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Mirrors > Home > ILE Home > Th. List > txopn | Unicode version |
Description: The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
txopn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2113 |
. . . . . 6
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2 | 1 | txbasex 12261 |
. . . . 5
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3 | bastg 12066 |
. . . . 5
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4 | 2, 3 | syl 14 |
. . . 4
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5 | 4 | adantr 272 |
. . 3
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6 | eqid 2113 |
. . . . . 6
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7 | xpeq1 4511 |
. . . . . . . 8
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8 | 7 | eqeq2d 2124 |
. . . . . . 7
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9 | xpeq2 4512 |
. . . . . . . 8
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10 | 9 | eqeq2d 2124 |
. . . . . . 7
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11 | 8, 10 | rspc2ev 2772 |
. . . . . 6
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12 | 6, 11 | mp3an3 1285 |
. . . . 5
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13 | xpexg 4611 |
. . . . . 6
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14 | eqid 2113 |
. . . . . . 7
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15 | 14 | elrnmpog 5835 |
. . . . . 6
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16 | 13, 15 | syl 14 |
. . . . 5
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17 | 12, 16 | mpbird 166 |
. . . 4
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18 | 17 | adantl 273 |
. . 3
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19 | 5, 18 | sseldd 3062 |
. 2
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20 | 1 | txval 12259 |
. . 3
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21 | 20 | adantr 272 |
. 2
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22 | 19, 21 | eleqtrrd 2192 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5989 df-2nd 5990 df-topgen 11977 df-tx 12257 |
This theorem is referenced by: txbasval 12271 neitx 12272 tx1cn 12273 tx2cn 12274 txlm 12283 |
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