Theorem dvrass 14384

Description: An associative law for division. (divassap 8981 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 1030	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> X e. B )
3		simpr2 1031	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> Y e. B )
4		simpr3 1032	. . . 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 2234	. . . . 5 |- ( invr ` R ) = ( invr ` R )
7		dvrass.b	. . . . 5 |- B = ( Base ` R )
8	5, 6, 7	ringinvcl 14370	. . . 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 14259	. . 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 1276	. 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 14256	. . . 4 |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
20	19	3adant3r3 1241	. . 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 14379	. 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 14379	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( Y ./ Z ) = ( Y .x. ( ( invr ` R ) ` Z ) ) )
23	22	oveq2d 6074	. 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 2277	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 1005   = wceq 1398   e. wcel 2205   ` cfv 5357  (class class class)co 6058   Basecbs 13296   .rcmulr 13375   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:  dvrcan3  14386