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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 21558 | . 2 ⊢ ℝfld ∈ Field | |
2 | isfld 20642 | . . . . 5 ⊢ (ℝfld ∈ Field ↔ (ℝfld ∈ DivRing ∧ ℝfld ∈ CRing)) | |
3 | 2 | simplbi 496 | . . . 4 ⊢ (ℝfld ∈ Field → ℝfld ∈ DivRing) |
4 | drngring 20638 | . . . 4 ⊢ (ℝfld ∈ DivRing → ℝfld ∈ Ring) | |
5 | 1, 3, 4 | mp2b 10 | . . 3 ⊢ ℝfld ∈ Ring |
6 | ringgrp 20185 | . . . . 5 ⊢ (ℝfld ∈ Ring → ℝfld ∈ Grp) | |
7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ ℝfld ∈ Grp |
8 | grpmnd 18904 | . . . . . 6 ⊢ (ℝfld ∈ Grp → ℝfld ∈ Mnd) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ ℝfld ∈ Mnd |
10 | retos 21557 | . . . . 5 ⊢ ℝfld ∈ Toset | |
11 | simpl 481 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → 𝑎 ∈ ℝ) | |
12 | simpr1 1191 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → 𝑏 ∈ ℝ) | |
13 | simpr2 1192 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → 𝑐 ∈ ℝ) | |
14 | simpr3 1193 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → 𝑎 ≤ 𝑏) | |
15 | 11, 12, 13, 14 | leadd1dd 11866 | . . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑎 ≤ 𝑏)) → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) |
16 | 15 | 3anassrs 1357 | . . . . . . . 8 ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) ∧ 𝑎 ≤ 𝑏) → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) |
17 | 16 | ex 411 | . . . . . . 7 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))) |
18 | 17 | 3impa 1107 | . . . . . 6 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))) |
19 | 18 | rgen3 3200 | . . . . 5 ⊢ ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ∀𝑐 ∈ ℝ (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) |
20 | rebase 21545 | . . . . . 6 ⊢ ℝ = (Base‘ℝfld) | |
21 | replusg 21549 | . . . . . 6 ⊢ + = (+g‘ℝfld) | |
22 | rele2 21553 | . . . . . 6 ⊢ ≤ = (le‘ℝfld) | |
23 | 20, 21, 22 | isomnd 32802 | . . . . 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 1338 | . . . 4 ⊢ ℝfld ∈ oMnd |
25 | isogrp 32803 | . . . 4 ⊢ (ℝfld ∈ oGrp ↔ (ℝfld ∈ Grp ∧ ℝfld ∈ oMnd)) | |
26 | 7, 24, 25 | mpbir2an 709 | . . 3 ⊢ ℝfld ∈ oGrp |
27 | mulge0 11770 | . . . . . 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 411 | . . . 4 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → ((0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))) |
30 | 29 | rgen2 3195 | . . 3 ⊢ ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)) |
31 | re0g 21551 | . . . 4 ⊢ 0 = (0g‘ℝfld) | |
32 | remulr 21550 | . . . 4 ⊢ · = (.r‘ℝfld) | |
33 | 20, 31, 32, 22 | isorng 33038 | . . 3 ⊢ (ℝfld ∈ oRing ↔ (ℝfld ∈ Ring ∧ ℝfld ∈ oGrp ∧ ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)))) |
34 | 5, 26, 30, 33 | mpbir3an 1338 | . 2 ⊢ ℝfld ∈ oRing |
35 | isofld 33041 | . 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 394 ∧ w3a 1084 ∈ wcel 2098 ∀wral 3058 class class class wbr 5152 (class class class)co 7426 ℝcr 11145 0cc0 11146 + caddc 11149 · cmul 11151 ≤ cle 11287 Tosetctos 18415 Mndcmnd 18701 Grpcgrp 18897 Ringcrg 20180 CRingccrg 20181 DivRingcdr 20631 Fieldcfield 20632 ℝfldcrefld 21543 oMndcomnd 32798 oGrpcogrp 32799 oRingcorng 33034 oFieldcofld 33035 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-addf 11225 ax-mulf 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-0g 17430 df-proset 18294 df-poset 18312 df-plt 18329 df-toset 18416 df-ps 18565 df-tsr 18566 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-subg 19085 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-dvr 20347 df-subrng 20490 df-subrg 20515 df-drng 20633 df-field 20634 df-cnfld 21287 df-refld 21544 df-omnd 32800 df-ogrp 32801 df-orng 33036 df-ofld 33037 |
This theorem is referenced by: nn0omnd 33081 rearchi 33082 rerrext 33643 cnrrext 33644 |
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