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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ofldlt1 | Structured version Visualization version GIF version |
Description: In an ordered field, the ring unity is strictly positive. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
Ref | Expression |
---|---|
orng0le1.1 | β’ 0 = (0gβπΉ) |
orng0le1.2 | β’ 1 = (1rβπΉ) |
ofld0lt1.3 | β’ < = (ltβπΉ) |
Ref | Expression |
---|---|
ofldlt1 | β’ (πΉ β oField β 0 < 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isofld 32151 | . . . 4 β’ (πΉ β oField β (πΉ β Field β§ πΉ β oRing)) | |
2 | 1 | simprbi 498 | . . 3 β’ (πΉ β oField β πΉ β oRing) |
3 | orng0le1.1 | . . . 4 β’ 0 = (0gβπΉ) | |
4 | orng0le1.2 | . . . 4 β’ 1 = (1rβπΉ) | |
5 | eqid 2733 | . . . 4 β’ (leβπΉ) = (leβπΉ) | |
6 | 3, 4, 5 | orng0le1 32161 | . . 3 β’ (πΉ β oRing β 0 (leβπΉ) 1 ) |
7 | 2, 6 | syl 17 | . 2 β’ (πΉ β oField β 0 (leβπΉ) 1 ) |
8 | ofldfld 32159 | . . . 4 β’ (πΉ β oField β πΉ β Field) | |
9 | isfld 20230 | . . . . 5 β’ (πΉ β Field β (πΉ β DivRing β§ πΉ β CRing)) | |
10 | 9 | simplbi 499 | . . . 4 β’ (πΉ β Field β πΉ β DivRing) |
11 | 3, 4 | drngunz 20237 | . . . 4 β’ (πΉ β DivRing β 1 β 0 ) |
12 | 8, 10, 11 | 3syl 18 | . . 3 β’ (πΉ β oField β 1 β 0 ) |
13 | 12 | necomd 2996 | . 2 β’ (πΉ β oField β 0 β 1 ) |
14 | 3 | fvexi 6860 | . . 3 β’ 0 β V |
15 | 4 | fvexi 6860 | . . 3 β’ 1 β V |
16 | ofld0lt1.3 | . . . 4 β’ < = (ltβπΉ) | |
17 | 5, 16 | pltval 18229 | . . 3 β’ ((πΉ β oField β§ 0 β V β§ 1 β V) β ( 0 < 1 β ( 0 (leβπΉ) 1 β§ 0 β 1 ))) |
18 | 14, 15, 17 | mp3an23 1454 | . 2 β’ (πΉ β oField β ( 0 < 1 β ( 0 (leβπΉ) 1 β§ 0 β 1 ))) |
19 | 7, 13, 18 | mpbir2and 712 | 1 β’ (πΉ β oField β 0 < 1 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2940 Vcvv 3447 class class class wbr 5109 βcfv 6500 lecple 17148 0gc0g 17329 ltcplt 18205 1rcur 19921 CRingccrg 19973 DivRingcdr 20219 Fieldcfield 20220 oRingcorng 32144 oFieldcofld 32145 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-mulr 17155 df-0g 17331 df-proset 18192 df-poset 18210 df-plt 18227 df-toset 18314 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-mgp 19905 df-ur 19922 df-ring 19974 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-drng 20221 df-field 20222 df-omnd 31963 df-ogrp 31964 df-orng 32146 df-ofld 32147 |
This theorem is referenced by: ofldchr 32163 isarchiofld 32166 |
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