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Mirrors > Home > ILE Home > Th. List > topgrpstrd | GIF version |
Description: A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
Ref | Expression |
---|---|
topgrpfn.w | β’ π = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(TopSetβndx), π½β©} |
topgrpfnd.b | β’ (π β π΅ β π) |
topgrpfnd.p | β’ (π β + β π) |
topgrpfnd.j | β’ (π β π½ β π) |
Ref | Expression |
---|---|
topgrpstrd | β’ (π β π Struct β¨1, 9β©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topgrpfn.w | . 2 β’ π = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(TopSetβndx), π½β©} | |
2 | topgrpfnd.b | . . 3 β’ (π β π΅ β π) | |
3 | topgrpfnd.p | . . 3 β’ (π β + β π) | |
4 | topgrpfnd.j | . . 3 β’ (π β π½ β π) | |
5 | 1nn 8949 | . . . 4 β’ 1 β β | |
6 | basendx 12541 | . . . 4 β’ (Baseβndx) = 1 | |
7 | 1lt2 9107 | . . . 4 β’ 1 < 2 | |
8 | 2nn 9099 | . . . 4 β’ 2 β β | |
9 | plusgndx 12593 | . . . 4 β’ (+gβndx) = 2 | |
10 | 2lt9 9141 | . . . 4 β’ 2 < 9 | |
11 | 9nn 9106 | . . . 4 β’ 9 β β | |
12 | tsetndx 12669 | . . . 4 β’ (TopSetβndx) = 9 | |
13 | 5, 6, 7, 8, 9, 10, 11, 12 | strle3g 12592 | . . 3 β’ ((π΅ β π β§ + β π β§ π½ β π) β {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(TopSetβndx), π½β©} Struct β¨1, 9β©) |
14 | 2, 3, 4, 13 | syl3anc 1249 | . 2 β’ (π β {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(TopSetβndx), π½β©} Struct β¨1, 9β©) |
15 | 1, 14 | eqbrtrid 4053 | 1 β’ (π β π Struct β¨1, 9β©) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1364 β wcel 2160 {ctp 3609 β¨cop 3610 class class class wbr 4018 βcfv 5231 1c1 7831 2c2 8989 9c9 8996 Struct cstr 12482 ndxcnx 12483 Basecbs 12486 +gcplusg 12561 TopSetcts 12567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-addcom 7930 ax-addass 7932 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-0id 7938 ax-rnegex 7939 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-apti 7945 ax-pre-ltadd 7946 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-tp 3615 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-inn 8939 df-2 8997 df-3 8998 df-4 8999 df-5 9000 df-6 9001 df-7 9002 df-8 9003 df-9 9004 df-n0 9196 df-z 9273 df-uz 9548 df-fz 10028 df-struct 12488 df-ndx 12489 df-slot 12490 df-base 12492 df-plusg 12574 df-tset 12580 |
This theorem is referenced by: topgrpbasd 12680 topgrpplusgd 12681 topgrptsetd 12682 |
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