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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 12423 | . . 3 TopOn | |
2 | topontop 12423 | . . 3 TopOn | |
3 | txtop 12671 | . . 3 | |
4 | 1, 2, 3 | syl2an 287 | . 2 TopOn TopOn |
5 | eqid 2157 | . . . . 5 | |
6 | eqid 2157 | . . . . 5 | |
7 | eqid 2157 | . . . . 5 | |
8 | 5, 6, 7 | txuni2 12667 | . . . 4 |
9 | toponuni 12424 | . . . . 5 TopOn | |
10 | toponuni 12424 | . . . . 5 TopOn | |
11 | xpeq12 4605 | . . . . 5 | |
12 | 9, 10, 11 | syl2an 287 | . . . 4 TopOn TopOn |
13 | 5 | txbasex 12668 | . . . . 5 TopOn TopOn |
14 | unitg 12473 | . . . . 5 | |
15 | 13, 14 | syl 14 | . . . 4 TopOn TopOn |
16 | 8, 12, 15 | 3eqtr4a 2216 | . . 3 TopOn TopOn |
17 | 5 | txval 12666 | . . . 4 TopOn TopOn |
18 | 17 | unieqd 3783 | . . 3 TopOn TopOn |
19 | 16, 18 | eqtr4d 2193 | . 2 TopOn TopOn |
20 | istopon 12422 | . 2 TopOn | |
21 | 4, 19, 20 | sylanbrc 414 | 1 TopOn TopOn TopOn |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 cvv 2712 cuni 3772 cxp 4584 crn 4587 cfv 5170 (class class class)co 5824 cmpo 5826 ctg 12377 ctop 12406 TopOnctopon 12419 ctx 12663 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-topgen 12383 df-top 12407 df-topon 12420 df-bases 12452 df-tx 12664 |
This theorem is referenced by: txuni 12674 tx1cn 12680 tx2cn 12681 txcnp 12682 txcnmpt 12684 txdis1cn 12689 txlm 12690 lmcn2 12691 cnmpt12 12698 cnmpt2c 12701 cnmpt21 12702 cnmpt2t 12704 cnmpt22 12705 cnmpt22f 12706 cnmpt2res 12708 cnmptcom 12709 txmetcn 12930 limccnp2lem 13056 limccnp2cntop 13057 dvcnp2cntop 13074 dvaddxxbr 13076 dvmulxxbr 13077 dvcoapbr 13082 |
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