![]() |
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 | β’ βfld β oField |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | refld 21508 | . 2 β’ βfld β Field | |
2 | isfld 20596 | . . . . 5 β’ (βfld β Field β (βfld β DivRing β§ βfld β CRing)) | |
3 | 2 | simplbi 497 | . . . 4 β’ (βfld β Field β βfld β DivRing) |
4 | drngring 20592 | . . . 4 β’ (βfld β DivRing β βfld β Ring) | |
5 | 1, 3, 4 | mp2b 10 | . . 3 β’ βfld β Ring |
6 | ringgrp 20141 | . . . . 5 β’ (βfld β Ring β βfld β Grp) | |
7 | 5, 6 | ax-mp 5 | . . . 4 β’ βfld β Grp |
8 | grpmnd 18868 | . . . . . 6 β’ (βfld β Grp β βfld β Mnd) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 β’ βfld β Mnd |
10 | retos 21507 | . . . . 5 β’ βfld β Toset | |
11 | simpl 482 | . . . . . . . . . 10 β’ ((π β β β§ (π β β β§ π β β β§ π β€ π)) β π β β) | |
12 | simpr1 1191 | . . . . . . . . . 10 β’ ((π β β β§ (π β β β§ π β β β§ π β€ π)) β π β β) | |
13 | simpr2 1192 | . . . . . . . . . 10 β’ ((π β β β§ (π β β β§ π β β β§ π β€ π)) β π β β) | |
14 | simpr3 1193 | . . . . . . . . . 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 11829 | . . . . . . . . 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 1357 | . . . . . . . 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 412 | . . . . . . 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 1107 | . . . . . 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 21495 | . . . . . 6 β’ β = (Baseββfld) | |
21 | replusg 21499 | . . . . . 6 β’ + = (+gββfld) | |
22 | rele2 21503 | . . . . . 6 β’ β€ = (leββfld) | |
23 | 20, 21, 22 | isomnd 32723 | . . . . 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 1338 | . . . 4 β’ βfld β oMnd |
25 | isogrp 32724 | . . . 4 β’ (βfld β oGrp β (βfld β Grp β§ βfld β oMnd)) | |
26 | 7, 24, 25 | mpbir2an 708 | . . 3 β’ βfld β oGrp |
27 | mulge0 11733 | . . . . . 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 657 | . . . . 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 412 | . . . 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 21501 | . . . 4 β’ 0 = (0gββfld) | |
32 | remulr 21500 | . . . 4 β’ Β· = (.rββfld) | |
33 | 20, 31, 32, 22 | isorng 32920 | . . 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 1338 | . 2 β’ βfld β oRing |
35 | isofld 32923 | . 2 β’ (βfld β oField β (βfld β Field β§ βfld β oRing)) | |
36 | 1, 34, 35 | mpbir2an 708 | 1 β’ βfld β oField |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 β wcel 2098 βwral 3055 class class class wbr 5141 (class class class)co 7404 βcr 11108 0cc0 11109 + caddc 11112 Β· cmul 11114 β€ cle 11250 Tosetctos 18379 Mndcmnd 18665 Grpcgrp 18861 Ringcrg 20136 CRingccrg 20137 DivRingcdr 20585 Fieldcfield 20586 βfldcrefld 21493 oMndcomnd 32719 oGrpcogrp 32720 oRingcorng 32916 oFieldcofld 32917 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-addf 11188 ax-mulf 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-0g 17394 df-proset 18258 df-poset 18276 df-plt 18293 df-toset 18380 df-ps 18529 df-tsr 18530 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-subg 19048 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-oppr 20234 df-dvdsr 20257 df-unit 20258 df-invr 20288 df-dvr 20301 df-subrng 20444 df-subrg 20469 df-drng 20587 df-field 20588 df-cnfld 21237 df-refld 21494 df-omnd 32721 df-ogrp 32722 df-orng 32918 df-ofld 32919 |
This theorem is referenced by: nn0omnd 32963 rearchi 32964 rerrext 33519 cnrrext 33520 |
Copyright terms: Public domain | W3C validator |