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Mirrors > Home > ILE Home > Th. List > srngstrd | GIF version |
Description: A constructed star ring is a structure. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.) |
Ref | Expression |
---|---|
srngstr.r | β’ π = ({β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(.rβndx), Β· β©} βͺ {β¨(*πβndx), β β©}) |
srngstrd.b | β’ (π β π΅ β π) |
srngstrd.p | β’ (π β + β π) |
srngstrd.m | β’ (π β Β· β π) |
srngstrd.s | β’ (π β β β π) |
Ref | Expression |
---|---|
srngstrd | β’ (π β π Struct β¨1, 4β©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srngstr.r | . 2 β’ π = ({β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(.rβndx), Β· β©} βͺ {β¨(*πβndx), β β©}) | |
2 | srngstrd.b | . . . 4 β’ (π β π΅ β π) | |
3 | srngstrd.p | . . . 4 β’ (π β + β π) | |
4 | srngstrd.m | . . . 4 β’ (π β Β· β π) | |
5 | eqid 2177 | . . . . 5 β’ {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(.rβndx), Β· β©} = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(.rβndx), Β· β©} | |
6 | 5 | rngstrg 12593 | . . . 4 β’ ((π΅ β π β§ + β π β§ Β· β π) β {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(.rβndx), Β· β©} Struct β¨1, 3β©) |
7 | 2, 3, 4, 6 | syl3anc 1238 | . . 3 β’ (π β {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(.rβndx), Β· β©} Struct β¨1, 3β©) |
8 | srngstrd.s | . . . 4 β’ (π β β β π) | |
9 | 4nn 9082 | . . . . 5 β’ 4 β β | |
10 | starvndx 12597 | . . . . 5 β’ (*πβndx) = 4 | |
11 | 9, 10 | strle1g 12565 | . . . 4 β’ ( β β π β {β¨(*πβndx), β β©} Struct β¨4, 4β©) |
12 | 8, 11 | syl 14 | . . 3 β’ (π β {β¨(*πβndx), β β©} Struct β¨4, 4β©) |
13 | 3lt4 9091 | . . . 4 β’ 3 < 4 | |
14 | 13 | a1i 9 | . . 3 β’ (π β 3 < 4) |
15 | 7, 12, 14 | strleund 12562 | . 2 β’ (π β ({β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(.rβndx), Β· β©} βͺ {β¨(*πβndx), β β©}) Struct β¨1, 4β©) |
16 | 1, 15 | eqbrtrid 4039 | 1 β’ (π β π Struct β¨1, 4β©) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1353 β wcel 2148 βͺ cun 3128 {csn 3593 {ctp 3595 β¨cop 3596 class class class wbr 4004 βcfv 5217 1c1 7812 < clt 7992 3c3 8971 4c4 8972 Struct cstr 12458 ndxcnx 12459 Basecbs 12462 +gcplusg 12536 .rcmulr 12537 *πcstv 12538 |
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 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-addcom 7911 ax-addass 7913 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-0id 7919 ax-rnegex 7920 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 |
This theorem depends on definitions: df-bi 117 df-3or 979 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-reu 2462 df-rab 2464 df-v 2740 df-sbc 2964 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-tp 3601 df-op 3602 df-uni 3811 df-int 3846 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-n0 9177 df-z 9254 df-uz 9529 df-fz 10009 df-struct 12464 df-ndx 12465 df-slot 12466 df-base 12468 df-plusg 12549 df-mulr 12550 df-starv 12551 |
This theorem is referenced by: srngbased 12605 srngplusgd 12606 srngmulrd 12607 srnginvld 12608 cnfldstr 13460 |
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