MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  abvdiv Structured version   Visualization version   Unicode version
Theorem abvdiv 18837
Description: The absolute value distributes under division. (Contributed by Mario Carneiro, 10-Sep-2014.)
Hypotheses
Ref Expression
abv0.a |- A = (AbsVal ` R )
abvneg.b |- B = ( Base ` R )
abvrec.z |- .0. = ( 0g ` R )
abvdiv.p |- ./ = (/r ` R )
Assertion
Ref Expression
abvdiv |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> ( F ` ( X ./ Y ) ) = ( ( F ` X ) / ( F ` Y ) ) )
Proof of Theorem abvdiv
StepHypRef Expression
1 simplr 792 . . . 4 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> F e. A )
2 simpr1 1067 . . . 4 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> X e. B )
3 simpll 790 . . . . 5 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> R e. DivRing )
4 simpr2 1068 . . . . 5 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> Y e. B )
5 simpr3 1069 . . . . 5 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> Y =/= .0. )
6 abvneg.b . . . . . 6 |- B = ( Base ` R )
7 abvrec.z . . . . . 6 |- .0. = ( 0g ` R )
8 eqid 2622 . . . . . 6 |- ( invr ` R ) = ( invr ` R )
96, 7, 8drnginvrcl 18764 . . . . 5 |- ( ( R e. DivRing /\ Y e. B /\ Y =/= .0. ) -> ( ( invr ` R ) ` Y ) e. B )
103, 4, 5, 9syl3anc 1326 . . . 4 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> ( ( invr ` R ) ` Y ) e. B )
11 abv0.a . . . . 5 |- A = (AbsVal ` R )
12 eqid 2622 . . . . 5 |- ( .r ` R ) = ( .r ` R )
1311, 6, 12abvmul 18829 . . . 4 |- ( ( F e. A /\ X e. B /\ ( ( invr ` R ) ` Y ) e. B ) -> ( F ` ( X ( .r ` R ) ( ( invr ` R ) ` Y ) ) ) = ( ( F ` X ) x. ( F ` ( ( invr ` R ) ` Y ) ) ) )
141, 2, 10, 13syl3anc 1326 . . 3 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> ( F ` ( X ( .r ` R ) ( ( invr ` R ) ` Y ) ) ) = ( ( F ` X ) x. ( F ` ( ( invr ` R ) ` Y ) ) ) )
1511, 6, 7, 8abvrec 18836 . . . . 5 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( F ` ( ( invr ` R ) ` Y ) ) = ( 1 / ( F ` Y ) ) )
16153adantr1 1220 . . . 4 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> ( F ` ( ( invr ` R ) ` Y ) ) = ( 1 / ( F ` Y ) ) )
1716oveq2d 6666 . . 3 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> ( ( F ` X ) x. ( F ` ( ( invr ` R ) ` Y ) ) ) = ( ( F ` X ) x. ( 1 / ( F ` Y ) ) ) )
1814, 17eqtrd 2656 . 2 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> ( F ` ( X ( .r ` R ) ( ( invr ` R ) ` Y ) ) ) = ( ( F ` X ) x. ( 1 / ( F ` Y ) ) ) )
19 eqid 2622 . . . . . . 7 |- (Unit ` R ) = (Unit ` R )
206, 19, 7drngunit 18752 . . . . . 6 |- ( R e. DivRing -> ( Y e. (Unit ` R ) <-> ( Y e. B /\ Y =/= .0. ) ) )
213, 20syl 17 . . . . 5 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> ( Y e. (Unit ` R ) <-> ( Y e. B /\ Y =/= .0. ) ) )
224, 5, 21mpbir2and 957 . . . 4 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> Y e. (Unit ` R ) )
23 abvdiv.p . . . . 5 |- ./ = (/r ` R )
246, 12, 19, 8, 23dvrval 18685 . . . 4 |- ( ( X e. B /\ Y e. (Unit ` R ) ) -> ( X ./ Y ) = ( X ( .r ` R ) ( ( invr ` R ) ` Y ) ) )
252, 22, 24syl2anc 693 . . 3 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> ( X ./ Y ) = ( X ( .r ` R ) ( ( invr ` R ) ` Y ) ) )
2625fveq2d 6195 . 2 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> ( F ` ( X ./ Y ) ) = ( F ` ( X ( .r ` R ) ( ( invr ` R ) ` Y ) ) ) )
2711, 6abvcl 18824 . . . . 5 |- ( ( F e. A /\ X e. B ) -> ( F ` X ) e. RR )
281, 2, 27syl2anc 693 . . . 4 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> ( F ` X ) e. RR )
2928recnd 10068 . . 3 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> ( F ` X ) e. CC )
3011, 6abvcl 18824 . . . . 5 |- ( ( F e. A /\ Y e. B ) -> ( F ` Y ) e. RR )
311, 4, 30syl2anc 693 . . . 4 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> ( F ` Y ) e. RR )
3231recnd 10068 . . 3 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> ( F ` Y ) e. CC )
3311, 6, 7abvne0 18827 . . . 4 |- ( ( F e. A /\ Y e. B /\ Y =/= .0. ) -> ( F ` Y ) =/= 0 )
341, 4, 5, 33syl3anc 1326 . . 3 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> ( F ` Y ) =/= 0 )
3529, 32, 34divrecd 10804 . 2 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> ( ( F ` X ) / ( F ` Y ) ) = ( ( F ` X ) x. ( 1 / ( F ` Y ) ) ) )
3618, 26, 353eqtr4d 2666 1 |- ( ( ( R e. DivRing /\ F e. A ) /\ ( X e. B /\ Y e. B /\ Y =/= .0. ) ) -> ( F ` ( X ./ Y ) ) = ( ( F ` X ) / ( F ` Y ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   / cdiv 10684   Basecbs 15857   .rcmulr 15942   0gc0g 16100  Unitcui 18639   invrcinvr 18671  /rcdvr 18682   DivRingcdr 18747  AbsValcabv 18816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-ico 12181  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-abv 18817
This theorem is referenced by:  ostthlem1  25316
  Copyright terms: Public domain W3C validator