![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > grpbaseg | GIF version |
Description: The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
grpfn.g | β’ πΊ = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©} |
Ref | Expression |
---|---|
grpbaseg | β’ ((π΅ β π β§ + β π) β π΅ = (BaseβπΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpfn.g | . 2 β’ πΊ = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©} | |
2 | df-plusg 12552 | . 2 β’ +g = Slot 2 | |
3 | 1lt2 9091 | . 2 β’ 1 < 2 | |
4 | 2nn 9083 | . 2 β’ 2 β β | |
5 | 1, 2, 3, 4 | 2strbasg 12581 | 1 β’ ((π΅ β π β§ + β π) β π΅ = (BaseβπΊ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 {cpr 3595 β¨cop 3597 βcfv 5218 2c2 8973 ndxcnx 12462 Basecbs 12465 +gcplusg 12539 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-i2m1 7919 ax-0lt1 7920 ax-0id 7922 ax-rnegex 7923 ax-pre-ltirr 7926 ax-pre-ltadd 7930 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-iota 5180 df-fun 5220 df-fv 5226 df-ov 5881 df-pnf 7997 df-mnf 7998 df-ltxr 8000 df-inn 8923 df-2 8981 df-ndx 12468 df-slot 12469 df-base 12471 df-plusg 12552 |
This theorem is referenced by: mgm1 12795 sgrp1 12822 mnd1 12853 mnd1id 12854 grppropstrg 12901 grp1 12982 grp1inv 12983 ring1 13242 |
Copyright terms: Public domain | W3C validator |