Theorem dvrvald 14092

Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)

Hypotheses

Ref	Expression
dvrvald.b	|- ( ph -> B = ( Base ` R ) )
dvrvald.t	|- ( ph -> .x. = ( .r ` R ) )
dvrvald.u	|- ( ph -> U = (Unit ` R ) )
dvrvald.i	|- ( ph -> I = ( invr ` R ) )
dvrvald.d	|- ( ph -> ./ = (/r ` R ) )
dvrvald.r	|- ( ph -> R e. Ring )
dvrvald.x	|- ( ph -> X e. B )
dvrvald.y	|- ( ph -> Y e. U )

Assertion

Ref	Expression
dvrvald	|- ( ph -> ( X ./ Y ) = ( X .x. ( I ` Y ) ) )

Proof of Theorem dvrvald

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvrvald.b	. . 3 |- ( ph -> B = ( Base ` R ) )
2		dvrvald.t	. . 3 |- ( ph -> .x. = ( .r ` R ) )
3		dvrvald.u	. . 3 |- ( ph -> U = (Unit ` R ) )
4		dvrvald.i	. . 3 |- ( ph -> I = ( invr ` R ) )
5		dvrvald.d	. . 3 |- ( ph -> ./ = (/r ` R ) )
6		dvrvald.r	. . . 4 |- ( ph -> R e. Ring )
7		ringsrg 14005	. . . 4 |- ( R e. Ring -> R e. SRing )
8	6, 7	syl 14	. . 3 |- ( ph -> R e. SRing )
9	1, 2, 3, 4, 5, 8	dvrfvald 14091	. 2 |- ( ph -> ./ = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) )
10		simpl 109	. . . 4 |- ( ( x = X /\ y = Y ) -> x = X )
11		fveq2 5626	. . . . 5 |- ( y = Y -> ( I ` y ) = ( I ` Y ) )
12	11	adantl 277	. . . 4 |- ( ( x = X /\ y = Y ) -> ( I ` y ) = ( I ` Y ) )
13	10, 12	oveq12d 6018	. . 3 |- ( ( x = X /\ y = Y ) -> ( x .x. ( I ` y ) ) = ( X .x. ( I ` Y ) ) )
14	13	adantl 277	. 2 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( x .x. ( I ` y ) ) = ( X .x. ( I ` Y ) ) )
15		dvrvald.x	. 2 |- ( ph -> X e. B )
16		dvrvald.y	. 2 |- ( ph -> Y e. U )
17	2	oveqd 6017	. . 3 |- ( ph -> ( X .x. ( I ` Y ) ) = ( X ( .r ` R ) ( I ` Y ) ) )
18	15, 1	eleqtrd 2308	. . . 4 |- ( ph -> X e. ( Base ` R ) )
19		eqidd 2230	. . . . 5 |- ( ph -> ( Base ` R ) = ( Base ` R ) )
20	16, 3	eleqtrd 2308	. . . . . . 7 |- ( ph -> Y e. (Unit ` R ) )
21		eqid 2229	. . . . . . . 8 |- (Unit ` R ) = (Unit ` R )
22		eqid 2229	. . . . . . . 8 |- ( invr ` R ) = ( invr ` R )
23	21, 22	unitinvcl 14081	. . . . . . 7 |- ( ( R e. Ring /\ Y e. (Unit ` R ) ) -> ( ( invr ` R ) ` Y ) e. (Unit ` R ) )
24	6, 20, 23	syl2anc 411	. . . . . 6 |- ( ph -> ( ( invr ` R ) ` Y ) e. (Unit ` R ) )
25	4	fveq1d 5628	. . . . . 6 |- ( ph -> ( I ` Y ) = ( ( invr ` R ) ` Y ) )
26	24, 25, 3	3eltr4d 2313	. . . . 5 |- ( ph -> ( I ` Y ) e. U )
27	19, 3, 8, 26	unitcld 14066	. . . 4 |- ( ph -> ( I ` Y ) e. ( Base ` R ) )
28		eqid 2229	. . . . 5 |- ( Base ` R ) = ( Base ` R )
29		eqid 2229	. . . . 5 |- ( .r ` R ) = ( .r ` R )
30	28, 29	ringcl 13971	. . . 4 |- ( ( R e. Ring /\ X e. ( Base ` R ) /\ ( I ` Y ) e. ( Base ` R ) ) -> ( X ( .r ` R ) ( I ` Y ) ) e. ( Base ` R ) )
31	6, 18, 27, 30	syl3anc 1271	. . 3 |- ( ph -> ( X ( .r ` R ) ( I ` Y ) ) e. ( Base ` R ) )
32	17, 31	eqeltrd 2306	. 2 |- ( ph -> ( X .x. ( I ` Y ) ) e. ( Base ` R ) )
33	9, 14, 15, 16, 32	ovmpod 6131	1 |- ( ph -> ( X ./ Y ) = ( X .x. ( I ` Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   ` cfv 5317  (class class class)co 6000   Basecbs 13027   .rcmulr 13106  SRingcsrg 13921   Ringcrg 13954  Unitcui 14045   invrcinvr 14078  /_rcdvr 14089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-tpos 6389  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-3 9166  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-iress 13035  df-plusg 13118  df-mulr 13119  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-grp 13531  df-minusg 13532  df-cmn 13818  df-abl 13819  df-mgp 13879  df-ur 13918  df-srg 13922  df-ring 13956  df-oppr 14026  df-dvdsr 14047  df-unit 14048  df-invr 14079  df-dvr 14090

This theorem is referenced by:  dvrcl  14093  unitdvcl  14094  dvrid  14095  dvr1  14096  dvrass  14097  dvrcan1  14098  dvrdir  14101  rdivmuldivd  14102  ringinvdv  14103  subrgdv  14196