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Mirrors > Home > MPE Home > Th. List > odrngstr | Structured version Visualization version GIF version |
Description: Functionality of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.) (Proof shortened by AV, 15-Sep-2021.) |
Ref | Expression |
---|---|
odrngstr.w | ⊢ 𝑊 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(TopSet‘ndx), 𝐽〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), 𝐷〉}) |
Ref | Expression |
---|---|
odrngstr | ⊢ 𝑊 Struct 〈1, ;12〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | odrngstr.w | . 2 ⊢ 𝑊 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(TopSet‘ndx), 𝐽〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), 𝐷〉}) | |
2 | eqid 2824 | . . . 4 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} | |
3 | 2 | rngstr 16358 | . . 3 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} Struct 〈1, 3〉 |
4 | 9nn 11454 | . . . 4 ⊢ 9 ∈ ℕ | |
5 | tsetndx 16398 | . . . 4 ⊢ (TopSet‘ndx) = 9 | |
6 | 9lt10 11953 | . . . 4 ⊢ 9 < ;10 | |
7 | 10nn 11836 | . . . 4 ⊢ ;10 ∈ ℕ | |
8 | plendx 16405 | . . . 4 ⊢ (le‘ndx) = ;10 | |
9 | 1nn0 11635 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
10 | 0nn0 11634 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
11 | 2nn 11423 | . . . . 5 ⊢ 2 ∈ ℕ | |
12 | 2pos 11460 | . . . . 5 ⊢ 0 < 2 | |
13 | 9, 10, 11, 12 | declt 11849 | . . . 4 ⊢ ;10 < ;12 |
14 | 9, 11 | decnncl 11841 | . . . 4 ⊢ ;12 ∈ ℕ |
15 | dsndx 16414 | . . . 4 ⊢ (dist‘ndx) = ;12 | |
16 | 4, 5, 6, 7, 8, 13, 14, 15 | strle3 16333 | . . 3 ⊢ {〈(TopSet‘ndx), 𝐽〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), 𝐷〉} Struct 〈9, ;12〉 |
17 | 3lt9 11561 | . . 3 ⊢ 3 < 9 | |
18 | 3, 16, 17 | strleun 16330 | . 2 ⊢ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(TopSet‘ndx), 𝐽〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), 𝐷〉}) Struct 〈1, ;12〉 |
19 | 1, 18 | eqbrtri 4893 | 1 ⊢ 𝑊 Struct 〈1, ;12〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∪ cun 3795 {ctp 4400 〈cop 4402 class class class wbr 4872 ‘cfv 6122 0cc0 10251 1c1 10252 2c2 11405 3c3 11406 9c9 11412 ;cdc 11820 Struct cstr 16217 ndxcnx 16218 Basecbs 16221 +gcplusg 16304 .rcmulr 16305 TopSetcts 16310 lecple 16311 distcds 16313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-oadd 7829 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-nn 11350 df-2 11413 df-3 11414 df-4 11415 df-5 11416 df-6 11417 df-7 11418 df-8 11419 df-9 11420 df-n0 11618 df-z 11704 df-dec 11821 df-uz 11968 df-fz 12619 df-struct 16223 df-ndx 16224 df-slot 16225 df-base 16227 df-plusg 16317 df-mulr 16318 df-tset 16323 df-ple 16324 df-ds 16326 |
This theorem is referenced by: odrngbas 16419 odrngplusg 16420 odrngmulr 16421 odrngtset 16422 odrngle 16423 odrngds 16424 xrsstr 20119 |
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