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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 22934 | . . 3 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top) | |
2 | topontop 22934 | . . 3 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top) | |
3 | txtop 23592 | . . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top) | |
4 | 1, 2, 3 | syl2an 596 | . 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ Top) |
5 | eqid 2734 | . . . . 5 ⊢ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) | |
6 | eqid 2734 | . . . . 5 ⊢ ∪ 𝑅 = ∪ 𝑅 | |
7 | eqid 2734 | . . . . 5 ⊢ ∪ 𝑆 = ∪ 𝑆 | |
8 | 5, 6, 7 | txuni2 23588 | . . . 4 ⊢ (∪ 𝑅 × ∪ 𝑆) = ∪ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) |
9 | toponuni 22935 | . . . . 5 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑅) | |
10 | toponuni 22935 | . . . . 5 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝑆) | |
11 | xpeq12 5713 | . . . . 5 ⊢ ((𝑋 = ∪ 𝑅 ∧ 𝑌 = ∪ 𝑆) → (𝑋 × 𝑌) = (∪ 𝑅 × ∪ 𝑆)) | |
12 | 9, 10, 11 | syl2an 596 | . . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = (∪ 𝑅 × ∪ 𝑆)) |
13 | 5 | txbasex 23589 | . . . . 5 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V) |
14 | unitg 22989 | . . . . 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 2800 | . . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
17 | 5 | txval 23587 | . . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
18 | 17 | unieqd 4924 | . . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∪ (𝑅 ×t 𝑆) = ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
19 | 16, 18 | eqtr4d 2777 | . 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆)) |
20 | istopon 22933 | . 2 ⊢ ((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆))) | |
21 | 4, 19, 20 | sylanbrc 583 | 1 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∪ cuni 4911 × cxp 5686 ran crn 5689 ‘cfv 6562 (class class class)co 7430 ∈ cmpo 7432 topGenctg 17483 Topctop 22914 TopOnctopon 22931 ×t ctx 23583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-topgen 17489 df-top 22915 df-topon 22932 df-bases 22968 df-tx 23585 |
This theorem is referenced by: txuni 23615 txcls 23627 tx1cn 23632 tx2cn 23633 txcnp 23643 txcnmpt 23647 txindis 23657 txdis1cn 23658 txlm 23671 lmcn2 23672 xkococn 23683 cnmpt12 23690 cnmpt2c 23693 cnmpt21 23694 cnmpt2t 23696 cnmpt22 23697 cnmpt22f 23698 cnmpt2res 23700 cnmptcom 23701 cnmpt2k 23711 ptunhmeo 23831 xpstopnlem1 23832 xkocnv 23837 xkohmeo 23838 txflf 24029 flfcnp2 24030 cnmpt2plusg 24111 tmdcn2 24112 indistgp 24123 clssubg 24132 qustgplem 24144 prdstmdd 24147 tsmsadd 24170 cnmpt2vsca 24218 txmetcn 24576 cnmpt2ds 24878 fsum2cn 24908 cnmpopc 24968 htpyco2 25024 phtpyco2 25035 cnmpt2ip 25295 limccnp2 25941 dvcnp2 25969 dvcnp2OLD 25970 dvaddbr 25988 dvmulbr 25989 dvmulbrOLD 25990 dvcobr 25997 dvcobrOLD 25998 lhop1lem 26066 taylthlem2 26430 taylthlem2OLD 26431 cxpcn3 26805 tpr2tp 33864 txsconnlem 35224 txsconn 35225 cvmlift2lem11 35297 cvmlift2lem12 35298 |
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