Theorem dvrcan5 29793

Description: Cancellation law for common factor in ratio. (divcan5 10727 analog.) (Contributed by Thierry Arnoux, 26-Oct-2016.)

Hypotheses

Ref	Expression
dvrcan5.b	|- B = ( Base ` R )
dvrcan5.o	|- U = (Unit ` R )
dvrcan5.d	|- ./ = (/r ` R )
dvrcan5.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
dvrcan5	|- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( ( X .x. Z ) ./ ( Y .x. Z ) ) = ( X ./ Y ) )

Proof of Theorem dvrcan5

Step	Hyp	Ref	Expression
1		dvrcan5.b	. . . . . . 7 |- B = ( Base ` R )
2		dvrcan5.o	. . . . . . 7 |- U = (Unit ` R )
3	1, 2	unitss 18660	. . . . . 6 |- U C_ B
4		simpr3 1069	. . . . . 6 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> Z e. U )
5	3, 4	sseldi 3601	. . . . 5 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> Z e. B )
6		dvrcan5.t	. . . . . . 7 |- .x. = ( .r ` R )
7	2, 6	unitmulcl 18664	. . . . . 6 |- ( ( R e. Ring /\ Y e. U /\ Z e. U ) -> ( Y .x. Z ) e. U )
8	7	3adant3r1 1274	. . . . 5 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( Y .x. Z ) e. U )
9		eqid 2622	. . . . . 6 |- ( invr ` R ) = ( invr ` R )
10		dvrcan5.d	. . . . . 6 |- ./ = (/r ` R )
11	1, 6, 2, 9, 10	dvrval 18685	. . . . 5 |- ( ( Z e. B /\ ( Y .x. Z ) e. U ) -> ( Z ./ ( Y .x. Z ) ) = ( Z .x. ( ( invr ` R ) ` ( Y .x. Z ) ) ) )
12	5, 8, 11	syl2anc 693	. . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( Z ./ ( Y .x. Z ) ) = ( Z .x. ( ( invr ` R ) ` ( Y .x. Z ) ) ) )
13		simpl 473	. . . . . 6 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> R e. Ring )
14		eqid 2622	. . . . . . 7 |- ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U )
15	2, 14	unitgrp 18667	. . . . . 6 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s U ) e. Grp )
16	13, 15	syl 17	. . . . 5 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( (mulGrp ` R ) ↾s U ) e. Grp )
17		simpr2 1068	. . . . 5 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> Y e. U )
18	2, 14	unitgrpbas 18666	. . . . . . 7 |- U = ( Base ` ( (mulGrp ` R ) ↾s U ) )
19		fvex 6201	. . . . . . . . 9 |- (Unit ` R ) e. _V
20	2, 19	eqeltri 2697	. . . . . . . 8 |- U e. _V
21		eqid 2622	. . . . . . . . . 10 |- (mulGrp ` R ) = (mulGrp ` R )
22	21, 6	mgpplusg 18493	. . . . . . . . 9 |- .x. = ( +g ` (mulGrp ` R ) )
23	14, 22	ressplusg 15993	. . . . . . . 8 |- ( U e. _V -> .x. = ( +g ` ( (mulGrp ` R ) ↾s U ) ) )
24	20, 23	ax-mp 5	. . . . . . 7 |- .x. = ( +g ` ( (mulGrp ` R ) ↾s U ) )
25	2, 14, 9	invrfval 18673	. . . . . . 7 |- ( invr ` R ) = ( invg ` ( (mulGrp ` R ) ↾s U ) )
26	18, 24, 25	grpinvadd 17493	. . . . . 6 |- ( ( ( (mulGrp ` R ) ↾s U ) e. Grp /\ Y e. U /\ Z e. U ) -> ( ( invr ` R ) ` ( Y .x. Z ) ) = ( ( ( invr ` R ) ` Z ) .x. ( ( invr ` R ) ` Y ) ) )
27	26	oveq2d 6666	. . . . 5 |- ( ( ( (mulGrp ` R ) ↾s U ) e. Grp /\ Y e. U /\ Z e. U ) -> ( Z .x. ( ( invr ` R ) ` ( Y .x. Z ) ) ) = ( Z .x. ( ( ( invr ` R ) ` Z ) .x. ( ( invr ` R ) ` Y ) ) ) )
28	16, 17, 4, 27	syl3anc 1326	. . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( Z .x. ( ( invr ` R ) ` ( Y .x. Z ) ) ) = ( Z .x. ( ( ( invr ` R ) ` Z ) .x. ( ( invr ` R ) ` Y ) ) ) )
29		eqid 2622	. . . . . . . 8 |- ( 1r ` R ) = ( 1r ` R )
30	2, 9, 6, 29	unitrinv 18678	. . . . . . 7 |- ( ( R e. Ring /\ Z e. U ) -> ( Z .x. ( ( invr ` R ) ` Z ) ) = ( 1r ` R ) )
31	30	oveq1d 6665	. . . . . 6 |- ( ( R e. Ring /\ Z e. U ) -> ( ( Z .x. ( ( invr ` R ) ` Z ) ) .x. ( ( invr ` R ) ` Y ) ) = ( ( 1r ` R ) .x. ( ( invr ` R ) ` Y ) ) )
32	31	3ad2antr3 1228	. . . . 5 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( ( Z .x. ( ( invr ` R ) ` Z ) ) .x. ( ( invr ` R ) ` Y ) ) = ( ( 1r ` R ) .x. ( ( invr ` R ) ` Y ) ) )
33	2, 9	unitinvcl 18674	. . . . . . . 8 |- ( ( R e. Ring /\ Z e. U ) -> ( ( invr ` R ) ` Z ) e. U )
34	33	3ad2antr3 1228	. . . . . . 7 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( ( invr ` R ) ` Z ) e. U )
35	3, 34	sseldi 3601	. . . . . 6 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( ( invr ` R ) ` Z ) e. B )
36	2, 9	unitinvcl 18674	. . . . . . . 8 |- ( ( R e. Ring /\ Y e. U ) -> ( ( invr ` R ) ` Y ) e. U )
37	36	3ad2antr2 1227	. . . . . . 7 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( ( invr ` R ) ` Y ) e. U )
38	3, 37	sseldi 3601	. . . . . 6 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( ( invr ` R ) ` Y ) e. B )
39	1, 6	ringass 18564	. . . . . 6 |- ( ( R e. Ring /\ ( Z e. B /\ ( ( invr ` R ) ` Z ) e. B /\ ( ( invr ` R ) ` Y ) e. B ) ) -> ( ( Z .x. ( ( invr ` R ) ` Z ) ) .x. ( ( invr ` R ) ` Y ) ) = ( Z .x. ( ( ( invr ` R ) ` Z ) .x. ( ( invr ` R ) ` Y ) ) ) )
40	13, 5, 35, 38, 39	syl13anc 1328	. . . . 5 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( ( Z .x. ( ( invr ` R ) ` Z ) ) .x. ( ( invr ` R ) ` Y ) ) = ( Z .x. ( ( ( invr ` R ) ` Z ) .x. ( ( invr ` R ) ` Y ) ) ) )
41	1, 6, 29	ringlidm 18571	. . . . . 6 |- ( ( R e. Ring /\ ( ( invr ` R ) ` Y ) e. B ) -> ( ( 1r ` R ) .x. ( ( invr ` R ) ` Y ) ) = ( ( invr ` R ) ` Y ) )
42	13, 38, 41	syl2anc 693	. . . . 5 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( ( 1r ` R ) .x. ( ( invr ` R ) ` Y ) ) = ( ( invr ` R ) ` Y ) )
43	32, 40, 42	3eqtr3d 2664	. . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( Z .x. ( ( ( invr ` R ) ` Z ) .x. ( ( invr ` R ) ` Y ) ) ) = ( ( invr ` R ) ` Y ) )
44	12, 28, 43	3eqtrd 2660	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( Z ./ ( Y .x. Z ) ) = ( ( invr ` R ) ` Y ) )
45	44	oveq2d 6666	. 2 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( X .x. ( Z ./ ( Y .x. Z ) ) ) = ( X .x. ( ( invr ` R ) ` Y ) ) )
46		simpr1 1067	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> X e. B )
47	1, 2, 10, 6	dvrass 18690	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Z e. B /\ ( Y .x. Z ) e. U ) ) -> ( ( X .x. Z ) ./ ( Y .x. Z ) ) = ( X .x. ( Z ./ ( Y .x. Z ) ) ) )
48	13, 46, 5, 8, 47	syl13anc 1328	. 2 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( ( X .x. Z ) ./ ( Y .x. Z ) ) = ( X .x. ( Z ./ ( Y .x. Z ) ) ) )
49	1, 6, 2, 9, 10	dvrval 18685	. . 3 |- ( ( X e. B /\ Y e. U ) -> ( X ./ Y ) = ( X .x. ( ( invr ` R ) ` Y ) ) )
50	46, 17, 49	syl2anc 693	. 2 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( X ./ Y ) = ( X .x. ( ( invr ` R ) ` Y ) ) )
51	45, 48, 50	3eqtr4d 2666	1 |- ( ( R e. Ring /\ ( X e. B /\ Y e. U /\ Z e. U ) ) -> ( ( X .x. Z ) ./ ( Y .x. Z ) ) = ( X ./ Y ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   .rcmulr 15942   Grpcgrp 17422  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547  Unitcui 18639   invrcinvr 18671  /_rcdvr 18682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683

This theorem is referenced by:  rhmdvd  29821