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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 unit is strictly positive. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
Ref | Expression |
---|---|
orng0le1.1 | ⊢ 0 = (0g‘𝐹) |
orng0le1.2 | ⊢ 1 = (1r‘𝐹) |
ofld0lt1.3 | ⊢ < = (lt‘𝐹) |
Ref | Expression |
---|---|
ofldlt1 | ⊢ (𝐹 ∈ oField → 0 < 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isofld 30142 | . . . 4 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing)) | |
2 | 1 | simprbi 484 | . . 3 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing) |
3 | orng0le1.1 | . . . 4 ⊢ 0 = (0g‘𝐹) | |
4 | orng0le1.2 | . . . 4 ⊢ 1 = (1r‘𝐹) | |
5 | eqid 2771 | . . . 4 ⊢ (le‘𝐹) = (le‘𝐹) | |
6 | 3, 4, 5 | orng0le1 30152 | . . 3 ⊢ (𝐹 ∈ oRing → 0 (le‘𝐹) 1 ) |
7 | 2, 6 | syl 17 | . 2 ⊢ (𝐹 ∈ oField → 0 (le‘𝐹) 1 ) |
8 | ofldfld 30150 | . . . 4 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Field) | |
9 | isfld 18966 | . . . . 5 ⊢ (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing)) | |
10 | 9 | simplbi 485 | . . . 4 ⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing) |
11 | 3, 4 | drngunz 18972 | . . . 4 ⊢ (𝐹 ∈ DivRing → 1 ≠ 0 ) |
12 | 8, 10, 11 | 3syl 18 | . . 3 ⊢ (𝐹 ∈ oField → 1 ≠ 0 ) |
13 | 12 | necomd 2998 | . 2 ⊢ (𝐹 ∈ oField → 0 ≠ 1 ) |
14 | fvex 6342 | . . . 4 ⊢ (0g‘𝐹) ∈ V | |
15 | 3, 14 | eqeltri 2846 | . . 3 ⊢ 0 ∈ V |
16 | fvex 6342 | . . . 4 ⊢ (1r‘𝐹) ∈ V | |
17 | 4, 16 | eqeltri 2846 | . . 3 ⊢ 1 ∈ V |
18 | ofld0lt1.3 | . . . 4 ⊢ < = (lt‘𝐹) | |
19 | 5, 18 | pltval 17168 | . . 3 ⊢ ((𝐹 ∈ oField ∧ 0 ∈ V ∧ 1 ∈ V) → ( 0 < 1 ↔ ( 0 (le‘𝐹) 1 ∧ 0 ≠ 1 ))) |
20 | 15, 17, 19 | mp3an23 1564 | . 2 ⊢ (𝐹 ∈ oField → ( 0 < 1 ↔ ( 0 (le‘𝐹) 1 ∧ 0 ≠ 1 ))) |
21 | 7, 13, 20 | mpbir2and 692 | 1 ⊢ (𝐹 ∈ oField → 0 < 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 Vcvv 3351 class class class wbr 4786 ‘cfv 6031 lecple 16156 0gc0g 16308 ltcplt 17149 1rcur 18709 CRingccrg 18756 DivRingcdr 18957 Fieldcfield 18958 oRingcorng 30135 oFieldcofld 30136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-tpos 7504 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-plusg 16162 df-mulr 16163 df-0g 16310 df-preset 17136 df-poset 17154 df-plt 17166 df-toset 17242 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-grp 17633 df-minusg 17634 df-mgp 18698 df-ur 18710 df-ring 18757 df-oppr 18831 df-dvdsr 18849 df-unit 18850 df-drng 18959 df-field 18960 df-omnd 30039 df-ogrp 30040 df-orng 30137 df-ofld 30138 |
This theorem is referenced by: ofldchr 30154 isarchiofld 30157 |
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