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 29802 | . . . 4 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing)) | |
2 | 1 | simprbi 480 | . . 3 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing) |
3 | orng0le1.1 | . . . 4 ⊢ 0 = (0g‘𝐹) | |
4 | orng0le1.2 | . . . 4 ⊢ 1 = (1r‘𝐹) | |
5 | eqid 2622 | . . . 4 ⊢ (le‘𝐹) = (le‘𝐹) | |
6 | 3, 4, 5 | orng0le1 29812 | . . 3 ⊢ (𝐹 ∈ oRing → 0 (le‘𝐹) 1 ) |
7 | 2, 6 | syl 17 | . 2 ⊢ (𝐹 ∈ oField → 0 (le‘𝐹) 1 ) |
8 | ofldfld 29810 | . . . 4 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Field) | |
9 | isfld 18756 | . . . . 5 ⊢ (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing)) | |
10 | 9 | simplbi 476 | . . . 4 ⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing) |
11 | 3, 4 | drngunz 18762 | . . . 4 ⊢ (𝐹 ∈ DivRing → 1 ≠ 0 ) |
12 | 8, 10, 11 | 3syl 18 | . . 3 ⊢ (𝐹 ∈ oField → 1 ≠ 0 ) |
13 | 12 | necomd 2849 | . 2 ⊢ (𝐹 ∈ oField → 0 ≠ 1 ) |
14 | fvex 6201 | . . . 4 ⊢ (0g‘𝐹) ∈ V | |
15 | 3, 14 | eqeltri 2697 | . . 3 ⊢ 0 ∈ V |
16 | fvex 6201 | . . . 4 ⊢ (1r‘𝐹) ∈ V | |
17 | 4, 16 | eqeltri 2697 | . . 3 ⊢ 1 ∈ V |
18 | ofld0lt1.3 | . . . 4 ⊢ < = (lt‘𝐹) | |
19 | 5, 18 | pltval 16960 | . . 3 ⊢ ((𝐹 ∈ oField ∧ 0 ∈ V ∧ 1 ∈ V) → ( 0 < 1 ↔ ( 0 (le‘𝐹) 1 ∧ 0 ≠ 1 ))) |
20 | 15, 17, 19 | mp3an23 1416 | . 2 ⊢ (𝐹 ∈ oField → ( 0 < 1 ↔ ( 0 (le‘𝐹) 1 ∧ 0 ≠ 1 ))) |
21 | 7, 13, 20 | mpbir2and 957 | 1 ⊢ (𝐹 ∈ oField → 0 < 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 class class class wbr 4653 ‘cfv 5888 lecple 15948 0gc0g 16100 ltcplt 16941 1rcur 18501 CRingccrg 18548 DivRingcdr 18747 Fieldcfield 18748 oRingcorng 29795 oFieldcofld 29796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-mulr 15955 df-0g 16102 df-preset 16928 df-poset 16946 df-plt 16958 df-toset 17034 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-drng 18749 df-field 18750 df-omnd 29699 df-ogrp 29700 df-orng 29797 df-ofld 29798 |
This theorem is referenced by: ofldchr 29814 isarchiofld 29817 |
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