Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > reofld | Structured version Visualization version GIF version |
Description: The real numbers form an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
Ref | Expression |
---|---|
reofld | ⊢ ℝfld ∈ oField |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | refld 20691 | . 2 ⊢ ℝfld ∈ Field | |
2 | isfld 19440 | . . . . 5 ⊢ (ℝfld ∈ Field ↔ (ℝfld ∈ DivRing ∧ ℝfld ∈ CRing)) | |
3 | 2 | simplbi 498 | . . . 4 ⊢ (ℝfld ∈ Field → ℝfld ∈ DivRing) |
4 | drngring 19438 | . . . 4 ⊢ (ℝfld ∈ DivRing → ℝfld ∈ Ring) | |
5 | 1, 3, 4 | mp2b 10 | . . 3 ⊢ ℝfld ∈ Ring |
6 | ringgrp 19231 | . . . . 5 ⊢ (ℝfld ∈ Ring → ℝfld ∈ Grp) | |
7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ ℝfld ∈ Grp |
8 | grpmnd 18048 | . . . . . 6 ⊢ (ℝfld ∈ Grp → ℝfld ∈ Mnd) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ ℝfld ∈ Mnd |
10 | retos 20690 | . . . . 5 ⊢ ℝfld ∈ Toset | |
11 | simpl 483 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → 𝑎 ∈ ℝ) | |
12 | simpr1 1186 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → 𝑏 ∈ ℝ) | |
13 | simpr2 1187 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → 𝑐 ∈ ℝ) | |
14 | simpr3 1188 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → 𝑎 ≤ 𝑏) | |
15 | 11, 12, 13, 14 | leadd1dd 11242 | . . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) |
16 | 15 | 3anassrs 1352 | . . . . . . . 8 ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) ∧ 𝑎 ≤ 𝑏) → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) |
17 | 16 | ex 413 | . . . . . . 7 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))) |
18 | 17 | 3impa 1102 | . . . . . 6 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))) |
19 | 18 | rgen3 3201 | . . . . 5 ⊢ ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ∀𝑐 ∈ ℝ (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) |
20 | rebase 20678 | . . . . . 6 ⊢ ℝ = (Base‘ℝfld) | |
21 | replusg 20682 | . . . . . 6 ⊢ + = (+g‘ℝfld) | |
22 | rele2 20686 | . . . . . 6 ⊢ ≤ = (le‘ℝfld) | |
23 | 20, 21, 22 | isomnd 30629 | . . . . 5 ⊢ (ℝfld ∈ oMnd ↔ (ℝfld ∈ Mnd ∧ ℝfld ∈ Toset ∧ ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ∀𝑐 ∈ ℝ (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))) |
24 | 9, 10, 19, 23 | mpbir3an 1333 | . . . 4 ⊢ ℝfld ∈ oMnd |
25 | isogrp 30630 | . . . 4 ⊢ (ℝfld ∈ oGrp ↔ (ℝfld ∈ Grp ∧ ℝfld ∈ oMnd)) | |
26 | 7, 24, 25 | mpbir2an 707 | . . 3 ⊢ ℝfld ∈ oGrp |
27 | mulge0 11146 | . . . . . 6 ⊢ (((𝑎 ∈ ℝ ∧ 0 ≤ 𝑎) ∧ (𝑏 ∈ ℝ ∧ 0 ≤ 𝑏)) → 0 ≤ (𝑎 · 𝑏)) | |
28 | 27 | an4s 656 | . . . . 5 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ (0 ≤ 𝑎 ∧ 0 ≤ 𝑏)) → 0 ≤ (𝑎 · 𝑏)) |
29 | 28 | ex 413 | . . . 4 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → ((0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))) |
30 | 29 | rgen2 3200 | . . 3 ⊢ ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)) |
31 | re0g 20684 | . . . 4 ⊢ 0 = (0g‘ℝfld) | |
32 | remulr 20683 | . . . 4 ⊢ · = (.r‘ℝfld) | |
33 | 20, 31, 32, 22 | isorng 30799 | . . 3 ⊢ (ℝfld ∈ oRing ↔ (ℝfld ∈ Ring ∧ ℝfld ∈ oGrp ∧ ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)))) |
34 | 5, 26, 30, 33 | mpbir3an 1333 | . 2 ⊢ ℝfld ∈ oRing |
35 | isofld 30802 | . 2 ⊢ (ℝfld ∈ oField ↔ (ℝfld ∈ Field ∧ ℝfld ∈ oRing)) | |
36 | 1, 34, 35 | mpbir2an 707 | 1 ⊢ ℝfld ∈ oField |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 ∈ wcel 2105 ∀wral 3135 class class class wbr 5057 (class class class)co 7145 ℝcr 10524 0cc0 10525 + caddc 10528 · cmul 10530 ≤ cle 10664 Tosetctos 17631 Mndcmnd 17899 Grpcgrp 18041 Ringcrg 19226 CRingccrg 19227 DivRingcdr 19431 Fieldcfield 19432 ℝfldcrefld 20676 oMndcomnd 30625 oGrpcogrp 30626 oRingcorng 30795 oFieldcofld 30796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-proset 17526 df-poset 17544 df-plt 17556 df-toset 17632 df-ps 17798 df-tsr 17799 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-subg 18214 df-cmn 18837 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-field 19434 df-subrg 19462 df-cnfld 20474 df-refld 20677 df-omnd 30627 df-ogrp 30628 df-orng 30797 df-ofld 30798 |
This theorem is referenced by: nn0omnd 30841 rearchi 30842 rerrext 31149 cnrrext 31150 |
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