Theorem dvrass 14152

Description: An associative law for division. (divassap 8869 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
dvrass.b	|- B = ( Base ` R )
dvrass.o	|- U = (Unit ` R )
dvrass.d	|- ./ = (/r ` R )
dvrass.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
dvrass	|- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( X .x. Y ) ./ Z ) = ( X .x. ( Y ./ Z ) ) )

Proof of Theorem dvrass

Step	Hyp	Ref	Expression
1		simpl 109	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> R e. Ring )
2		simpr1 1029	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> X e. B )
3		simpr2 1030	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> Y e. B )
4		simpr3 1031	. . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> Z e. U )
5		dvrass.o	. . . . 5 |- U = (Unit ` R )
6		eqid 2231	. . . . 5 |- ( invr ` R ) = ( invr ` R )
7		dvrass.b	. . . . 5 |- B = ( Base ` R )
8	5, 6, 7	ringinvcl 14138	. . . 4 |- ( ( R e. Ring /\ Z e. U ) -> ( ( invr ` R ) ` Z ) e. B )
9	4, 8	syldan 282	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( invr ` R ) ` Z ) e. B )
10		dvrass.t	. . . 4 |- .x. = ( .r ` R )
11	7, 10	ringass 14028	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ ( ( invr ` R ) ` Z ) e. B ) ) -> ( ( X .x. Y ) .x. ( ( invr ` R ) ` Z ) ) = ( X .x. ( Y .x. ( ( invr ` R ) ` Z ) ) ) )
12	1, 2, 3, 9, 11	syl13anc 1275	. 2 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( X .x. Y ) .x. ( ( invr ` R ) ` Z ) ) = ( X .x. ( Y .x. ( ( invr ` R ) ` Z ) ) ) )
13	7	a1i 9	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> B = ( Base ` R ) )
14	10	a1i 9	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> .x. = ( .r ` R ) )
15	5	a1i 9	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> U = (Unit ` R ) )
16	6	a1i 9	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( invr ` R ) = ( invr ` R ) )
17		dvrass.d	. . . 4 |- ./ = (/r ` R )
18	17	a1i 9	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ./ = (/r ` R ) )
19	7, 10	ringcl 14025	. . . 4 |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
20	19	3adant3r3 1240	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( X .x. Y ) e. B )
21	13, 14, 15, 16, 18, 1, 20, 4	dvrvald 14147	. 2 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( X .x. Y ) ./ Z ) = ( ( X .x. Y ) .x. ( ( invr ` R ) ` Z ) ) )
22	13, 14, 15, 16, 18, 1, 3, 4	dvrvald 14147	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( Y ./ Z ) = ( Y .x. ( ( invr ` R ) ` Z ) ) )
23	22	oveq2d 6033	. 2 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( X .x. ( Y ./ Z ) ) = ( X .x. ( Y .x. ( ( invr ` R ) ` Z ) ) ) )
24	12, 21, 23	3eqtr4d 2274	1 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( X .x. Y ) ./ Z ) = ( X .x. ( Y ./ Z ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   Basecbs 13081   .rcmulr 13160   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:  dvrcan3  14154