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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 11326 | . . 3 ⊢ 1 ∈ ℕ | |
3 | basendx 16247 | . . 3 ⊢ (Base‘ndx) = 1 | |
4 | 1lt2 11490 | . . 3 ⊢ 1 < 2 | |
5 | 2nn 11385 | . . 3 ⊢ 2 ∈ ℕ | |
6 | plusgndx 16296 | . . 3 ⊢ (+g‘ndx) = 2 | |
7 | 2lt3 11491 | . . 3 ⊢ 2 < 3 | |
8 | 3nn 11391 | . . 3 ⊢ 3 ∈ ℕ | |
9 | mulrndx 16316 | . . 3 ⊢ (.r‘ndx) = 3 | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | strle3 16295 | . 2 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} Struct 〈1, 3〉 |
11 | 1, 10 | eqbrtri 4865 | 1 ⊢ 𝑅 Struct 〈1, 3〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 {ctp 4373 〈cop 4375 class class class wbr 4844 ‘cfv 6102 1c1 10226 2c2 11367 3c3 11368 Struct cstr 16179 ndxcnx 16180 Basecbs 16183 +gcplusg 16266 .rcmulr 16267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-int 4669 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-om 7301 df-1st 7402 df-2nd 7403 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-1o 7800 df-oadd 7804 df-er 7983 df-en 8197 df-dom 8198 df-sdom 8199 df-fin 8200 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-nn 11314 df-2 11375 df-3 11376 df-n0 11580 df-z 11666 df-uz 11930 df-fz 12580 df-struct 16185 df-ndx 16186 df-slot 16187 df-base 16189 df-plusg 16279 df-mulr 16280 |
This theorem is referenced by: rngbase 16321 rngplusg 16322 rngmulr 16323 srngfn 16328 ipsstr 16344 odrngstr 16380 psrvalstr 19685 algstr 38527 |
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