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Mirrors > Home > ILE Home > Th. List > txtopon | 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 13145 | . . 3 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top) | |
2 | topontop 13145 | . . 3 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top) | |
3 | txtop 13393 | . . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top) | |
4 | 1, 2, 3 | syl2an 289 | . 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ Top) |
5 | eqid 2177 | . . . . 5 ⊢ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) | |
6 | eqid 2177 | . . . . 5 ⊢ ∪ 𝑅 = ∪ 𝑅 | |
7 | eqid 2177 | . . . . 5 ⊢ ∪ 𝑆 = ∪ 𝑆 | |
8 | 5, 6, 7 | txuni2 13389 | . . . 4 ⊢ (∪ 𝑅 × ∪ 𝑆) = ∪ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) |
9 | toponuni 13146 | . . . . 5 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑅) | |
10 | toponuni 13146 | . . . . 5 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝑆) | |
11 | xpeq12 4641 | . . . . 5 ⊢ ((𝑋 = ∪ 𝑅 ∧ 𝑌 = ∪ 𝑆) → (𝑋 × 𝑌) = (∪ 𝑅 × ∪ 𝑆)) | |
12 | 9, 10, 11 | syl2an 289 | . . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = (∪ 𝑅 × ∪ 𝑆)) |
13 | 5 | txbasex 13390 | . . . . 5 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V) |
14 | unitg 13195 | . . . . 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 14 | . . . 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 2236 | . . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
17 | 5 | txval 13388 | . . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
18 | 17 | unieqd 3818 | . . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∪ (𝑅 ×t 𝑆) = ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
19 | 16, 18 | eqtr4d 2213 | . 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆)) |
20 | istopon 13144 | . 2 ⊢ ((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆))) | |
21 | 4, 19, 20 | sylanbrc 417 | 1 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∪ cuni 3807 × cxp 4620 ran crn 4623 ‘cfv 5211 (class class class)co 5868 ∈ cmpo 5870 topGenctg 12638 Topctop 13128 TopOnctopon 13141 ×t ctx 13385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-topgen 12644 df-top 13129 df-topon 13142 df-bases 13174 df-tx 13386 |
This theorem is referenced by: txuni 13396 tx1cn 13402 tx2cn 13403 txcnp 13404 txcnmpt 13406 txdis1cn 13411 txlm 13412 lmcn2 13413 cnmpt12 13420 cnmpt2c 13423 cnmpt21 13424 cnmpt2t 13426 cnmpt22 13427 cnmpt22f 13428 cnmpt2res 13430 cnmptcom 13431 txmetcn 13652 limccnp2lem 13778 limccnp2cntop 13779 dvcnp2cntop 13796 dvaddxxbr 13798 dvmulxxbr 13799 dvcoapbr 13804 |
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