Theorem rdivmuldivd 14177

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 14041	. . . . 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 14167	. . . 4 |- ( ph -> ( X ./ Y ) = ( X .x. ( ( invr ` R ) ` Y ) ) )
15	14	oveq1d 6033	. . 3 |- ( ph -> ( ( X ./ Y ) .x. ( Z ./ W ) ) = ( ( X .x. ( ( invr ` R ) ` Y ) ) .x. ( Z ./ W ) ) )
16		ringsrg 14079	. . . . . . 7 |- ( R e. Ring -> R e. SRing )
17	11, 16	syl 14	. . . . . 6 |- ( ph -> R e. SRing )
18	2, 6, 17	unitssd 14142	. . . . 5 |- ( ph -> U C_ B )
19		eqid 2231	. . . . . . 7 |- ( invr ` R ) = ( invr ` R )
20	5, 19	unitinvcl 14156	. . . . . 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 3228	. . . 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 14168	. . . . 5 |- ( ( R e. Ring /\ Z e. B /\ W e. U ) -> ( Z ./ W ) e. B )
26	11, 23, 24, 25	syl3anc 1273	. . . 4 |- ( ph -> ( Z ./ W ) e. B )
27	1, 3	ringass 14048	. . . 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 1275	. . 3 |- ( ph -> ( ( X .x. ( ( invr ` R ) ` Y ) ) .x. ( Z ./ W ) ) = ( X .x. ( ( ( invr ` R ) ` Y ) .x. ( Z ./ W ) ) ) )
29	1, 3	crngcom 14046	. . . . 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 1273	. . . 4 |- ( ph -> ( ( ( invr ` R ) ` Y ) .x. ( Z ./ W ) ) = ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) )
31	30	oveq2d 6034	. . 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 14149	. . . . . . . 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 14148	. . . . . . . 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 13679	. . . . . . 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 1273	. . . . . 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 14155	. . . . . . 7 |- ( ph -> ( invr ` R ) = ( invg ` ( (mulGrp ` R ) ↾s U ) ) )
46	45	fveq1d 5641	. . . . . 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 5641	. . . . . . 7 |- ( ph -> ( ( invr ` R ) ` W ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` W ) )
48	45	fveq1d 5641	. . . . . . 7 |- ( ph -> ( ( invr ` R ) ` Y ) = ( ( invg ` ( (mulGrp ` R ) ↾s U ) ) ` Y ) )
49	47, 48	oveq12d 6036	. . . . . 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 13159	. . . . . . . . . . . 12 |- Base Fn _V
52	10	elexd 2816	. . . . . . . . . . . 12 |- ( ph -> R e. _V )
53		funfvex 5656	. . . . . . . . . . . . 13 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
54	53	funfni 5432	. . . . . . . . . . . 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 4229	. . . . . . . . 9 |- ( ph -> U e. _V )
58		ressex 13166	. . . . . . . . . 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 13956	. . . . . . . . . 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 13246	. . . . . . . . 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 13963	. . . . . . . . . 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 5643	. . . . . . . 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 6035	. . . . . 6 |- ( ph -> ( Y .x. W ) = ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) )
73	72	fveq2d 5643	. . . . 5 |- ( ph -> ( ( invr ` R ) ` ( Y .x. W ) ) = ( ( invr ` R ) ` ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) ) )
74	71	oveqd 6035	. . . . 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 6034	. . 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 14045	. . . . 5 |- ( ( R e. Ring /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
78	11, 12, 23, 77	syl3anc 1273	. . . 4 |- ( ph -> ( X .x. Z ) e. B )
79	5, 3	unitmulcl 14146	. . . . 5 |- ( ( R e. Ring /\ Y e. U /\ W e. U ) -> ( Y .x. W ) e. U )
80	11, 13, 24, 79	syl3anc 1273	. . . 4 |- ( ph -> ( Y .x. W ) e. U )
81	2, 4, 6, 7, 9, 11, 78, 80	dvrvald 14167	. . 3 |- ( ph -> ( ( X .x. Z ) ./ ( Y .x. W ) ) = ( ( X .x. Z ) .x. ( ( invr ` R ) ` ( Y .x. W ) ) ) )
82	5, 19	unitinvcl 14156	. . . . . . . . 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 3228	. . . . . . 7 |- ( ph -> ( ( invr ` R ) ` W ) e. B )
85	1, 3	ringass 14048	. . . . . . 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 1275	. . . . . 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 14167	. . . . . . 7 |- ( ph -> ( Z ./ W ) = ( Z .x. ( ( invr ` R ) ` W ) ) )
88	87	oveq2d 6034	. . . . . 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 6033	. . . 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 14048	. . . . 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 1275	. . . 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 14048	. . . . 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 1275	. . . 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 1397   e. wcel 2202   _Vcvv 2802   Fn wfn 5321   ` cfv 5326  (class class class)co 6018   Basecbs 13100   ↾_s cress 13101   +g cplusg 13178   .rcmulr 13179   Grpcgrp 13601   invgcminusg 13602  mulGrpcmgp 13952  SRingcsrg 13995   Ringcrg 14028   CRingccrg 14029  Unitcui 14119   invrcinvr 14153  /_rcdvr 14164

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-tpos 6411  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13103  df-slot 13104  df-base 13106  df-sets 13107  df-iress 13108  df-plusg 13191  df-mulr 13192  df-0g 13359  df-mgm 13457  df-sgrp 13503  df-mnd 13518  df-grp 13604  df-minusg 13605  df-cmn 13891  df-abl 13892  df-mgp 13953  df-ur 13992  df-srg 13996  df-ring 14030  df-cring 14031  df-oppr 14100  df-dvdsr 14121  df-unit 14122  df-invr 14154  df-dvr 14165

This theorem is referenced by: (None)