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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 22919 | . . 3 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top) | |
| 2 | topontop 22919 | . . 3 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top) | |
| 3 | txtop 23577 | . . 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 2737 | . . . . 5 ⊢ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) | |
| 6 | eqid 2737 | . . . . 5 ⊢ ∪ 𝑅 = ∪ 𝑅 | |
| 7 | eqid 2737 | . . . . 5 ⊢ ∪ 𝑆 = ∪ 𝑆 | |
| 8 | 5, 6, 7 | txuni2 23573 | . . . 4 ⊢ (∪ 𝑅 × ∪ 𝑆) = ∪ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) |
| 9 | toponuni 22920 | . . . . 5 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑅) | |
| 10 | toponuni 22920 | . . . . 5 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝑆) | |
| 11 | xpeq12 5710 | . . . . 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 23574 | . . . . 5 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V) |
| 14 | unitg 22974 | . . . . 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 2803 | . . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
| 17 | 5 | txval 23572 | . . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
| 18 | 17 | unieqd 4920 | . . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∪ (𝑅 ×t 𝑆) = ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
| 19 | 16, 18 | eqtr4d 2780 | . 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆)) |
| 20 | istopon 22918 | . 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 1540 ∈ wcel 2108 Vcvv 3480 ∪ cuni 4907 × cxp 5683 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 topGenctg 17482 Topctop 22899 TopOnctopon 22916 ×t ctx 23568 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-topgen 17488 df-top 22900 df-topon 22917 df-bases 22953 df-tx 23570 |
| This theorem is referenced by: txuni 23600 txcls 23612 tx1cn 23617 tx2cn 23618 txcnp 23628 txcnmpt 23632 txindis 23642 txdis1cn 23643 txlm 23656 lmcn2 23657 xkococn 23668 cnmpt12 23675 cnmpt2c 23678 cnmpt21 23679 cnmpt2t 23681 cnmpt22 23682 cnmpt22f 23683 cnmpt2res 23685 cnmptcom 23686 cnmpt2k 23696 ptunhmeo 23816 xpstopnlem1 23817 xkocnv 23822 xkohmeo 23823 txflf 24014 flfcnp2 24015 cnmpt2plusg 24096 tmdcn2 24097 indistgp 24108 clssubg 24117 qustgplem 24129 prdstmdd 24132 tsmsadd 24155 cnmpt2vsca 24203 txmetcn 24561 cnmpt2ds 24865 fsum2cn 24895 cnmpopc 24955 htpyco2 25011 phtpyco2 25022 cnmpt2ip 25282 limccnp2 25927 dvcnp2 25955 dvcnp2OLD 25956 dvaddbr 25974 dvmulbr 25975 dvmulbrOLD 25976 dvcobr 25983 dvcobrOLD 25984 lhop1lem 26052 taylthlem2 26416 taylthlem2OLD 26417 cxpcn3 26791 tpr2tp 33903 txsconnlem 35245 txsconn 35246 cvmlift2lem11 35318 cvmlift2lem12 35319 |
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