Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rdivmuldivd | Structured version Visualization version Unicode version |
Description: Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 30-Oct-2017.) |
Ref | Expression |
---|---|
dvrdir.b | |
dvrdir.u | Unit |
dvrdir.p | |
dvrdir.t | /r |
rdivmuldivd.p | |
rdivmuldivd.r | |
rdivmuldivd.a | |
rdivmuldivd.b | |
rdivmuldivd.c | |
rdivmuldivd.d |
Ref | Expression |
---|---|
rdivmuldivd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdivmuldivd.a | . . . 4 | |
2 | rdivmuldivd.b | . . . 4 | |
3 | dvrdir.b | . . . . . 6 | |
4 | rdivmuldivd.p | . . . . . 6 | |
5 | dvrdir.u | . . . . . 6 Unit | |
6 | eqid 2622 | . . . . . 6 | |
7 | dvrdir.t | . . . . . 6 /r | |
8 | 3, 4, 5, 6, 7 | dvrval 18685 | . . . . 5 |
9 | 8 | oveq1d 6665 | . . . 4 |
10 | 1, 2, 9 | syl2anc 693 | . . 3 |
11 | rdivmuldivd.r | . . . . 5 | |
12 | crngring 18558 | . . . . 5 | |
13 | 11, 12 | syl 17 | . . . 4 |
14 | 3, 5 | unitss 18660 | . . . . 5 |
15 | 5, 6 | unitinvcl 18674 | . . . . . 6 |
16 | 13, 2, 15 | syl2anc 693 | . . . . 5 |
17 | 14, 16 | sseldi 3601 | . . . 4 |
18 | rdivmuldivd.c | . . . . 5 | |
19 | rdivmuldivd.d | . . . . 5 | |
20 | 3, 5, 7 | dvrcl 18686 | . . . . 5 |
21 | 13, 18, 19, 20 | syl3anc 1326 | . . . 4 |
22 | 3, 4 | ringass 18564 | . . . 4 |
23 | 13, 1, 17, 21, 22 | syl13anc 1328 | . . 3 |
24 | 3, 4 | crngcom 18562 | . . . . 5 |
25 | 11, 17, 21, 24 | syl3anc 1326 | . . . 4 |
26 | 25 | oveq2d 6666 | . . 3 |
27 | 10, 23, 26 | 3eqtrd 2660 | . 2 |
28 | eqid 2622 | . . . . . . . 8 mulGrp ↾s mulGrp ↾s | |
29 | 5, 28 | unitgrp 18667 | . . . . . . 7 mulGrp ↾s |
30 | 13, 29 | syl 17 | . . . . . 6 mulGrp ↾s |
31 | 5, 28 | unitgrpbas 18666 | . . . . . . 7 mulGrp ↾s |
32 | eqid 2622 | . . . . . . 7 mulGrp ↾s mulGrp ↾s | |
33 | 5, 28, 6 | invrfval 18673 | . . . . . . 7 mulGrp ↾s |
34 | 31, 32, 33 | grpinvadd 17493 | . . . . . 6 mulGrp ↾s mulGrp ↾s mulGrp ↾s |
35 | 30, 2, 19, 34 | syl3anc 1326 | . . . . 5 mulGrp ↾s mulGrp ↾s |
36 | fvex 6201 | . . . . . . . . . . 11 Unit | |
37 | 5, 36 | eqeltri 2697 | . . . . . . . . . 10 |
38 | eqid 2622 | . . . . . . . . . . 11 ↾s ↾s | |
39 | eqid 2622 | . . . . . . . . . . 11 mulGrp mulGrp | |
40 | 38, 39 | mgpress 18500 | . . . . . . . . . 10 mulGrp ↾s mulGrp ↾s |
41 | 13, 37, 40 | sylancl 694 | . . . . . . . . 9 mulGrp ↾s mulGrp ↾s |
42 | 41 | fveq2d 6195 | . . . . . . . 8 mulGrp ↾s mulGrp ↾s |
43 | eqid 2622 | . . . . . . . . 9 mulGrp ↾s mulGrp ↾s | |
44 | 38, 4 | ressmulr 16006 | . . . . . . . . . 10 ↾s |
45 | 37, 44 | ax-mp 5 | . . . . . . . . 9 ↾s |
46 | 43, 45 | mgpplusg 18493 | . . . . . . . 8 mulGrp ↾s |
47 | 42, 46 | syl6reqr 2675 | . . . . . . 7 mulGrp ↾s |
48 | 47 | oveqd 6667 | . . . . . 6 mulGrp ↾s |
49 | 48 | fveq2d 6195 | . . . . 5 mulGrp ↾s |
50 | 47 | oveqd 6667 | . . . . 5 mulGrp ↾s |
51 | 35, 49, 50 | 3eqtr4d 2666 | . . . 4 |
52 | 51 | oveq2d 6666 | . . 3 |
53 | 3, 4 | ringcl 18561 | . . . . 5 |
54 | 13, 1, 18, 53 | syl3anc 1326 | . . . 4 |
55 | 5, 4 | unitmulcl 18664 | . . . . 5 |
56 | 13, 2, 19, 55 | syl3anc 1326 | . . . 4 |
57 | 3, 4, 5, 6, 7 | dvrval 18685 | . . . 4 |
58 | 54, 56, 57 | syl2anc 693 | . . 3 |
59 | 5, 6 | unitinvcl 18674 | . . . . . . . . 9 |
60 | 13, 19, 59 | syl2anc 693 | . . . . . . . 8 |
61 | 14, 60 | sseldi 3601 | . . . . . . 7 |
62 | 3, 4 | ringass 18564 | . . . . . . 7 |
63 | 13, 1, 18, 61, 62 | syl13anc 1328 | . . . . . 6 |
64 | 3, 4, 5, 6, 7 | dvrval 18685 | . . . . . . . 8 |
65 | 18, 19, 64 | syl2anc 693 | . . . . . . 7 |
66 | 65 | oveq2d 6666 | . . . . . 6 |
67 | 63, 66 | eqtr4d 2659 | . . . . 5 |
68 | 67 | oveq1d 6665 | . . . 4 |
69 | 3, 4 | ringass 18564 | . . . . 5 |
70 | 13, 54, 61, 17, 69 | syl13anc 1328 | . . . 4 |
71 | 3, 4 | ringass 18564 | . . . . 5 |
72 | 13, 1, 21, 17, 71 | syl13anc 1328 | . . . 4 |
73 | 68, 70, 72 | 3eqtr3rd 2665 | . . 3 |
74 | 52, 58, 73 | 3eqtr4rd 2667 | . 2 |
75 | 27, 74 | eqtrd 2656 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cvv 3200 cfv 5888 (class class class)co 6650 cbs 15857 ↾s cress 15858 cplusg 15941 cmulr 15942 cgrp 17422 mulGrpcmgp 18489 crg 18547 ccrg 18548 Unitcui 18639 cinvr 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 |
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