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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 8943 | . . . 4 β’ 1 β β | |
6 | basendx 12530 | . . . 4 β’ (Baseβndx) = 1 | |
7 | 1lt2 9101 | . . . 4 β’ 1 < 2 | |
8 | 2nn 9093 | . . . 4 β’ 2 β β | |
9 | plusgndx 12582 | . . . 4 β’ (+gβndx) = 2 | |
10 | 2lt9 9135 | . . . 4 β’ 2 < 9 | |
11 | 9nn 9100 | . . . 4 β’ 9 β β | |
12 | tsetndx 12658 | . . . 4 β’ (TopSetβndx) = 9 | |
13 | 5, 6, 7, 8, 9, 10, 11, 12 | strle3g 12581 | . . 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 1248 | . 2 β’ (π β {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(TopSetβndx), π½β©} Struct β¨1, 9β©) |
15 | 1, 14 | eqbrtrid 4050 | 1 β’ (π β π Struct β¨1, 9β©) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1363 β wcel 2158 {ctp 3606 β¨cop 3607 class class class wbr 4015 βcfv 5228 1c1 7825 2c2 8983 9c9 8990 Struct cstr 12471 ndxcnx 12472 Basecbs 12475 +gcplusg 12550 TopSetcts 12556 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-tp 3612 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-5 8994 df-6 8995 df-7 8996 df-8 8997 df-9 8998 df-n0 9190 df-z 9267 df-uz 9542 df-fz 10022 df-struct 12477 df-ndx 12478 df-slot 12479 df-base 12481 df-plusg 12563 df-tset 12569 |
This theorem is referenced by: topgrpbasd 12669 topgrpplusgd 12670 topgrptsetd 12671 |
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