| 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 21637 | . 2 ⊢ ℝfld ∈ Field | |
| 2 | isfld 20740 | . . . . 5 ⊢ (ℝfld ∈ Field ↔ (ℝfld ∈ DivRing ∧ ℝfld ∈ CRing)) | |
| 3 | 2 | simplbi 497 | . . . 4 ⊢ (ℝfld ∈ Field → ℝfld ∈ DivRing) |
| 4 | drngring 20736 | . . . 4 ⊢ (ℝfld ∈ DivRing → ℝfld ∈ Ring) | |
| 5 | 1, 3, 4 | mp2b 10 | . . 3 ⊢ ℝfld ∈ Ring |
| 6 | ringgrp 20235 | . . . . 5 ⊢ (ℝfld ∈ Ring → ℝfld ∈ Grp) | |
| 7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ ℝfld ∈ Grp |
| 8 | grpmnd 18958 | . . . . . 6 ⊢ (ℝfld ∈ Grp → ℝfld ∈ Mnd) | |
| 9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ ℝfld ∈ Mnd |
| 10 | retos 21636 | . . . . 5 ⊢ ℝfld ∈ Toset | |
| 11 | simpl 482 | . . . . . . . . . 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 11877 | . . . . . . . . 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 412 | . . . . . . 7 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))) |
| 18 | 17 | 3impa 1110 | . . . . . 6 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))) |
| 19 | 18 | rgen3 3204 | . . . . 5 ⊢ ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ∀𝑐 ∈ ℝ (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) |
| 20 | rebase 21624 | . . . . . 6 ⊢ ℝ = (Base‘ℝfld) | |
| 21 | replusg 21628 | . . . . . 6 ⊢ + = (+g‘ℝfld) | |
| 22 | rele2 21632 | . . . . . 6 ⊢ ≤ = (le‘ℝfld) | |
| 23 | 20, 21, 22 | isomnd 33078 | . . . . 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 33079 | . . . 4 ⊢ (ℝfld ∈ oGrp ↔ (ℝfld ∈ Grp ∧ ℝfld ∈ oMnd)) | |
| 26 | 7, 24, 25 | mpbir2an 711 | . . 3 ⊢ ℝfld ∈ oGrp |
| 27 | mulge0 11781 | . . . . . 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 412 | . . . 4 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → ((0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))) |
| 30 | 29 | rgen2 3199 | . . 3 ⊢ ∀𝑎 ∈ ℝ ∀𝑏 ∈ ℝ ((0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)) |
| 31 | re0g 21630 | . . . 4 ⊢ 0 = (0g‘ℝfld) | |
| 32 | remulr 21629 | . . . 4 ⊢ · = (.r‘ℝfld) | |
| 33 | 20, 31, 32, 22 | isorng 33329 | . . 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 33332 | . 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 395 ∧ w3a 1087 ∈ wcel 2108 ∀wral 3061 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 + caddc 11158 · cmul 11160 ≤ cle 11296 Tosetctos 18461 Mndcmnd 18747 Grpcgrp 18951 Ringcrg 20230 CRingccrg 20231 DivRingcdr 20729 Fieldcfield 20730 ℝfldcrefld 21622 oMndcomnd 33074 oGrpcogrp 33075 oRingcorng 33325 oFieldcofld 33326 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-proset 18340 df-poset 18359 df-plt 18375 df-toset 18462 df-ps 18611 df-tsr 18612 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-subrng 20546 df-subrg 20570 df-drng 20731 df-field 20732 df-cnfld 21365 df-refld 21623 df-omnd 33076 df-ogrp 33077 df-orng 33327 df-ofld 33328 |
| This theorem is referenced by: nn0omnd 33373 rearchi 33374 rerrext 34010 cnrrext 34011 |
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