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 20766 | . 2 ⊢ ℝfld ∈ Field | |
2 | isfld 19514 | . . . . 5 ⊢ (ℝfld ∈ Field ↔ (ℝfld ∈ DivRing ∧ ℝfld ∈ CRing)) | |
3 | 2 | simplbi 500 | . . . 4 ⊢ (ℝfld ∈ Field → ℝfld ∈ DivRing) |
4 | drngring 19512 | . . . 4 ⊢ (ℝfld ∈ DivRing → ℝfld ∈ Ring) | |
5 | 1, 3, 4 | mp2b 10 | . . 3 ⊢ ℝfld ∈ Ring |
6 | ringgrp 19305 | . . . . 5 ⊢ (ℝfld ∈ Ring → ℝfld ∈ Grp) | |
7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ ℝfld ∈ Grp |
8 | grpmnd 18113 | . . . . . 6 ⊢ (ℝfld ∈ Grp → ℝfld ∈ Mnd) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ ℝfld ∈ Mnd |
10 | retos 20765 | . . . . 5 ⊢ ℝfld ∈ Toset | |
11 | simpl 485 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → 𝑎 ∈ ℝ) | |
12 | simpr1 1190 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → 𝑏 ∈ ℝ) | |
13 | simpr2 1191 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → 𝑐 ∈ ℝ) | |
14 | simpr3 1192 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → 𝑎 ≤ 𝑏) | |
15 | 11, 12, 13, 14 | leadd1dd 11257 | . . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) |
16 | 15 | 3anassrs 1356 | . . . . . . . 8 ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) ∧ 𝑎 ≤ 𝑏) → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) |
17 | 16 | ex 415 | . . . . . . 7 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))) |
18 | 17 | 3impa 1106 | . . . . . 6 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))) |
19 | 18 | rgen3 3207 | . . . . 5 ⊢ ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ∀𝑐 ∈ ℝ (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) |
20 | rebase 20753 | . . . . . 6 ⊢ ℝ = (Base‘ℝfld) | |
21 | replusg 20757 | . . . . . 6 ⊢ + = (+g‘ℝfld) | |
22 | rele2 20761 | . . . . . 6 ⊢ ≤ = (le‘ℝfld) | |
23 | 20, 21, 22 | isomnd 30706 | . . . . 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 1337 | . . . 4 ⊢ ℝfld ∈ oMnd |
25 | isogrp 30707 | . . . 4 ⊢ (ℝfld ∈ oGrp ↔ (ℝfld ∈ Grp ∧ ℝfld ∈ oMnd)) | |
26 | 7, 24, 25 | mpbir2an 709 | . . 3 ⊢ ℝfld ∈ oGrp |
27 | mulge0 11161 | . . . . . 6 ⊢ (((𝑎 ∈ ℝ ∧ 0 ≤ 𝑎) ∧ (𝑏 ∈ ℝ ∧ 0 ≤ 𝑏)) → 0 ≤ (𝑎 · 𝑏)) | |
28 | 27 | an4s 658 | . . . . 5 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ (0 ≤ 𝑎 ∧ 0 ≤ 𝑏)) → 0 ≤ (𝑎 · 𝑏)) |
29 | 28 | ex 415 | . . . 4 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → ((0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))) |
30 | 29 | rgen2 3206 | . . 3 ⊢ ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)) |
31 | re0g 20759 | . . . 4 ⊢ 0 = (0g‘ℝfld) | |
32 | remulr 20758 | . . . 4 ⊢ · = (.r‘ℝfld) | |
33 | 20, 31, 32, 22 | isorng 30876 | . . 3 ⊢ (ℝfld ∈ oRing ↔ (ℝfld ∈ Ring ∧ ℝfld ∈ oGrp ∧ ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)))) |
34 | 5, 26, 30, 33 | mpbir3an 1337 | . 2 ⊢ ℝfld ∈ oRing |
35 | isofld 30879 | . 2 ⊢ (ℝfld ∈ oField ↔ (ℝfld ∈ Field ∧ ℝfld ∈ oRing)) | |
36 | 1, 34, 35 | mpbir2an 709 | 1 ⊢ ℝfld ∈ oField |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2113 ∀wral 3141 class class class wbr 5069 (class class class)co 7159 ℝcr 10539 0cc0 10540 + caddc 10543 · cmul 10545 ≤ cle 10679 Tosetctos 17646 Mndcmnd 17914 Grpcgrp 18106 Ringcrg 19300 CRingccrg 19301 DivRingcdr 19505 Fieldcfield 19506 ℝfldcrefld 20751 oMndcomnd 30702 oGrpcogrp 30703 oRingcorng 30872 oFieldcofld 30873 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-0g 16718 df-proset 17541 df-poset 17559 df-plt 17571 df-toset 17647 df-ps 17813 df-tsr 17814 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-subg 18279 df-cmn 18911 df-mgp 19243 df-ur 19255 df-ring 19302 df-cring 19303 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-dvr 19436 df-drng 19507 df-field 19508 df-subrg 19536 df-cnfld 20549 df-refld 20752 df-omnd 30704 df-ogrp 30705 df-orng 30874 df-ofld 30875 |
This theorem is referenced by: nn0omnd 30918 rearchi 30919 rerrext 31254 cnrrext 31255 |
Copyright terms: Public domain | W3C validator |