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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 29133 | . . . 4 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing)) | |
2 | 1 | simprbi 479 | . . 3 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing) |
3 | orng0le1.1 | . . . 4 ⊢ 0 = (0g‘𝐹) | |
4 | orng0le1.2 | . . . 4 ⊢ 1 = (1r‘𝐹) | |
5 | eqid 2610 | . . . 4 ⊢ (le‘𝐹) = (le‘𝐹) | |
6 | 3, 4, 5 | orng0le1 29143 | . . 3 ⊢ (𝐹 ∈ oRing → 0 (le‘𝐹) 1 ) |
7 | 2, 6 | syl 17 | . 2 ⊢ (𝐹 ∈ oField → 0 (le‘𝐹) 1 ) |
8 | ofldfld 29141 | . . . 4 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Field) | |
9 | isfld 18579 | . . . . 5 ⊢ (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing)) | |
10 | 9 | simplbi 475 | . . . 4 ⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing) |
11 | 3, 4 | drngunz 18585 | . . . 4 ⊢ (𝐹 ∈ DivRing → 1 ≠ 0 ) |
12 | 8, 10, 11 | 3syl 18 | . . 3 ⊢ (𝐹 ∈ oField → 1 ≠ 0 ) |
13 | 12 | necomd 2837 | . 2 ⊢ (𝐹 ∈ oField → 0 ≠ 1 ) |
14 | fvex 6113 | . . . 4 ⊢ (0g‘𝐹) ∈ V | |
15 | 3, 14 | eqeltri 2684 | . . 3 ⊢ 0 ∈ V |
16 | fvex 6113 | . . . 4 ⊢ (1r‘𝐹) ∈ V | |
17 | 4, 16 | eqeltri 2684 | . . 3 ⊢ 1 ∈ V |
18 | ofld0lt1.3 | . . . 4 ⊢ < = (lt‘𝐹) | |
19 | 5, 18 | pltval 16783 | . . 3 ⊢ ((𝐹 ∈ oField ∧ 0 ∈ V ∧ 1 ∈ V) → ( 0 < 1 ↔ ( 0 (le‘𝐹) 1 ∧ 0 ≠ 1 ))) |
20 | 15, 17, 19 | mp3an23 1408 | . 2 ⊢ (𝐹 ∈ oField → ( 0 < 1 ↔ ( 0 (le‘𝐹) 1 ∧ 0 ≠ 1 ))) |
21 | 7, 13, 20 | mpbir2and 959 | 1 ⊢ (𝐹 ∈ oField → 0 < 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 class class class wbr 4583 ‘cfv 5804 lecple 15775 0gc0g 15923 ltcplt 16764 1rcur 18324 CRingccrg 18371 DivRingcdr 18570 Fieldcfield 18571 oRingcorng 29126 oFieldcofld 29127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-mulr 15782 df-0g 15925 df-preset 16751 df-poset 16769 df-plt 16781 df-toset 16857 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-drng 18572 df-field 18573 df-omnd 29030 df-ogrp 29031 df-orng 29128 df-ofld 29129 |
This theorem is referenced by: ofldchr 29145 isarchiofld 29148 |
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