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Mirrors > Home > MPE Home > Th. List > rngstr | Structured version Visualization version GIF version |
Description: A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 29-Aug-2015.) |
Ref | Expression |
---|---|
rngfn.r | ⊢ 𝑅 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} |
Ref | Expression |
---|---|
rngstr | ⊢ 𝑅 Struct 〈1, 3〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngfn.r | . 2 ⊢ 𝑅 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} | |
2 | 1nn 11982 | . . 3 ⊢ 1 ∈ ℕ | |
3 | basendx 16917 | . . 3 ⊢ (Base‘ndx) = 1 | |
4 | 1lt2 12142 | . . 3 ⊢ 1 < 2 | |
5 | 2nn 12044 | . . 3 ⊢ 2 ∈ ℕ | |
6 | plusgndx 16984 | . . 3 ⊢ (+g‘ndx) = 2 | |
7 | 2lt3 12143 | . . 3 ⊢ 2 < 3 | |
8 | 3nn 12050 | . . 3 ⊢ 3 ∈ ℕ | |
9 | mulrndx 16999 | . . 3 ⊢ (.r‘ndx) = 3 | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | strle3 16857 | . 2 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} Struct 〈1, 3〉 |
11 | 1, 10 | eqbrtri 5100 | 1 ⊢ 𝑅 Struct 〈1, 3〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 {ctp 4571 〈cop 4573 class class class wbr 5079 ‘cfv 6431 1c1 10871 2c2 12026 3c3 12027 Struct cstr 16843 ndxcnx 16890 Basecbs 16908 +gcplusg 16958 .rcmulr 16959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8479 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-n0 12232 df-z 12318 df-uz 12580 df-fz 13237 df-struct 16844 df-slot 16879 df-ndx 16891 df-base 16909 df-plusg 16971 df-mulr 16972 |
This theorem is referenced by: rngbase 17005 rngplusg 17006 rngmulr 17007 srngstr 17015 ipsstr 17042 odrngstr 17109 psrvalstr 21115 idlsrgstr 31641 algstr 40997 |
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