Theorem rdivmuldivd 14222

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 2232	. . . . 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 14086	. . . . 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 14212	. . . 4 |- ( ph -> ( X ./ Y ) = ( X .x. ( ( invr ` R ) ` Y ) ) )
15	14	oveq1d 6043	. . 3 |- ( ph -> ( ( X ./ Y ) .x. ( Z ./ W ) ) = ( ( X .x. ( ( invr ` R ) ` Y ) ) .x. ( Z ./ W ) ) )
16		ringsrg 14124	. . . . . . 7 |- ( R e. Ring -> R e. SRing )
17	11, 16	syl 14	. . . . . 6 |- ( ph -> R e. SRing )
18	2, 6, 17	unitssd 14187	. . . . 5 |- ( ph -> U C_ B )
19		eqid 2231	. . . . . . 7 |- ( invr ` R ) = ( invr ` R )
20	5, 19	unitinvcl 14201	. . . . . 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 3229	. . . 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 14213	. . . . 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 14093	. . . 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 14091	. . . . 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 6044	. . 3 |- ( ph -> ( X .x. ( ( ( invr ` R ) ` Y ) .x. ( Z ./ W ) ) ) = ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) )
32	15, 28, 31	3eqtrd 2268	. 2 |- ( ph -> ( ( X ./ Y ) .x. ( Z ./ W ) ) = ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) )
33		eqid 2231	. . . . . . . . 9 |- ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U )
34	5, 33	unitgrp 14194	. . . . . . . 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 2232	. . . . . . . . 9 |- ( ph -> ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U ) )
37	6, 36, 17	unitgrpbasd 14193	. . . . . . . 8 |- ( ph -> U = ( Base ` ( (mulGrp ` R ) ↾s U ) ) )
38	13, 37	eleqtrd 2310	. . . . . . 7 |- ( ph -> Y e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) )
39	24, 37	eleqtrd 2310	. . . . . . 7 |- ( ph -> W e. ( Base ` ( (mulGrp ` R ) ↾s U ) ) )
40		eqid 2231	. . . . . . . 8 |- ( Base ` ( (mulGrp ` R ) ↾s U ) ) = ( Base ` ( (mulGrp ` R ) ↾s U ) )
41		eqid 2231	. . . . . . . 8 |- ( +g ` ( (mulGrp ` R ) ↾s U ) ) = ( +g ` ( (mulGrp ` R ) ↾s U ) )
42		eqid 2231	. . . . . . . 8 |- ( invg ` ( (mulGrp ` R ) ↾s U ) ) = ( invg ` ( (mulGrp ` R ) ↾s U ) )
43	40, 41, 42	grpinvadd 13724	. . . . . . 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 14200	. . . . . . 7 |- ( ph -> ( invr ` R ) = ( invg ` ( (mulGrp ` R ) ↾s U ) ) )
46	45	fveq1d 5650	. . . . . 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 5650	. . . . . . 7 |- ( ph -> ( ( invr ` R ) ` W ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` W ) )
48	45	fveq1d 5650	. . . . . . 7 |- ( ph -> ( ( invr ` R ) ` Y ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` Y ) )
49	47, 48	oveq12d 6046	. . . . . 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 2274	. . . . 5 |- ( ph -> ( ( invr ` R ) ` ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) ) = ( ( ( invr ` R ) ` W ) ( +g ` ( (mulGrp ` R ) ↾s U ) ) ( ( invr ` R ) ` Y ) ) )
51		basfn 13204	. . . . . . . . . . . 12 |- Base Fn _V
52	10	elexd 2817	. . . . . . . . . . . 12 |- ( ph -> R e. _V )
53		funfvex 5665	. . . . . . . . . . . . 13 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
54	53	funfni 5439	. . . . . . . . . . . 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 2318	. . . . . . . . . 10 |- ( ph -> B e. _V )
57	56, 18	ssexd 4234	. . . . . . . . 9 |- ( ph -> U e. _V )
58		ressex 13211	. . . . . . . . . 10 |- ( ( R e. CRing /\ U e. _V ) -> ( R ↾s U ) e. _V )
59		eqid 2231	. . . . . . . . . . 11 |- (mulGrp ` ( R ↾s U ) ) = (mulGrp ` ( R ↾s U ) )
60		eqid 2231	. . . . . . . . . . 11 |- ( .r ` ( R ↾s U ) ) = ( .r ` ( R ↾s U ) )
61	59, 60	mgpplusgg 14001	. . . . . . . . . 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 2231	. . . . . . . . . 10 |- ( R ↾s U ) = ( R ↾s U )
65	64, 3	ressmulrg 13291	. . . . . . . . 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 2231	. . . . . . . . . . 11 |- (mulGrp ` R ) = (mulGrp ` R )
68	64, 67	mgpress 14008	. . . . . . . . . 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 5652	. . . . . . . 8 |- ( ph -> ( +g ` ( (mulGrp ` R ) ↾s U ) ) = ( +g ` (mulGrp ` ( R ↾s U ) ) ) )
71	63, 66, 70	3eqtr4d 2274	. . . . . . 7 |- ( ph -> .x. = ( +g ` ( (mulGrp ` R ) ↾s U ) ) )
72	71	oveqd 6045	. . . . . 6 |- ( ph -> ( Y .x. W ) = ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) )
73	72	fveq2d 5652	. . . . 5 |- ( ph -> ( ( invr ` R ) ` ( Y .x. W ) ) = ( ( invr ` R ) ` ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) ) )
74	71	oveqd 6045	. . . . 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 2274	. . . 4 |- ( ph -> ( ( invr ` R ) ` ( Y .x. W ) ) = ( ( ( invr ` R ) ` W ) .x. ( ( invr ` R ) ` Y ) ) )
76	75	oveq2d 6044	. . 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 14090	. . . . 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 14191	. . . . 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 14212	. . 3 |- ( ph -> ( ( X .x. Z ) ./ ( Y .x. W ) ) = ( ( X .x. Z ) .x. ( ( invr ` R ) ` ( Y .x. W ) ) ) )
82	5, 19	unitinvcl 14201	. . . . . . . . 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 3229	. . . . . . 7 |- ( ph -> ( ( invr ` R ) ` W ) e. B )
85	1, 3	ringass 14093	. . . . . . 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 14212	. . . . . . 7 |- ( ph -> ( Z ./ W ) = ( Z .x. ( ( invr ` R ) ` W ) ) )
88	87	oveq2d 6044	. . . . . 6 |- ( ph -> ( X .x. ( Z ./ W ) ) = ( X .x. ( Z .x. ( ( invr ` R ) ` W ) ) ) )
89	86, 88	eqtr4d 2267	. . . . 5 |- ( ph -> ( ( X .x. Z ) .x. ( ( invr ` R ) ` W ) ) = ( X .x. ( Z ./ W ) ) )
90	89	oveq1d 6043	. . . 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 14093	. . . . 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 14093	. . . . 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 2273	. . 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 2275	. 2 |- ( ph -> ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) = ( ( X .x. Z ) ./ ( Y .x. W ) ) )
97	32, 96	eqtrd 2264	1 |- ( ph -> ( ( X ./ Y ) .x. ( Z ./ W ) ) = ( ( X .x. Z ) ./ ( Y .x. W ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   Basecbs 13145   ↾_s cress 13146   +g cplusg 13223   .rcmulr 13224   Grpcgrp 13646   invgcminusg 13647  mulGrpcmgp 13997  SRingcsrg 14040   Ringcrg 14073   CRingccrg 14074  Unitcui 14164   invrcinvr 14198  /_rcdvr 14209

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-lttrn 8189  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-tpos 6454  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-3 9245  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-iress 13153  df-plusg 13236  df-mulr 13237  df-0g 13404  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-minusg 13650  df-cmn 13936  df-abl 13937  df-mgp 13998  df-ur 14037  df-srg 14041  df-ring 14075  df-cring 14076  df-oppr 14145  df-dvdsr 14166  df-unit 14167  df-invr 14199  df-dvr 14210

This theorem is referenced by: (None)