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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 | β’ βfld β oField |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | refld 21039 | . 2 β’ βfld β Field | |
2 | isfld 20208 | . . . . 5 β’ (βfld β Field β (βfld β DivRing β§ βfld β CRing)) | |
3 | 2 | simplbi 499 | . . . 4 β’ (βfld β Field β βfld β DivRing) |
4 | drngring 20204 | . . . 4 β’ (βfld β DivRing β βfld β Ring) | |
5 | 1, 3, 4 | mp2b 10 | . . 3 β’ βfld β Ring |
6 | ringgrp 19974 | . . . . 5 β’ (βfld β Ring β βfld β Grp) | |
7 | 5, 6 | ax-mp 5 | . . . 4 β’ βfld β Grp |
8 | grpmnd 18760 | . . . . . 6 β’ (βfld β Grp β βfld β Mnd) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 β’ βfld β Mnd |
10 | retos 21038 | . . . . 5 β’ βfld β Toset | |
11 | simpl 484 | . . . . . . . . . 10 β’ ((π β β β§ (π β β β§ π β β β§ π β€ π)) β π β β) | |
12 | simpr1 1195 | . . . . . . . . . 10 β’ ((π β β β§ (π β β β§ π β β β§ π β€ π)) β π β β) | |
13 | simpr2 1196 | . . . . . . . . . 10 β’ ((π β β β§ (π β β β§ π β β β§ π β€ π)) β π β β) | |
14 | simpr3 1197 | . . . . . . . . . 10 β’ ((π β β β§ (π β β β§ π β β β§ π β€ π)) β π β€ π) | |
15 | 11, 12, 13, 14 | leadd1dd 11774 | . . . . . . . . 9 β’ ((π β β β§ (π β β β§ π β β β§ π β€ π)) β (π + π) β€ (π + π)) |
16 | 15 | 3anassrs 1361 | . . . . . . . 8 β’ ((((π β β β§ π β β) β§ π β β) β§ π β€ π) β (π + π) β€ (π + π)) |
17 | 16 | ex 414 | . . . . . . 7 β’ (((π β β β§ π β β) β§ π β β) β (π β€ π β (π + π) β€ (π + π))) |
18 | 17 | 3impa 1111 | . . . . . 6 β’ ((π β β β§ π β β β§ π β β) β (π β€ π β (π + π) β€ (π + π))) |
19 | 18 | rgen3 3196 | . . . . 5 β’ βπ β β βπ β β βπ β β (π β€ π β (π + π) β€ (π + π)) |
20 | rebase 21026 | . . . . . 6 β’ β = (Baseββfld) | |
21 | replusg 21030 | . . . . . 6 β’ + = (+gββfld) | |
22 | rele2 21034 | . . . . . 6 β’ β€ = (leββfld) | |
23 | 20, 21, 22 | isomnd 31958 | . . . . 5 β’ (βfld β oMnd β (βfld β Mnd β§ βfld β Toset β§ βπ β β βπ β β βπ β β (π β€ π β (π + π) β€ (π + π)))) |
24 | 9, 10, 19, 23 | mpbir3an 1342 | . . . 4 β’ βfld β oMnd |
25 | isogrp 31959 | . . . 4 β’ (βfld β oGrp β (βfld β Grp β§ βfld β oMnd)) | |
26 | 7, 24, 25 | mpbir2an 710 | . . 3 β’ βfld β oGrp |
27 | mulge0 11678 | . . . . . 6 β’ (((π β β β§ 0 β€ π) β§ (π β β β§ 0 β€ π)) β 0 β€ (π Β· π)) | |
28 | 27 | an4s 659 | . . . . 5 β’ (((π β β β§ π β β) β§ (0 β€ π β§ 0 β€ π)) β 0 β€ (π Β· π)) |
29 | 28 | ex 414 | . . . 4 β’ ((π β β β§ π β β) β ((0 β€ π β§ 0 β€ π) β 0 β€ (π Β· π))) |
30 | 29 | rgen2 3191 | . . 3 β’ βπ β β βπ β β ((0 β€ π β§ 0 β€ π) β 0 β€ (π Β· π)) |
31 | re0g 21032 | . . . 4 β’ 0 = (0gββfld) | |
32 | remulr 21031 | . . . 4 β’ Β· = (.rββfld) | |
33 | 20, 31, 32, 22 | isorng 32141 | . . 3 β’ (βfld β oRing β (βfld β Ring β§ βfld β oGrp β§ βπ β β βπ β β ((0 β€ π β§ 0 β€ π) β 0 β€ (π Β· π)))) |
34 | 5, 26, 30, 33 | mpbir3an 1342 | . 2 β’ βfld β oRing |
35 | isofld 32144 | . 2 β’ (βfld β oField β (βfld β Field β§ βfld β oRing)) | |
36 | 1, 34, 35 | mpbir2an 710 | 1 β’ βfld β oField |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 β wcel 2107 βwral 3061 class class class wbr 5106 (class class class)co 7358 βcr 11055 0cc0 11056 + caddc 11059 Β· cmul 11061 β€ cle 11195 Tosetctos 18310 Mndcmnd 18561 Grpcgrp 18753 Ringcrg 19969 CRingccrg 19970 DivRingcdr 20197 Fieldcfield 20198 βfldcrefld 21024 oMndcomnd 31954 oGrpcogrp 31955 oRingcorng 32137 oFieldcofld 32138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-addf 11135 ax-mulf 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-0g 17328 df-proset 18189 df-poset 18207 df-plt 18224 df-toset 18311 df-ps 18460 df-tsr 18461 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-subg 18930 df-cmn 19569 df-mgp 19902 df-ur 19919 df-ring 19971 df-cring 19972 df-oppr 20054 df-dvdsr 20075 df-unit 20076 df-invr 20106 df-dvr 20117 df-drng 20199 df-field 20200 df-subrg 20234 df-cnfld 20813 df-refld 21025 df-omnd 31956 df-ogrp 31957 df-orng 32139 df-ofld 32140 |
This theorem is referenced by: nn0omnd 32184 rearchi 32185 rerrext 32647 cnrrext 32648 |
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