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Mirrors > Home > ILE Home > Th. List > topgrptsetd | Unicode version |
Description: The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
Ref | Expression |
---|---|
topgrpfn.w | TopSet |
topgrpfnd.b | |
topgrpfnd.p | |
topgrpfnd.j |
Ref | Expression |
---|---|
topgrptsetd | TopSet |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsetslid 12119 | . 2 TopSet Slot TopSet TopSet | |
2 | topgrpfn.w | . . 3 TopSet | |
3 | topgrpfnd.b | . . 3 | |
4 | topgrpfnd.p | . . 3 | |
5 | topgrpfnd.j | . . 3 | |
6 | 2, 3, 4, 5 | topgrpstrd 12120 | . 2 Struct |
7 | 1 | simpri 112 | . . . . 5 TopSet |
8 | opexg 4150 | . . . . 5 TopSet TopSet | |
9 | 7, 5, 8 | sylancr 410 | . . . 4 TopSet |
10 | tpid3g 3638 | . . . 4 TopSet TopSet TopSet | |
11 | 9, 10 | syl 14 | . . 3 TopSet TopSet |
12 | 11, 2 | eleqtrrdi 2233 | . 2 TopSet |
13 | 1, 6, 5, 12 | opelstrsl 12065 | 1 TopSet |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wcel 1480 cvv 2686 ctp 3529 cop 3530 cfv 5123 c1 7628 cn 8727 c9 8785 cnx 11966 Slot cslot 11968 cbs 11969 cplusg 12031 TopSetcts 12037 |
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-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7718 ax-resscn 7719 ax-1cn 7720 ax-1re 7721 ax-icn 7722 ax-addcl 7723 ax-addrcl 7724 ax-mulcl 7725 ax-addcom 7727 ax-addass 7729 ax-distr 7731 ax-i2m1 7732 ax-0lt1 7733 ax-0id 7735 ax-rnegex 7736 ax-cnre 7738 ax-pre-ltirr 7739 ax-pre-ltwlin 7740 ax-pre-lttrn 7741 ax-pre-apti 7742 ax-pre-ltadd 7743 |
This theorem depends on definitions: df-bi 116 df-3or 963 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-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-tp 3535 df-op 3536 df-uni 3737 df-int 3772 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-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7809 df-mnf 7810 df-xr 7811 df-ltxr 7812 df-le 7813 df-sub 7942 df-neg 7943 df-inn 8728 df-2 8786 df-3 8787 df-4 8788 df-5 8789 df-6 8790 df-7 8791 df-8 8792 df-9 8793 df-n0 8985 df-z 9062 df-uz 9334 df-fz 9798 df-struct 11971 df-ndx 11972 df-slot 11973 df-base 11975 df-plusg 12044 df-tset 12050 |
This theorem is referenced by: (None) |
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