Theorem dvrvald 14167

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 14079	. . . 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 14166	. 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 6036	. . 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 6035	. . 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 14156	. . . . . . 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 14141	. . . 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 14045	. . . 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 6149	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 6018   Basecbs 13100   .rcmulr 13179  SRingcsrg 13995   Ringcrg 14028  Unitcui 14119   invrcinvr 14153  /_rcdvr 14164

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 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-lttrn 8146  ax-pre-ltadd 8148

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 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-tpos 6411  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13103  df-slot 13104  df-base 13106  df-sets 13107  df-iress 13108  df-plusg 13191  df-mulr 13192  df-0g 13359  df-mgm 13457  df-sgrp 13503  df-mnd 13518  df-grp 13604  df-minusg 13605  df-cmn 13891  df-abl 13892  df-mgp 13953  df-ur 13992  df-srg 13996  df-ring 14030  df-oppr 14100  df-dvdsr 14121  df-unit 14122  df-invr 14154  df-dvr 14165

This theorem is referenced by:  dvrcl  14168  unitdvcl  14169  dvrid  14170  dvr1  14171  dvrass  14172  dvrcan1  14173  dvrdir  14176  rdivmuldivd  14177  ringinvdv  14178  subrgdv  14271