Theorem rdivmuldivd 14289

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 2233	. . . . 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 14153	. . . . 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 14279	. . . 4 |- ( ph -> ( X ./ Y ) = ( X .x. ( ( invr ` R ) ` Y ) ) )
15	14	oveq1d 6065	. . 3 |- ( ph -> ( ( X ./ Y ) .x. ( Z ./ W ) ) = ( ( X .x. ( ( invr ` R ) ` Y ) ) .x. ( Z ./ W ) ) )
16		ringsrg 14191	. . . . . . 7 |- ( R e. Ring -> R e. SRing )
17	11, 16	syl 14	. . . . . 6 |- ( ph -> R e. SRing )
18	2, 6, 17	unitssd 14254	. . . . 5 |- ( ph -> U C_ B )
19		eqid 2232	. . . . . . 7 |- ( invr ` R ) = ( invr ` R )
20	5, 19	unitinvcl 14268	. . . . . 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 3239	. . . 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 14280	. . . . 5 |- ( ( R e. Ring /\ Z e. B /\ W e. U ) -> ( Z ./ W ) e. B )
26	11, 23, 24, 25	syl3anc 1274	. . . 4 |- ( ph -> ( Z ./ W ) e. B )
27	1, 3	ringass 14160	. . . 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 1276	. . 3 |- ( ph -> ( ( X .x. ( ( invr ` R ) ` Y ) ) .x. ( Z ./ W ) ) = ( X .x. ( ( ( invr ` R ) ` Y ) .x. ( Z ./ W ) ) ) )
29	1, 3	crngcom 14158	. . . . 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 1274	. . . 4 |- ( ph -> ( ( ( invr ` R ) ` Y ) .x. ( Z ./ W ) ) = ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) )
31	30	oveq2d 6066	. . 3 |- ( ph -> ( X .x. ( ( ( invr ` R ) ` Y ) .x. ( Z ./ W ) ) ) = ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) )
32	15, 28, 31	3eqtrd 2269	. 2 |- ( ph -> ( ( X ./ Y ) .x. ( Z ./ W ) ) = ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) )
33		eqid 2232	. . . . . . . . 9 |- ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U )
34	5, 33	unitgrp 14261	. . . . . . . 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 2233	. . . . . . . . 9 |- ( ph -> ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U ) )
37	6, 36, 17	unitgrpbasd 14260	. . . . . . . 8 |- ( ph -> U = ( Base ` ( (mulGrp ` R ) ↾s U ) ) )
38	13, 37	eleqtrd 2311	. . . . . . 7 |- ( ph -> Y e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) )
39	24, 37	eleqtrd 2311	. . . . . . 7 |- ( ph -> W e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) )
40		eqid 2232	. . . . . . . 8 |- ( Base ` ( (mulGrp ` R ) ↾s U ) ) = ( Base ` ( (mulGrp ` R ) ↾s U ) )
41		eqid 2232	. . . . . . . 8 |- ( +g ` ( (mulGrp ` R ) ↾s U ) ) = ( +g ` ( (mulGrp ` R ) ↾s U ) )
42		eqid 2232	. . . . . . . 8 |- ( invg ` ( (mulGrp ` R ) ↾s U ) ) = ( invg ` ( (mulGrp ` R ) ↾s U ) )
43	40, 41, 42	grpinvadd 13791	. . . . . . 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 1274	. . . . . 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 14267	. . . . . . 7 |- ( ph -> ( invr ` R ) = ( invg ` ( (mulGrp ` R ) ↾s U ) ) )
46	45	fveq1d 5672	. . . . . 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 5672	. . . . . . 7 |- ( ph -> ( ( invr ` R ) ` W ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` W ) )
48	45	fveq1d 5672	. . . . . . 7 |- ( ph -> ( ( invr ` R ) ` Y ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` Y ) )
49	47, 48	oveq12d 6068	. . . . . 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 2275	. . . . 5 |- ( ph -> ( ( invr ` R ) ` ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) ) = ( ( ( invr ` R ) ` W ) ( +g ` ( (mulGrp ` R ) ↾s U ) ) ( ( invr ` R ) ` Y ) ) )
51		basfn 13271	. . . . . . . . . . . 12 |- Base Fn _V
52	10	elexd 2827	. . . . . . . . . . . 12 |- ( ph -> R e. _V )
53		funfvex 5687	. . . . . . . . . . . . 13 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
54	53	funfni 5458	. . . . . . . . . . . 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 2319	. . . . . . . . . 10 |- ( ph -> B e. _V )
57	56, 18	ssexd 4250	. . . . . . . . 9 |- ( ph -> U e. _V )
58		ressex 13278	. . . . . . . . . 10 |- ( ( R e. CRing /\ U e. _V ) -> ( R ↾s U ) e. _V )
59		eqid 2232	. . . . . . . . . . 11 |- (mulGrp ` ( R ↾s U ) ) = (mulGrp ` ( R ↾s U ) )
60		eqid 2232	. . . . . . . . . . 11 |- ( .r ` ( R ↾s U ) ) = ( .r ` ( R ↾s U ) )
61	59, 60	mgpplusgg 14068	. . . . . . . . . 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 2232	. . . . . . . . . 10 |- ( R ↾s U ) = ( R ↾s U )
65	64, 3	ressmulrg 13358	. . . . . . . . 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 2232	. . . . . . . . . . 11 |- (mulGrp ` R ) = (mulGrp ` R )
68	64, 67	mgpress 14075	. . . . . . . . . 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 5674	. . . . . . . 8 |- ( ph -> ( +g ` ( (mulGrp ` R ) ↾s U ) ) = ( +g ` (mulGrp ` ( R ↾s U ) ) ) )
71	63, 66, 70	3eqtr4d 2275	. . . . . . 7 |- ( ph -> .x. = ( +g ` ( (mulGrp ` R ) ↾s U ) ) )
72	71	oveqd 6067	. . . . . 6 |- ( ph -> ( Y .x. W ) = ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) )
73	72	fveq2d 5674	. . . . 5 |- ( ph -> ( ( invr ` R ) ` ( Y .x. W ) ) = ( ( invr ` R ) ` ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) ) )
74	71	oveqd 6067	. . . . 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 2275	. . . 4 |- ( ph -> ( ( invr ` R ) ` ( Y .x. W ) ) = ( ( ( invr ` R ) ` W ) .x. ( ( invr ` R ) ` Y ) ) )
76	75	oveq2d 6066	. . 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 14157	. . . . 5 |- ( ( R e. Ring /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
78	11, 12, 23, 77	syl3anc 1274	. . . 4 |- ( ph -> ( X .x. Z ) e. B )
79	5, 3	unitmulcl 14258	. . . . 5 |- ( ( R e. Ring /\ Y e. U /\ W e. U ) -> ( Y .x. W ) e. U )
80	11, 13, 24, 79	syl3anc 1274	. . . 4 |- ( ph -> ( Y .x. W ) e. U )
81	2, 4, 6, 7, 9, 11, 78, 80	dvrvald 14279	. . 3 |- ( ph -> ( ( X .x. Z ) ./ ( Y .x. W ) ) = ( ( X .x. Z ) .x. ( ( invr ` R ) ` ( Y .x. W ) ) ) )
82	5, 19	unitinvcl 14268	. . . . . . . . 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 3239	. . . . . . 7 |- ( ph -> ( ( invr ` R ) ` W ) e. B )
85	1, 3	ringass 14160	. . . . . . 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 1276	. . . . . 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 14279	. . . . . . 7 |- ( ph -> ( Z ./ W ) = ( Z .x. ( ( invr ` R ) ` W ) ) )
88	87	oveq2d 6066	. . . . . 6 |- ( ph -> ( X .x. ( Z ./ W ) ) = ( X .x. ( Z .x. ( ( invr ` R ) ` W ) ) ) )
89	86, 88	eqtr4d 2268	. . . . 5 |- ( ph -> ( ( X .x. Z ) .x. ( ( invr ` R ) ` W ) ) = ( X .x. ( Z ./ W ) ) )
90	89	oveq1d 6065	. . . 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 14160	. . . . 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 1276	. . . 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 14160	. . . . 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 1276	. . . 4 |- ( ph -> ( ( X .x. ( Z ./ W ) ) .x. ( ( invr ` R ) ` Y ) ) = ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) )
95	90, 92, 94	3eqtr3rd 2274	. . 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 2276	. 2 |- ( ph -> ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) = ( ( X .x. Z ) ./ ( Y .x. W ) ) )
97	32, 96	eqtrd 2265	1 |- ( ph -> ( ( X ./ Y ) .x. ( Z ./ W ) ) = ( ( X .x. Z ) ./ ( Y .x. W ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   _Vcvv 2813   Fn wfn 5347   ` cfv 5352  (class class class)co 6050   Basecbs 13212   ↾_s cress 13213   +g cplusg 13290   .rcmulr 13291   Grpcgrp 13713   invgcminusg 13714  mulGrpcmgp 14064  SRingcsrg 14107   Ringcrg 14140   CRingccrg 14141  Unitcui 14231   invrcinvr 14265  /_rcdvr 14276

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-tpos 6476  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-cmn 14003  df-abl 14004  df-mgp 14065  df-ur 14104  df-srg 14108  df-ring 14142  df-cring 14143  df-oppr 14212  df-dvdsr 14233  df-unit 14234  df-invr 14266  df-dvr 14277

This theorem is referenced by: (None)