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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 |
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Ref | Expression |
---|---|
rngstrg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngfn.r |
. 2
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2 | 1nn 8948 |
. . 3
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3 | basendx 12535 |
. . 3
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4 | 1lt2 9106 |
. . 3
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5 | 2nn 9098 |
. . 3
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6 | plusgndx 12587 |
. . 3
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7 | 2lt3 9107 |
. . 3
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8 | 3nn 9099 |
. . 3
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9 | mulrndx 12607 |
. . 3
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10 | 2, 3, 4, 5, 6, 7, 8, 9 | strle3g 12586 |
. 2
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11 | 1, 10 | eqbrtrid 4053 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-addcom 7929 ax-addass 7931 ax-distr 7933 ax-i2m1 7934 ax-0lt1 7935 ax-0id 7937 ax-rnegex 7938 ax-cnre 7940 ax-pre-ltirr 7941 ax-pre-ltwlin 7942 ax-pre-lttrn 7943 ax-pre-apti 7944 ax-pre-ltadd 7945 |
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 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-sub 8148 df-neg 8149 df-inn 8938 df-2 8996 df-3 8997 df-n0 9195 df-z 9272 df-uz 9547 df-fz 10027 df-struct 12482 df-ndx 12483 df-slot 12484 df-base 12486 df-plusg 12568 df-mulr 12569 |
This theorem is referenced by: rngbaseg 12613 rngplusgg 12614 rngmulrg 12615 srngstrd 12623 ipsstrd 12653 |
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