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Mirrors > Home > MPE Home > Th. List > qrngdiv | Structured version Visualization version GIF version |
Description: The division operation in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.) |
Ref | Expression |
---|---|
qrng.q | β’ π = (βfld βΎs β) |
Ref | Expression |
---|---|
qrngdiv | β’ ((π β β β§ π β β β§ π β 0) β (π(/rβπ)π) = (π / π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qsubdrg 20756 | . . . 4 β’ (β β (SubRingββfld) β§ (βfld βΎs β) β DivRing) | |
2 | 1 | simpli 484 | . . 3 β’ β β (SubRingββfld) |
3 | simp1 1135 | . . 3 β’ ((π β β β§ π β β β§ π β 0) β π β β) | |
4 | 3simpc 1149 | . . . 4 β’ ((π β β β§ π β β β§ π β 0) β (π β β β§ π β 0)) | |
5 | eldifsn 4734 | . . . 4 β’ (π β (β β {0}) β (π β β β§ π β 0)) | |
6 | 4, 5 | sylibr 233 | . . 3 β’ ((π β β β§ π β β β§ π β 0) β π β (β β {0})) |
7 | qrng.q | . . . 4 β’ π = (βfld βΎs β) | |
8 | cnflddiv 20734 | . . . 4 β’ / = (/rββfld) | |
9 | 7 | qrngbas 26873 | . . . . 5 β’ β = (Baseβπ) |
10 | 7 | qrng0 26875 | . . . . 5 β’ 0 = (0gβπ) |
11 | 7 | qdrng 26874 | . . . . 5 β’ π β DivRing |
12 | 9, 10, 11 | drngui 20099 | . . . 4 β’ (β β {0}) = (Unitβπ) |
13 | eqid 2736 | . . . 4 β’ (/rβπ) = (/rβπ) | |
14 | 7, 8, 12, 13 | subrgdv 20146 | . . 3 β’ ((β β (SubRingββfld) β§ π β β β§ π β (β β {0})) β (π / π) = (π(/rβπ)π)) |
15 | 2, 3, 6, 14 | mp3an2i 1465 | . 2 β’ ((π β β β§ π β β β§ π β 0) β (π / π) = (π(/rβπ)π)) |
16 | 15 | eqcomd 2742 | 1 β’ ((π β β β§ π β β β§ π β 0) β (π(/rβπ)π) = (π / π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1086 = wceq 1540 β wcel 2105 β wne 2940 β cdif 3895 {csn 4573 βcfv 6479 (class class class)co 7337 0cc0 10972 / cdiv 11733 βcq 12789 βΎs cress 17038 /rcdvr 20019 DivRingcdr 20093 SubRingcsubrg 20125 βfldccnfld 20703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-addf 11051 ax-mulf 11052 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-tpos 8112 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-div 11734 df-nn 12075 df-2 12137 df-3 12138 df-4 12139 df-5 12140 df-6 12141 df-7 12142 df-8 12143 df-9 12144 df-n0 12335 df-z 12421 df-dec 12539 df-uz 12684 df-q 12790 df-fz 13341 df-struct 16945 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-mulr 17073 df-starv 17074 df-tset 17078 df-ple 17079 df-ds 17081 df-unif 17082 df-0g 17249 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-grp 18676 df-minusg 18677 df-subg 18848 df-cmn 19483 df-mgp 19816 df-ur 19833 df-ring 19880 df-cring 19881 df-oppr 19957 df-dvdsr 19978 df-unit 19979 df-invr 20009 df-dvr 20020 df-drng 20095 df-subrg 20127 df-cnfld 20704 |
This theorem is referenced by: ostthlem1 26881 |
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