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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 | ⊢ 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 33180 | . . . 4 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing)) | |
2 | 1 | simprbi 495 | . . 3 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing) |
3 | orng0le1.1 | . . . 4 ⊢ 0 = (0g‘𝐹) | |
4 | orng0le1.2 | . . . 4 ⊢ 1 = (1r‘𝐹) | |
5 | eqid 2726 | . . . 4 ⊢ (le‘𝐹) = (le‘𝐹) | |
6 | 3, 4, 5 | orng0le1 33190 | . . 3 ⊢ (𝐹 ∈ oRing → 0 (le‘𝐹) 1 ) |
7 | 2, 6 | syl 17 | . 2 ⊢ (𝐹 ∈ oField → 0 (le‘𝐹) 1 ) |
8 | ofldfld 33188 | . . . 4 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Field) | |
9 | isfld 20718 | . . . . 5 ⊢ (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing)) | |
10 | 9 | simplbi 496 | . . . 4 ⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing) |
11 | 3, 4 | drngunz 20725 | . . . 4 ⊢ (𝐹 ∈ DivRing → 1 ≠ 0 ) |
12 | 8, 10, 11 | 3syl 18 | . . 3 ⊢ (𝐹 ∈ oField → 1 ≠ 0 ) |
13 | 12 | necomd 2986 | . 2 ⊢ (𝐹 ∈ oField → 0 ≠ 1 ) |
14 | 3 | fvexi 6915 | . . 3 ⊢ 0 ∈ V |
15 | 4 | fvexi 6915 | . . 3 ⊢ 1 ∈ V |
16 | ofld0lt1.3 | . . . 4 ⊢ < = (lt‘𝐹) | |
17 | 5, 16 | pltval 18357 | . . 3 ⊢ ((𝐹 ∈ oField ∧ 0 ∈ V ∧ 1 ∈ V) → ( 0 < 1 ↔ ( 0 (le‘𝐹) 1 ∧ 0 ≠ 1 ))) |
18 | 14, 15, 17 | mp3an23 1450 | . 2 ⊢ (𝐹 ∈ oField → ( 0 < 1 ↔ ( 0 (le‘𝐹) 1 ∧ 0 ≠ 1 ))) |
19 | 7, 13, 18 | mpbir2and 711 | 1 ⊢ (𝐹 ∈ oField → 0 < 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 Vcvv 3462 class class class wbr 5153 ‘cfv 6554 lecple 17273 0gc0g 17454 ltcplt 18333 1rcur 20164 CRingccrg 20217 DivRingcdr 20707 Fieldcfield 20708 oRingcorng 33173 oFieldcofld 33174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-tpos 8241 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-plusg 17279 df-mulr 17280 df-0g 17456 df-proset 18320 df-poset 18338 df-plt 18355 df-toset 18442 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-minusg 18932 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-oppr 20316 df-dvdsr 20339 df-unit 20340 df-drng 20709 df-field 20710 df-omnd 32934 df-ogrp 32935 df-orng 33175 df-ofld 33176 |
This theorem is referenced by: ofldchr 33192 isarchiofld 33195 |
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