Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 20435 | . 2 ⊢ ℝfld ∈ Field | |
2 | isfld 19630 | . . . . 5 ⊢ (ℝfld ∈ Field ↔ (ℝfld ∈ DivRing ∧ ℝfld ∈ CRing)) | |
3 | 2 | simplbi 501 | . . . 4 ⊢ (ℝfld ∈ Field → ℝfld ∈ DivRing) |
4 | drngring 19628 | . . . 4 ⊢ (ℝfld ∈ DivRing → ℝfld ∈ Ring) | |
5 | 1, 3, 4 | mp2b 10 | . . 3 ⊢ ℝfld ∈ Ring |
6 | ringgrp 19421 | . . . . 5 ⊢ (ℝfld ∈ Ring → ℝfld ∈ Grp) | |
7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ ℝfld ∈ Grp |
8 | grpmnd 18226 | . . . . . 6 ⊢ (ℝfld ∈ Grp → ℝfld ∈ Mnd) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ ℝfld ∈ Mnd |
10 | retos 20434 | . . . . 5 ⊢ ℝfld ∈ Toset | |
11 | simpl 486 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → 𝑎 ∈ ℝ) | |
12 | simpr1 1195 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → 𝑏 ∈ ℝ) | |
13 | simpr2 1196 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → 𝑐 ∈ ℝ) | |
14 | simpr3 1197 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → 𝑎 ≤ 𝑏) | |
15 | 11, 12, 13, 14 | leadd1dd 11332 | . . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) |
16 | 15 | 3anassrs 1361 | . . . . . . . 8 ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) ∧ 𝑎 ≤ 𝑏) → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) |
17 | 16 | ex 416 | . . . . . . 7 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))) |
18 | 17 | 3impa 1111 | . . . . . 6 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))) |
19 | 18 | rgen3 3116 | . . . . 5 ⊢ ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ∀𝑐 ∈ ℝ (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) |
20 | rebase 20422 | . . . . . 6 ⊢ ℝ = (Base‘ℝfld) | |
21 | replusg 20426 | . . . . . 6 ⊢ + = (+g‘ℝfld) | |
22 | rele2 20430 | . . . . . 6 ⊢ ≤ = (le‘ℝfld) | |
23 | 20, 21, 22 | isomnd 30904 | . . . . 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 1342 | . . . 4 ⊢ ℝfld ∈ oMnd |
25 | isogrp 30905 | . . . 4 ⊢ (ℝfld ∈ oGrp ↔ (ℝfld ∈ Grp ∧ ℝfld ∈ oMnd)) | |
26 | 7, 24, 25 | mpbir2an 711 | . . 3 ⊢ ℝfld ∈ oGrp |
27 | mulge0 11236 | . . . . . 6 ⊢ (((𝑎 ∈ ℝ ∧ 0 ≤ 𝑎) ∧ (𝑏 ∈ ℝ ∧ 0 ≤ 𝑏)) → 0 ≤ (𝑎 · 𝑏)) | |
28 | 27 | an4s 660 | . . . . 5 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ (0 ≤ 𝑎 ∧ 0 ≤ 𝑏)) → 0 ≤ (𝑎 · 𝑏)) |
29 | 28 | ex 416 | . . . 4 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → ((0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))) |
30 | 29 | rgen2 3115 | . . 3 ⊢ ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)) |
31 | re0g 20428 | . . . 4 ⊢ 0 = (0g‘ℝfld) | |
32 | remulr 20427 | . . . 4 ⊢ · = (.r‘ℝfld) | |
33 | 20, 31, 32, 22 | isorng 31075 | . . 3 ⊢ (ℝfld ∈ oRing ↔ (ℝfld ∈ Ring ∧ ℝfld ∈ oGrp ∧ ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)))) |
34 | 5, 26, 30, 33 | mpbir3an 1342 | . 2 ⊢ ℝfld ∈ oRing |
35 | isofld 31078 | . 2 ⊢ (ℝfld ∈ oField ↔ (ℝfld ∈ Field ∧ ℝfld ∈ oRing)) | |
36 | 1, 34, 35 | mpbir2an 711 | 1 ⊢ ℝfld ∈ oField |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 ∈ wcel 2114 ∀wral 3053 class class class wbr 5030 (class class class)co 7170 ℝcr 10614 0cc0 10615 + caddc 10618 · cmul 10620 ≤ cle 10754 Tosetctos 17759 Mndcmnd 18027 Grpcgrp 18219 Ringcrg 19416 CRingccrg 19417 DivRingcdr 19621 Fieldcfield 19622 ℝfldcrefld 20420 oMndcomnd 30900 oGrpcogrp 30901 oRingcorng 31071 oFieldcofld 31072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-addf 10694 ax-mulf 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-tpos 7921 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-starv 16683 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-0g 16818 df-proset 17654 df-poset 17672 df-plt 17684 df-toset 17760 df-ps 17926 df-tsr 17927 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-grp 18222 df-minusg 18223 df-subg 18394 df-cmn 19026 df-mgp 19359 df-ur 19371 df-ring 19418 df-cring 19419 df-oppr 19495 df-dvdsr 19513 df-unit 19514 df-invr 19544 df-dvr 19555 df-drng 19623 df-field 19624 df-subrg 19652 df-cnfld 20218 df-refld 20421 df-omnd 30902 df-ogrp 30903 df-orng 31073 df-ofld 31074 |
This theorem is referenced by: nn0omnd 31117 rearchi 31118 rerrext 31529 cnrrext 31530 |
Copyright terms: Public domain | W3C validator |