Theorem rdivmuldivd 13318

Description: Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 30-Oct-2017.)

Hypotheses

Ref	Expression
dvrdir.b	|- B = ( Base ` R )
dvrdir.u	|- U = (Unit ` R )
dvrdir.p	|- .+ = ( +g ` R )
dvrdir.t	|- ./ = (/r ` R )
rdivmuldivd.p	|- .x. = ( .r ` R )
rdivmuldivd.r	|- ( ph -> R e. CRing )
rdivmuldivd.a	|- ( ph -> X e. B )
rdivmuldivd.b	|- ( ph -> Y e. U )
rdivmuldivd.c	|- ( ph -> Z e. B )
rdivmuldivd.d	|- ( ph -> W e. U )

Assertion

Ref	Expression
rdivmuldivd	|- ( ph -> ( ( X ./ Y ) .x. ( Z ./ W ) ) = ( ( X .x. Z ) ./ ( Y .x. W ) ) )

Proof of Theorem rdivmuldivd

Step	Hyp	Ref	Expression
1		dvrdir.b	. . . . . 6 |- B = ( Base ` R )
2	1	a1i 9	. . . . 5 |- ( ph -> B = ( Base ` R ) )
3		rdivmuldivd.p	. . . . . 6 |- .x. = ( .r ` R )
4	3	a1i 9	. . . . 5 |- ( ph -> .x. = ( .r ` R ) )
5		dvrdir.u	. . . . . 6 |- U = (Unit ` R )
6	5	a1i 9	. . . . 5 |- ( ph -> U = (Unit ` R ) )
7		eqidd 2178	. . . . 5 |- ( ph -> ( invr ` R ) = ( invr ` R ) )
8		dvrdir.t	. . . . . 6 |- ./ = (/r ` R )
9	8	a1i 9	. . . . 5 |- ( ph -> ./ = (/r ` R ) )
10		rdivmuldivd.r	. . . . . 6 |- ( ph -> R e. CRing )
11	10	crngringd 13197	. . . . 5 |- ( ph -> R e. Ring )
12		rdivmuldivd.a	. . . . 5 |- ( ph -> X e. B )
13		rdivmuldivd.b	. . . . 5 |- ( ph -> Y e. U )
14	2, 4, 6, 7, 9, 11, 12, 13	dvrvald 13308	. . . 4 |- ( ph -> ( X ./ Y ) = ( X .x. ( ( invr ` R ) ` Y ) ) )
15	14	oveq1d 5892	. . 3 |- ( ph -> ( ( X ./ Y ) .x. ( Z ./ W ) ) = ( ( X .x. ( ( invr ` R ) ` Y ) ) .x. ( Z ./ W ) ) )
16		ringsrg 13229	. . . . . . 7 |- ( R e. Ring -> R e. SRing )
17	11, 16	syl 14	. . . . . 6 |- ( ph -> R e. SRing )
18	2, 6, 17	unitssd 13283	. . . . 5 |- ( ph -> U C_ B )
19		eqid 2177	. . . . . . 7 |- ( invr ` R ) = ( invr ` R )
20	5, 19	unitinvcl 13297	. . . . . 6 |- ( ( R e. Ring /\ Y e. U ) -> ( ( invr ` R ) ` Y ) e. U )
21	11, 13, 20	syl2anc 411	. . . . 5 |- ( ph -> ( ( invr ` R ) ` Y ) e. U )
22	18, 21	sseldd 3158	. . . 4 |- ( ph -> ( ( invr ` R ) ` Y ) e. B )
23		rdivmuldivd.c	. . . . 5 |- ( ph -> Z e. B )
24		rdivmuldivd.d	. . . . 5 |- ( ph -> W e. U )
25	1, 5, 8	dvrcl 13309	. . . . 5 |- ( ( R e. Ring /\ Z e. B /\ W e. U ) -> ( Z ./ W ) e. B )
26	11, 23, 24, 25	syl3anc 1238	. . . 4 |- ( ph -> ( Z ./ W ) e. B )
27	1, 3	ringass 13204	. . . 4 |- ( ( R e. Ring /\ ( X e. B /\ ( ( invr ` R ) ` Y ) e. B /\ ( Z ./ W ) e. B ) ) -> ( ( X .x. ( ( invr ` R ) ` Y ) ) .x. ( Z ./ W ) ) = ( X .x. ( ( ( invr ` R ) ` Y ) .x. ( Z ./ W ) ) ) )
28	11, 12, 22, 26, 27	syl13anc 1240	. . 3 |- ( ph -> ( ( X .x. ( ( invr ` R ) ` Y ) ) .x. ( Z ./ W ) ) = ( X .x. ( ( ( invr ` R ) ` Y ) .x. ( Z ./ W ) ) ) )
29	1, 3	crngcom 13202	. . . . 5 |- ( ( R e. CRing /\ ( ( invr ` R ) ` Y ) e. B /\ ( Z ./ W ) e. B ) -> ( ( ( invr ` R ) ` Y ) .x. ( Z ./ W ) ) = ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) )
30	10, 22, 26, 29	syl3anc 1238	. . . 4 |- ( ph -> ( ( ( invr ` R ) ` Y ) .x. ( Z ./ W ) ) = ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) )
31	30	oveq2d 5893	. . 3 |- ( ph -> ( X .x. ( ( ( invr ` R ) ` Y ) .x. ( Z ./ W ) ) ) = ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) )
32	15, 28, 31	3eqtrd 2214	. 2 |- ( ph -> ( ( X ./ Y ) .x. ( Z ./ W ) ) = ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) )
33		eqid 2177	. . . . . . . . 9 |- ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U )
34	5, 33	unitgrp 13290	. . . . . . . 8 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s U ) e. Grp )
35	11, 34	syl 14	. . . . . . 7 |- ( ph -> ( (mulGrp ` R ) ↾s U ) e. Grp )
36		eqidd 2178	. . . . . . . . 9 |- ( ph -> ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U ) )
37	6, 36, 17	unitgrpbasd 13289	. . . . . . . 8 |- ( ph -> U = ( Base ` ( (mulGrp ` R ) ↾s U ) ) )
38	13, 37	eleqtrd 2256	. . . . . . 7 |- ( ph -> Y e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) )
39	24, 37	eleqtrd 2256	. . . . . . 7 |- ( ph -> W e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) )
40		eqid 2177	. . . . . . . 8 |- ( Base ` ( (mulGrp ` R ) ↾s U ) ) = ( Base ` ( (mulGrp ` R ) ↾s U ) )
41		eqid 2177	. . . . . . . 8 |- ( +g ` ( (mulGrp ` R ) ↾s U ) ) = ( +g ` ( (mulGrp ` R ) ↾s U ) )
42		eqid 2177	. . . . . . . 8 |- ( invg ` ( (mulGrp ` R ) ↾s U ) ) = ( invg ` ( (mulGrp ` R ) ↾s U ) )
43	40, 41, 42	grpinvadd 12953	. . . . . . 7 |- ( ( ( (mulGrp ` R ) ↾s U ) e. Grp /\ Y e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) /\ W e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) ) -> ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) ) = ( ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` W ) ( +g ` ( (mulGrp ` R ) ↾s U ) ) ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` Y ) ) )
44	35, 38, 39, 43	syl3anc 1238	. . . . . 6 |- ( ph -> ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) ) = ( ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` W ) ( +g ` ( (mulGrp ` R ) ↾s U ) ) ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` Y ) ) )
45	6, 36, 7, 11	invrfvald 13296	. . . . . . 7 |- ( ph -> ( invr ` R ) = ( invg ` ( (mulGrp ` R ) ↾s U ) ) )
46	45	fveq1d 5519	. . . . . 6 |- ( ph -> ( ( invr ` R ) ` ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) ) )
47	45	fveq1d 5519	. . . . . . 7 |- ( ph -> ( ( invr ` R ) ` W ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` W ) )
48	45	fveq1d 5519	. . . . . . 7 |- ( ph -> ( ( invr ` R ) ` Y ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` Y ) )
49	47, 48	oveq12d 5895	. . . . . 6 |- ( ph -> ( ( ( invr ` R ) ` W ) ( +g ` ( (mulGrp ` R ) ↾s U ) ) ( ( invr ` R ) ` Y ) ) = ( ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` W ) ( +g ` ( (mulGrp ` R ) ↾s U ) ) ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` Y ) ) )
50	44, 46, 49	3eqtr4d 2220	. . . . 5 |- ( ph -> ( ( invr ` R ) ` ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) ) = ( ( ( invr ` R ) ` W ) ( +g ` ( (mulGrp ` R ) ↾s U ) ) ( ( invr ` R ) ` Y ) ) )
51		basfn 12522	. . . . . . . . . . . 12 |- Base Fn _V
52	10	elexd 2752	. . . . . . . . . . . 12 |- ( ph -> R e. _V )
53		funfvex 5534	. . . . . . . . . . . . 13 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
54	53	funfni 5318	. . . . . . . . . . . 12 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
55	51, 52, 54	sylancr 414	. . . . . . . . . . 11 |- ( ph -> ( Base ` R ) e. _V )
56	1, 55	eqeltrid 2264	. . . . . . . . . 10 |- ( ph -> B e. _V )
57	56, 18	ssexd 4145	. . . . . . . . 9 |- ( ph -> U e. _V )
58		ressex 12527	. . . . . . . . . 10 |- ( ( R e. CRing /\ U e. _V ) -> ( R ↾s U ) e. _V )
59		eqid 2177	. . . . . . . . . . 11 |- (mulGrp ` ( R ↾s U ) ) = (mulGrp ` ( R ↾s U ) )
60		eqid 2177	. . . . . . . . . . 11 |- ( .r ` ( R ↾s U ) ) = ( .r ` ( R ↾s U ) )
61	59, 60	mgpplusgg 13139	. . . . . . . . . 10 |- ( ( R ↾s U ) e. _V -> ( .r ` ( R ↾s U ) ) = ( +g ` (mulGrp ` ( R ↾s U ) ) ) )
62	58, 61	syl 14	. . . . . . . . 9 |- ( ( R e. CRing /\ U e. _V ) -> ( .r ` ( R ↾s U ) ) = ( +g ` (mulGrp ` ( R ↾s U ) ) ) )
63	10, 57, 62	syl2anc 411	. . . . . . . 8 |- ( ph -> ( .r ` ( R ↾s U ) ) = ( +g ` (mulGrp ` ( R ↾s U ) ) ) )
64		eqid 2177	. . . . . . . . . 10 |- ( R ↾s U ) = ( R ↾s U )
65	64, 3	ressmulrg 12605	. . . . . . . . 9 |- ( ( U e. _V /\ R e. CRing ) -> .x. = ( .r ` ( R ↾s U ) ) )
66	57, 10, 65	syl2anc 411	. . . . . . . 8 |- ( ph -> .x. = ( .r ` ( R ↾s U ) ) )
67		eqid 2177	. . . . . . . . . . 11 |- (mulGrp ` R ) = (mulGrp ` R )
68	64, 67	mgpress 13146	. . . . . . . . . 10 |- ( ( R e. Ring /\ U e. _V ) -> ( (mulGrp ` R ) ↾s U ) = (mulGrp ` ( R ↾s U ) ) )
69	11, 57, 68	syl2anc 411	. . . . . . . . 9 |- ( ph -> ( (mulGrp ` R ) ↾s U ) = (mulGrp ` ( R ↾s U ) ) )
70	69	fveq2d 5521	. . . . . . . 8 |- ( ph -> ( +g ` ( (mulGrp ` R ) ↾s U ) ) = ( +g ` (mulGrp ` ( R ↾s U ) ) ) )
71	63, 66, 70	3eqtr4d 2220	. . . . . . 7 |- ( ph -> .x. = ( +g ` ( (mulGrp ` R ) ↾s U ) ) )
72	71	oveqd 5894	. . . . . 6 |- ( ph -> ( Y .x. W ) = ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) )
73	72	fveq2d 5521	. . . . 5 |- ( ph -> ( ( invr ` R ) ` ( Y .x. W ) ) = ( ( invr ` R ) ` ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) ) )
74	71	oveqd 5894	. . . . 5 |- ( ph -> ( ( ( invr ` R ) ` W ) .x. ( ( invr ` R ) ` Y ) ) = ( ( ( invr ` R ) ` W ) ( +g ` ( (mulGrp ` R ) ↾s U ) ) ( ( invr ` R ) ` Y ) ) )
75	50, 73, 74	3eqtr4d 2220	. . . 4 |- ( ph -> ( ( invr ` R ) ` ( Y .x. W ) ) = ( ( ( invr ` R ) ` W ) .x. ( ( invr ` R ) ` Y ) ) )
76	75	oveq2d 5893	. . 3 |- ( ph -> ( ( X .x. Z ) .x. ( ( invr ` R ) ` ( Y .x. W ) ) ) = ( ( X .x. Z ) .x. ( ( ( invr ` R ) ` W ) .x. ( ( invr ` R ) ` Y ) ) ) )
77	1, 3	ringcl 13201	. . . . 5 |- ( ( R e. Ring /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
78	11, 12, 23, 77	syl3anc 1238	. . . 4 |- ( ph -> ( X .x. Z ) e. B )
79	5, 3	unitmulcl 13287	. . . . 5 |- ( ( R e. Ring /\ Y e. U /\ W e. U ) -> ( Y .x. W ) e. U )
80	11, 13, 24, 79	syl3anc 1238	. . . 4 |- ( ph -> ( Y .x. W ) e. U )
81	2, 4, 6, 7, 9, 11, 78, 80	dvrvald 13308	. . 3 |- ( ph -> ( ( X .x. Z ) ./ ( Y .x. W ) ) = ( ( X .x. Z ) .x. ( ( invr ` R ) ` ( Y .x. W ) ) ) )
82	5, 19	unitinvcl 13297	. . . . . . . . 9 |- ( ( R e. Ring /\ W e. U ) -> ( ( invr ` R ) ` W ) e. U )
83	11, 24, 82	syl2anc 411	. . . . . . . 8 |- ( ph -> ( ( invr ` R ) ` W ) e. U )
84	18, 83	sseldd 3158	. . . . . . 7 |- ( ph -> ( ( invr ` R ) ` W ) e. B )
85	1, 3	ringass 13204	. . . . . . 7 |- ( ( R e. Ring /\ ( X e. B /\ Z e. B /\ ( ( invr ` R ) ` W ) e. B ) ) -> ( ( X .x. Z ) .x. ( ( invr ` R ) ` W ) ) = ( X .x. ( Z .x. ( ( invr ` R ) ` W ) ) ) )
86	11, 12, 23, 84, 85	syl13anc 1240	. . . . . 6 |- ( ph -> ( ( X .x. Z ) .x. ( ( invr ` R ) ` W ) ) = ( X .x. ( Z .x. ( ( invr ` R ) ` W ) ) ) )
87	2, 4, 6, 7, 9, 11, 23, 24	dvrvald 13308	. . . . . . 7 |- ( ph -> ( Z ./ W ) = ( Z .x. ( ( invr ` R ) ` W ) ) )
88	87	oveq2d 5893	. . . . . 6 |- ( ph -> ( X .x. ( Z ./ W ) ) = ( X .x. ( Z .x. ( ( invr ` R ) ` W ) ) ) )
89	86, 88	eqtr4d 2213	. . . . 5 |- ( ph -> ( ( X .x. Z ) .x. ( ( invr ` R ) ` W ) ) = ( X .x. ( Z ./ W ) ) )
90	89	oveq1d 5892	. . . 4 |- ( ph -> ( ( ( X .x. Z ) .x. ( ( invr ` R ) ` W ) ) .x. ( ( invr ` R ) ` Y ) ) = ( ( X .x. ( Z ./ W ) ) .x. ( ( invr ` R ) ` Y ) ) )
91	1, 3	ringass 13204	. . . . 5 |- ( ( R e. Ring /\ ( ( X .x. Z ) e. B /\ ( ( invr ` R ) ` W ) e. B /\ ( ( invr ` R ) ` Y ) e. B ) ) -> ( ( ( X .x. Z ) .x. ( ( invr ` R ) ` W ) ) .x. ( ( invr ` R ) ` Y ) ) = ( ( X .x. Z ) .x. ( ( ( invr ` R ) ` W ) .x. ( ( invr ` R ) ` Y ) ) ) )
92	11, 78, 84, 22, 91	syl13anc 1240	. . . 4 |- ( ph -> ( ( ( X .x. Z ) .x. ( ( invr ` R ) ` W ) ) .x. ( ( invr ` R ) ` Y ) ) = ( ( X .x. Z ) .x. ( ( ( invr ` R ) ` W ) .x. ( ( invr ` R ) ` Y ) ) ) )
93	1, 3	ringass 13204	. . . . 5 |- ( ( R e. Ring /\ ( X e. B /\ ( Z ./ W ) e. B /\ ( ( invr ` R ) ` Y ) e. B ) ) -> ( ( X .x. ( Z ./ W ) ) .x. ( ( invr ` R ) ` Y ) ) = ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) )
94	11, 12, 26, 22, 93	syl13anc 1240	. . . 4 |- ( ph -> ( ( X .x. ( Z ./ W ) ) .x. ( ( invr ` R ) ` Y ) ) = ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) )
95	90, 92, 94	3eqtr3rd 2219	. . 3 |- ( ph -> ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) = ( ( X .x. Z ) .x. ( ( ( invr ` R ) ` W ) .x. ( ( invr ` R ) ` Y ) ) ) )
96	76, 81, 95	3eqtr4rd 2221	. 2 |- ( ph -> ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) = ( ( X .x. Z ) ./ ( Y .x. W ) ) )
97	32, 96	eqtrd 2210	1 |- ( ph -> ( ( X ./ Y ) .x. ( Z ./ W ) ) = ( ( X .x. Z ) ./ ( Y .x. W ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2739   Fn wfn 5213   ` cfv 5218  (class class class)co 5877   Basecbs 12464   ↾_s cress 12465   +g cplusg 12538   .rcmulr 12539   Grpcgrp 12882   invgcminusg 12883  mulGrpcmgp 13135  SRingcsrg 13151   Ringcrg 13184   CRingccrg 13185  Unitcui 13261   invrcinvr 13294  /_rcdvr 13305

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-tpos 6248  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-iress 12472  df-plusg 12551  df-mulr 12552  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886  df-cmn 13095  df-abl 13096  df-mgp 13136  df-ur 13148  df-srg 13152  df-ring 13186  df-cring 13187  df-oppr 13245  df-dvdsr 13263  df-unit 13264  df-invr 13295  df-dvr 13306

This theorem is referenced by: (None)