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Mirrors > Home > ILE Home > Th. List > rngstrg | Unicode version |
Description: A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Jim Kingdon, 3-Feb-2023.) |
Ref | Expression |
---|---|
rngfn.r |
Ref | Expression |
---|---|
rngstrg | Struct |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngfn.r | . 2 | |
2 | 1nn 8859 | . . 3 | |
3 | basendx 12391 | . . 3 | |
4 | 1lt2 9017 | . . 3 | |
5 | 2nn 9009 | . . 3 | |
6 | plusgndx 12430 | . . 3 | |
7 | 2lt3 9018 | . . 3 | |
8 | 3nn 9010 | . . 3 | |
9 | mulrndx 12447 | . . 3 | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | strle3g 12429 | . 2 Struct |
11 | 1, 10 | eqbrtrid 4011 | 1 Struct |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 967 wceq 1342 wcel 2135 ctp 3572 cop 3573 class class class wbr 3976 cfv 5182 c1 7745 c2 8899 c3 8900 Struct cstr 12333 cnx 12334 cbs 12337 cplusg 12399 cmulr 12400 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-tp 3578 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-2 8907 df-3 8908 df-n0 9106 df-z 9183 df-uz 9458 df-fz 9936 df-struct 12339 df-ndx 12340 df-slot 12341 df-base 12343 df-plusg 12412 df-mulr 12413 |
This theorem is referenced by: rngbaseg 12453 rngplusgg 12454 rngmulrg 12455 srngstrd 12459 ipsstrd 12478 |
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