Theorem dvrvald 14147

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 14059	. . . 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 14146	. 2 |- ( ph -> ./ = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) )
10		simpl 109	. . . 4 |- ( ( x = X /\ y = Y ) -> x = X )
11		fveq2 5639	. . . . 5 |- ( y = Y -> ( I ` y ) = ( I ` Y ) )
12	11	adantl 277	. . . 4 |- ( ( x = X /\ y = Y ) -> ( I ` y ) = ( I ` Y ) )
13	10, 12	oveq12d 6035	. . 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 6034	. . 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 14136	. . . . . . 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 5641	. . . . . 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 14121	. . . 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 14025	. . . 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 1273	. . 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 6148	1 |- ( ph -> ( X ./ Y ) = ( X .x. ( I ` Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   Basecbs 13081   .rcmulr 13160  SRingcsrg 13975   Ringcrg 14008  Unitcui 14099   invrcinvr 14133  /_rcdvr 14144

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-tpos 6410  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-cmn 13872  df-abl 13873  df-mgp 13933  df-ur 13972  df-srg 13976  df-ring 14010  df-oppr 14080  df-dvdsr 14101  df-unit 14102  df-invr 14134  df-dvr 14145

This theorem is referenced by:  dvrcl  14148  unitdvcl  14149  dvrid  14150  dvr1  14151  dvrass  14152  dvrcan1  14153  dvrdir  14156  rdivmuldivd  14157  ringinvdv  14158  subrgdv  14251