Theorem dvrvald 14229

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 14141	. . . 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 14228	. 2 |- ( ph -> ./ = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) )
10		simpl 109	. . . 4 |- ( ( x = X /\ y = Y ) -> x = X )
11		fveq2 5648	. . . . 5 |- ( y = Y -> ( I ` y ) = ( I ` Y ) )
12	11	adantl 277	. . . 4 |- ( ( x = X /\ y = Y ) -> ( I ` y ) = ( I ` Y ) )
13	10, 12	oveq12d 6046	. . 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 6045	. . 3 |- ( ph -> ( X .x. ( I ` Y ) ) = ( X ( .r ` R ) ( I ` Y ) ) )
18	15, 1	eleqtrd 2310	. . . 4 |- ( ph -> X e. ( Base ` R ) )
19		eqidd 2232	. . . . 5 |- ( ph -> ( Base ` R ) = ( Base ` R ) )
20	16, 3	eleqtrd 2310	. . . . . . 7 |- ( ph -> Y e. (Unit ` R ) )
21		eqid 2231	. . . . . . . 8 |- (Unit ` R ) = (Unit ` R )
22		eqid 2231	. . . . . . . 8 |- ( invr ` R ) = ( invr ` R )
23	21, 22	unitinvcl 14218	. . . . . . 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 5650	. . . . . 6 |- ( ph -> ( I ` Y ) = ( ( invr ` R ) ` Y ) )
26	24, 25, 3	3eltr4d 2315	. . . . 5 |- ( ph -> ( I ` Y ) e. U )
27	19, 3, 8, 26	unitcld 14203	. . . 4 |- ( ph -> ( I ` Y ) e. ( Base ` R ) )
28		eqid 2231	. . . . 5 |- ( Base ` R ) = ( Base ` R )
29		eqid 2231	. . . . 5 |- ( .r ` R ) = ( .r ` R )
30	28, 29	ringcl 14107	. . . 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 1274	. . 3 |- ( ph -> ( X ( .r ` R ) ( I ` Y ) ) e. ( Base ` R ) )
32	17, 31	eqeltrd 2308	. 2 |- ( ph -> ( X .x. ( I ` Y ) ) e. ( Base ` R ) )
33	9, 14, 15, 16, 32	ovmpod 6159	1 |- ( ph -> ( X ./ Y ) = ( X .x. ( I ` Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   ` cfv 5333  (class class class)co 6028   Basecbs 13162   .rcmulr 13241  SRingcsrg 14057   Ringcrg 14090  Unitcui 14181   invrcinvr 14215  /_rcdvr 14226

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-tpos 6454  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-mulr 13254  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-cmn 13953  df-abl 13954  df-mgp 14015  df-ur 14054  df-srg 14058  df-ring 14092  df-oppr 14162  df-dvdsr 14183  df-unit 14184  df-invr 14216  df-dvr 14227

This theorem is referenced by:  dvrcl  14230  unitdvcl  14231  dvrid  14232  dvr1  14233  dvrass  14234  dvrcan1  14235  dvrdir  14238  rdivmuldivd  14239  ringinvdv  14240  subrgdv  14333