![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2737 | . . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} | |
3 | 2 | rngstr 17186 | . . 3 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} Struct ⟨1, 3⟩ |
4 | 9nn 12258 | . . . 4 ⊢ 9 ∈ ℕ | |
5 | tsetndx 17240 | . . . 4 ⊢ (TopSet‘ndx) = 9 | |
6 | 9lt10 12756 | . . . 4 ⊢ 9 < ;10 | |
7 | 10nn 12641 | . . . 4 ⊢ ;10 ∈ ℕ | |
8 | plendx 17254 | . . . 4 ⊢ (le‘ndx) = ;10 | |
9 | 1nn0 12436 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
10 | 0nn0 12435 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
11 | 2nn 12233 | . . . . 5 ⊢ 2 ∈ ℕ | |
12 | 2pos 12263 | . . . . 5 ⊢ 0 < 2 | |
13 | 9, 10, 11, 12 | declt 12653 | . . . 4 ⊢ ;10 < ;12 |
14 | 9, 11 | decnncl 12645 | . . . 4 ⊢ ;12 ∈ ℕ |
15 | dsndx 17273 | . . . 4 ⊢ (dist‘ndx) = ;12 | |
16 | 4, 5, 6, 7, 8, 13, 14, 15 | strle3 17039 | . . 3 ⊢ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩} Struct ⟨9, ;12⟩ |
17 | 3lt9 12364 | . . 3 ⊢ 3 < 9 | |
18 | 3, 16, 17 | strleun 17036 | . 2 ⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩}) Struct ⟨1, ;12⟩ |
19 | 1, 18 | eqbrtri 5131 | 1 ⊢ 𝑊 Struct ⟨1, ;12⟩ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∪ cun 3913 {ctp 4595 ⟨cop 4597 class class class wbr 5110 ‘cfv 6501 0cc0 11058 1c1 11059 2c2 12215 3c3 12216 9c9 12222 ;cdc 12625 Struct cstr 17025 ndxcnx 17072 Basecbs 17090 +gcplusg 17140 .rcmulr 17141 TopSetcts 17146 lecple 17147 distcds 17149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-fz 13432 df-struct 17026 df-slot 17061 df-ndx 17073 df-base 17091 df-plusg 17153 df-mulr 17154 df-tset 17159 df-ple 17160 df-ds 17162 |
This theorem is referenced by: odrngbas 17292 odrngplusg 17293 odrngmulr 17294 odrngtset 17295 odrngle 17296 odrngds 17297 xrsstr 20827 |
Copyright terms: Public domain | W3C validator |