Theorem dvrvald 14379

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 14290	. . . 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 14378	. 2 |- ( ph -> ./ = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) )
10		simpl 109	. . . 4 |- ( ( x = X /\ y = Y ) -> x = X )
11		fveq2 5675	. . . . 5 |- ( y = Y -> ( I ` y ) = ( I ` Y ) )
12	11	adantl 277	. . . 4 |- ( ( x = X /\ y = Y ) -> ( I ` y ) = ( I ` Y ) )
13	10, 12	oveq12d 6076	. . 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 6075	. . 3 |- ( ph -> ( X .x. ( I ` Y ) ) = ( X ( .r ` R ) ( I ` Y ) ) )
18	15, 1	eleqtrd 2313	. . . 4 |- ( ph -> X e. ( Base ` R ) )
19		eqidd 2235	. . . . 5 |- ( ph -> ( Base ` R ) = ( Base ` R ) )
20	16, 3	eleqtrd 2313	. . . . . . 7 |- ( ph -> Y e. (Unit ` R ) )
21		eqid 2234	. . . . . . . 8 |- (Unit ` R ) = (Unit ` R )
22		eqid 2234	. . . . . . . 8 |- ( invr ` R ) = ( invr ` R )
23	21, 22	unitinvcl 14368	. . . . . . 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 5677	. . . . . 6 |- ( ph -> ( I ` Y ) = ( ( invr ` R ) ` Y ) )
26	24, 25, 3	3eltr4d 2318	. . . . 5 |- ( ph -> ( I ` Y ) e. U )
27	19, 3, 8, 26	unitcld 14353	. . . 4 |- ( ph -> ( I ` Y ) e. ( Base ` R ) )
28		eqid 2234	. . . . 5 |- ( Base ` R ) = ( Base ` R )
29		eqid 2234	. . . . 5 |- ( .r ` R ) = ( .r ` R )
30	28, 29	ringcl 14256	. . . 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 2311	. 2 |- ( ph -> ( X .x. ( I ` Y ) ) e. ( Base ` R ) )
33	9, 14, 15, 16, 32	ovmpod 6189	1 |- ( ph -> ( X ./ Y ) = ( X .x. ( I ` Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   ` cfv 5357  (class class class)co 6058   Basecbs 13296   .rcmulr 13375  SRingcsrg 14206   Ringcrg 14239  Unitcui 14331   invrcinvr 14365  /_rcdvr 14376

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-tpos 6489  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-cmn 14039  df-abl 14040  df-mgp 14160  df-ur 14203  df-srg 14207  df-ring 14241  df-oppr 14311  df-dvdsr 14333  df-unit 14334  df-invr 14366  df-dvr 14377

This theorem is referenced by:  dvrcl  14380  unitdvcl  14381  dvrid  14382  dvr1  14383  dvrass  14384  dvrcan1  14385  dvrdir  14388  rdivmuldivd  14389  ringinvdv  14390  subrgdv  14484