Theorem rdivmuldivd 29791

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		rdivmuldivd.a	. . . 4 |- ( ph -> X e. B )
2		rdivmuldivd.b	. . . 4 |- ( ph -> Y e. U )
3		dvrdir.b	. . . . . 6 |- B = ( Base ` R )
4		rdivmuldivd.p	. . . . . 6 |- .x. = ( .r ` R )
5		dvrdir.u	. . . . . 6 |- U = (Unit ` R )
6		eqid 2622	. . . . . 6 |- ( invr ` R ) = ( invr ` R )
7		dvrdir.t	. . . . . 6 |- ./ = (/r ` R )
8	3, 4, 5, 6, 7	dvrval 18685	. . . . 5 |- ( ( X e. B /\ Y e. U ) -> ( X ./ Y ) = ( X .x. ( ( invr ` R ) ` Y ) ) )
9	8	oveq1d 6665	. . . 4 |- ( ( X e. B /\ Y e. U ) -> ( ( X ./ Y ) .x. ( Z ./ W ) ) = ( ( X .x. ( ( invr ` R ) ` Y ) ) .x. ( Z ./ W ) ) )
10	1, 2, 9	syl2anc 693	. . 3 |- ( ph -> ( ( X ./ Y ) .x. ( Z ./ W ) ) = ( ( X .x. ( ( invr ` R ) ` Y ) ) .x. ( Z ./ W ) ) )
11		rdivmuldivd.r	. . . . 5 |- ( ph -> R e. CRing )
12		crngring 18558	. . . . 5 |- ( R e. CRing -> R e. Ring )
13	11, 12	syl 17	. . . 4 |- ( ph -> R e. Ring )
14	3, 5	unitss 18660	. . . . 5 |- U C_ B
15	5, 6	unitinvcl 18674	. . . . . 6 |- ( ( R e. Ring /\ Y e. U ) -> ( ( invr ` R ) ` Y ) e. U )
16	13, 2, 15	syl2anc 693	. . . . 5 |- ( ph -> ( ( invr ` R ) ` Y ) e. U )
17	14, 16	sseldi 3601	. . . 4 |- ( ph -> ( ( invr ` R ) ` Y ) e. B )
18		rdivmuldivd.c	. . . . 5 |- ( ph -> Z e. B )
19		rdivmuldivd.d	. . . . 5 |- ( ph -> W e. U )
20	3, 5, 7	dvrcl 18686	. . . . 5 |- ( ( R e. Ring /\ Z e. B /\ W e. U ) -> ( Z ./ W ) e. B )
21	13, 18, 19, 20	syl3anc 1326	. . . 4 |- ( ph -> ( Z ./ W ) e. B )
22	3, 4	ringass 18564	. . . 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 ) ) ) )
23	13, 1, 17, 21, 22	syl13anc 1328	. . 3 |- ( ph -> ( ( X .x. ( ( invr ` R ) ` Y ) ) .x. ( Z ./ W ) ) = ( X .x. ( ( ( invr ` R ) ` Y ) .x. ( Z ./ W ) ) ) )
24	3, 4	crngcom 18562	. . . . 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 ) ) )
25	11, 17, 21, 24	syl3anc 1326	. . . 4 |- ( ph -> ( ( ( invr ` R ) ` Y ) .x. ( Z ./ W ) ) = ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) )
26	25	oveq2d 6666	. . 3 |- ( ph -> ( X .x. ( ( ( invr ` R ) ` Y ) .x. ( Z ./ W ) ) ) = ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) )
27	10, 23, 26	3eqtrd 2660	. 2 |- ( ph -> ( ( X ./ Y ) .x. ( Z ./ W ) ) = ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) )
28		eqid 2622	. . . . . . . 8 |- ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s U )
29	5, 28	unitgrp 18667	. . . . . . 7 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s U ) e. Grp )
30	13, 29	syl 17	. . . . . 6 |- ( ph -> ( (mulGrp ` R ) ↾s U ) e. Grp )
31	5, 28	unitgrpbas 18666	. . . . . . 7 |- U = ( Base ` ( (mulGrp ` R ) ↾s U ) )
32		eqid 2622	. . . . . . 7 |- ( +g ` ( (mulGrp ` R ) ↾s U ) ) = ( +g ` ( (mulGrp ` R ) ↾s U ) )
33	5, 28, 6	invrfval 18673	. . . . . . 7 |- ( invr ` R ) = ( invg ` ( (mulGrp ` R ) ↾s U ) )
34	31, 32, 33	grpinvadd 17493	. . . . . 6 |- ( ( ( (mulGrp ` R ) ↾s U ) e. Grp /\ Y e. U /\ W e. U ) -> ( ( invr ` R ) ` ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) ) = ( ( ( invr ` R ) ` W ) ( +g ` ( (mulGrp ` R ) ↾s U ) ) ( ( invr ` R ) ` Y ) ) )
35	30, 2, 19, 34	syl3anc 1326	. . . . 5 |- ( ph -> ( ( invr ` R ) ` ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) ) = ( ( ( invr ` R ) ` W ) ( +g ` ( (mulGrp ` R ) ↾s U ) ) ( ( invr ` R ) ` Y ) ) )
36		fvex 6201	. . . . . . . . . . 11 |- (Unit ` R ) e. _V
37	5, 36	eqeltri 2697	. . . . . . . . . 10 |- U e. _V
38		eqid 2622	. . . . . . . . . . 11 |- ( R ↾s U ) = ( R ↾s U )
39		eqid 2622	. . . . . . . . . . 11 |- (mulGrp ` R ) = (mulGrp ` R )
40	38, 39	mgpress 18500	. . . . . . . . . 10 |- ( ( R e. Ring /\ U e. _V ) -> ( (mulGrp ` R ) ↾s U ) = (mulGrp ` ( R ↾s U ) ) )
41	13, 37, 40	sylancl 694	. . . . . . . . 9 |- ( ph -> ( (mulGrp ` R ) ↾s U ) = (mulGrp ` ( R ↾s U ) ) )
42	41	fveq2d 6195	. . . . . . . 8 |- ( ph -> ( +g ` ( (mulGrp ` R ) ↾s U ) ) = ( +g ` (mulGrp ` ( R ↾s U ) ) ) )
43		eqid 2622	. . . . . . . . 9 |- (mulGrp ` ( R ↾s U ) ) = (mulGrp ` ( R ↾s U ) )
44	38, 4	ressmulr 16006	. . . . . . . . . 10 |- ( U e. _V -> .x. = ( .r ` ( R ↾s U ) ) )
45	37, 44	ax-mp 5	. . . . . . . . 9 |- .x. = ( .r ` ( R ↾s U ) )
46	43, 45	mgpplusg 18493	. . . . . . . 8 |- .x. = ( +g ` (mulGrp ` ( R ↾s U ) ) )
47	42, 46	syl6reqr 2675	. . . . . . 7 |- ( ph -> .x. = ( +g ` ( (mulGrp ` R ) ↾s U ) ) )
48	47	oveqd 6667	. . . . . 6 |- ( ph -> ( Y .x. W ) = ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) )
49	48	fveq2d 6195	. . . . 5 |- ( ph -> ( ( invr ` R ) ` ( Y .x. W ) ) = ( ( invr ` R ) ` ( Y ( +g ` ( (mulGrp ` R ) ↾s U ) ) W ) ) )
50	47	oveqd 6667	. . . . 5 |- ( ph -> ( ( ( invr ` R ) ` W ) .x. ( ( invr ` R ) ` Y ) ) = ( ( ( invr ` R ) ` W ) ( +g ` ( (mulGrp ` R ) ↾s U ) ) ( ( invr ` R ) ` Y ) ) )
51	35, 49, 50	3eqtr4d 2666	. . . 4 |- ( ph -> ( ( invr ` R ) ` ( Y .x. W ) ) = ( ( ( invr ` R ) ` W ) .x. ( ( invr ` R ) ` Y ) ) )
52	51	oveq2d 6666	. . 3 |- ( ph -> ( ( X .x. Z ) .x. ( ( invr ` R ) ` ( Y .x. W ) ) ) = ( ( X .x. Z ) .x. ( ( ( invr ` R ) ` W ) .x. ( ( invr ` R ) ` Y ) ) ) )
53	3, 4	ringcl 18561	. . . . 5 |- ( ( R e. Ring /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
54	13, 1, 18, 53	syl3anc 1326	. . . 4 |- ( ph -> ( X .x. Z ) e. B )
55	5, 4	unitmulcl 18664	. . . . 5 |- ( ( R e. Ring /\ Y e. U /\ W e. U ) -> ( Y .x. W ) e. U )
56	13, 2, 19, 55	syl3anc 1326	. . . 4 |- ( ph -> ( Y .x. W ) e. U )
57	3, 4, 5, 6, 7	dvrval 18685	. . . 4 |- ( ( ( X .x. Z ) e. B /\ ( Y .x. W ) e. U ) -> ( ( X .x. Z ) ./ ( Y .x. W ) ) = ( ( X .x. Z ) .x. ( ( invr ` R ) ` ( Y .x. W ) ) ) )
58	54, 56, 57	syl2anc 693	. . 3 |- ( ph -> ( ( X .x. Z ) ./ ( Y .x. W ) ) = ( ( X .x. Z ) .x. ( ( invr ` R ) ` ( Y .x. W ) ) ) )
59	5, 6	unitinvcl 18674	. . . . . . . . 9 |- ( ( R e. Ring /\ W e. U ) -> ( ( invr ` R ) ` W ) e. U )
60	13, 19, 59	syl2anc 693	. . . . . . . 8 |- ( ph -> ( ( invr ` R ) ` W ) e. U )
61	14, 60	sseldi 3601	. . . . . . 7 |- ( ph -> ( ( invr ` R ) ` W ) e. B )
62	3, 4	ringass 18564	. . . . . . 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 ) ) ) )
63	13, 1, 18, 61, 62	syl13anc 1328	. . . . . 6 |- ( ph -> ( ( X .x. Z ) .x. ( ( invr ` R ) ` W ) ) = ( X .x. ( Z .x. ( ( invr ` R ) ` W ) ) ) )
64	3, 4, 5, 6, 7	dvrval 18685	. . . . . . . 8 |- ( ( Z e. B /\ W e. U ) -> ( Z ./ W ) = ( Z .x. ( ( invr ` R ) ` W ) ) )
65	18, 19, 64	syl2anc 693	. . . . . . 7 |- ( ph -> ( Z ./ W ) = ( Z .x. ( ( invr ` R ) ` W ) ) )
66	65	oveq2d 6666	. . . . . 6 |- ( ph -> ( X .x. ( Z ./ W ) ) = ( X .x. ( Z .x. ( ( invr ` R ) ` W ) ) ) )
67	63, 66	eqtr4d 2659	. . . . 5 |- ( ph -> ( ( X .x. Z ) .x. ( ( invr ` R ) ` W ) ) = ( X .x. ( Z ./ W ) ) )
68	67	oveq1d 6665	. . . 4 |- ( ph -> ( ( ( X .x. Z ) .x. ( ( invr ` R ) ` W ) ) .x. ( ( invr ` R ) ` Y ) ) = ( ( X .x. ( Z ./ W ) ) .x. ( ( invr ` R ) ` Y ) ) )
69	3, 4	ringass 18564	. . . . 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 ) ) ) )
70	13, 54, 61, 17, 69	syl13anc 1328	. . . 4 |- ( ph -> ( ( ( X .x. Z ) .x. ( ( invr ` R ) ` W ) ) .x. ( ( invr ` R ) ` Y ) ) = ( ( X .x. Z ) .x. ( ( ( invr ` R ) ` W ) .x. ( ( invr ` R ) ` Y ) ) ) )
71	3, 4	ringass 18564	. . . . 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 ) ) ) )
72	13, 1, 21, 17, 71	syl13anc 1328	. . . 4 |- ( ph -> ( ( X .x. ( Z ./ W ) ) .x. ( ( invr ` R ) ` Y ) ) = ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) )
73	68, 70, 72	3eqtr3rd 2665	. . 3 |- ( ph -> ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) = ( ( X .x. Z ) .x. ( ( ( invr ` R ) ` W ) .x. ( ( invr ` R ) ` Y ) ) ) )
74	52, 58, 73	3eqtr4rd 2667	. 2 |- ( ph -> ( X .x. ( ( Z ./ W ) .x. ( ( invr ` R ) ` Y ) ) ) = ( ( X .x. Z ) ./ ( Y .x. W ) ) )
75	27, 74	eqtrd 2656	1 |- ( ph -> ( ( X ./ Y ) .x. ( Z ./ W ) ) = ( ( X .x. Z ) ./ ( Y .x. W ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = 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   Ringcrg 18547   CRingccrg 18548  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-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683

This theorem is referenced by:  qqhrhm  30033