831 | . . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
17 | 5 | txval 21589 | . . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
18 | 17 | unieqd 4585 | . . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∪ (𝑅 ×t 𝑆) = ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
19 | 16, 18 | eqtr4d 2808 | . 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆)) |
20 | istopon 20938 | . 2 ⊢ ((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆))) | |
21 | 4, 19, 20 | sylanbrc 566 | 1 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ∪ cuni 4575 × cxp 5248 ran crn 5251 ‘cfv 6032 (class class class)co 6794 ↦ cmpt2 6796 topGenctg 16307 Topctop 20919 TopOnctopon 20936 ×t ctx 21585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-fv 6040 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-1st 7316 df-2nd 7317 df-topgen 16313 df-top 20920 df-topon 20937 df-bases 20972 df-tx 21587 |
This theorem is referenced by: txuni 21617 txcls 21629 tx1cn 21634 tx2cn 21635 txcnp 21645 txcnmpt 21649 txindis 21659 txdis1cn 21660 txlm 21673 lmcn2 21674 xkococn 21685 cnmpt12 21692 cnmpt2c 21695 cnmpt21 21696 cnmpt2t 21698 cnmpt22 21699 cnmpt22f 21700 cnmpt2res 21702 cnmptcom 21703 cnmpt2k 21713 ptunhmeo 21833 xpstopnlem1 21834 xkocnv 21839 xkohmeo 21840 txflf 22031 flfcnp2 22032 cnmpt2plusg 22113 tmdcn2 22114 indistgp 22125 clssubg 22133 qustgplem 22145 prdstmdd 22148 tsmsadd 22171 cnmpt2vsca 22219 txmetcn 22574 cnmpt2ds 22867 fsum2cn 22895 cnmpt2pc 22948 htpyco2 22999 phtpyco2 23010 cnmpt2ip 23267 limccnp2 23877 dvcnp2 23904 dvaddbr 23922 dvmulbr 23923 dvcobr 23930 lhop1lem 23997 taylthlem2 24349 cxpcn3 24711 tpr2tp 30291 txsconnlem 31561 txsconn 31562 cvmlift2lem11 31634 cvmlift2lem12 31635 |
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