Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > txtopon | Unicode 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 | TopOn TopOn TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 12184 | . . 3 TopOn | |
2 | topontop 12184 | . . 3 TopOn | |
3 | txtop 12432 | . . 3 | |
4 | 1, 2, 3 | syl2an 287 | . 2 TopOn TopOn |
5 | eqid 2139 | . . . . 5 | |
6 | eqid 2139 | . . . . 5 | |
7 | eqid 2139 | . . . . 5 | |
8 | 5, 6, 7 | txuni2 12428 | . . . 4 |
9 | toponuni 12185 | . . . . 5 TopOn | |
10 | toponuni 12185 | . . . . 5 TopOn | |
11 | xpeq12 4558 | . . . . 5 | |
12 | 9, 10, 11 | syl2an 287 | . . . 4 TopOn TopOn |
13 | 5 | txbasex 12429 | . . . . 5 TopOn TopOn |
14 | unitg 12234 | . . . . 5 | |
15 | 13, 14 | syl 14 | . . . 4 TopOn TopOn |
16 | 8, 12, 15 | 3eqtr4a 2198 | . . 3 TopOn TopOn |
17 | 5 | txval 12427 | . . . 4 TopOn TopOn |
18 | 17 | unieqd 3747 | . . 3 TopOn TopOn |
19 | 16, 18 | eqtr4d 2175 | . 2 TopOn TopOn |
20 | istopon 12183 | . 2 TopOn | |
21 | 4, 19, 20 | sylanbrc 413 | 1 TopOn TopOn TopOn |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cvv 2686 cuni 3736 cxp 4537 crn 4540 cfv 5123 (class class class)co 5774 cmpo 5776 ctg 12138 ctop 12167 TopOnctopon 12180 ctx 12424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-topgen 12144 df-top 12168 df-topon 12181 df-bases 12213 df-tx 12425 |
This theorem is referenced by: txuni 12435 tx1cn 12441 tx2cn 12442 txcnp 12443 txcnmpt 12445 txdis1cn 12450 txlm 12451 lmcn2 12452 cnmpt12 12459 cnmpt2c 12462 cnmpt21 12463 cnmpt2t 12465 cnmpt22 12466 cnmpt22f 12467 cnmpt2res 12469 cnmptcom 12470 txmetcn 12691 limccnp2lem 12817 limccnp2cntop 12818 dvcnp2cntop 12835 dvaddxxbr 12837 dvmulxxbr 12838 dvcoapbr 12843 |
Copyright terms: Public domain | W3C validator |