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Mirrors > Home > MPE Home > Th. List > ip2subdi | Structured version Visualization version Unicode version |
Description: Distributive law for inner product subtraction. (Contributed by Mario Carneiro, 8-Oct-2015.) |
Ref | Expression |
---|---|
phlsrng.f | Scalar |
phllmhm.h | |
phllmhm.v | |
ipsubdir.m | |
ipsubdir.s | |
ip2subdi.p | |
ip2subdi.1 | |
ip2subdi.2 | |
ip2subdi.3 | |
ip2subdi.4 | |
ip2subdi.5 |
Ref | Expression |
---|---|
ip2subdi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . 4 | |
2 | ip2subdi.p | . . . 4 | |
3 | ipsubdir.s | . . . 4 | |
4 | ip2subdi.1 | . . . . . . 7 | |
5 | phllmod 19975 | . . . . . . 7 | |
6 | 4, 5 | syl 17 | . . . . . 6 |
7 | phlsrng.f | . . . . . . 7 Scalar | |
8 | 7 | lmodring 18871 | . . . . . 6 |
9 | 6, 8 | syl 17 | . . . . 5 |
10 | ringabl 18580 | . . . . 5 | |
11 | 9, 10 | syl 17 | . . . 4 |
12 | ip2subdi.2 | . . . . 5 | |
13 | ip2subdi.4 | . . . . 5 | |
14 | phllmhm.h | . . . . . 6 | |
15 | phllmhm.v | . . . . . 6 | |
16 | 7, 14, 15, 1 | ipcl 19978 | . . . . 5 |
17 | 4, 12, 13, 16 | syl3anc 1326 | . . . 4 |
18 | ip2subdi.5 | . . . . 5 | |
19 | 7, 14, 15, 1 | ipcl 19978 | . . . . 5 |
20 | 4, 12, 18, 19 | syl3anc 1326 | . . . 4 |
21 | ip2subdi.3 | . . . . 5 | |
22 | 7, 14, 15, 1 | ipcl 19978 | . . . . 5 |
23 | 4, 21, 13, 22 | syl3anc 1326 | . . . 4 |
24 | 1, 2, 3, 11, 17, 20, 23 | ablsubsub4 18224 | . . 3 |
25 | 24 | oveq1d 6665 | . 2 |
26 | ipsubdir.m | . . . . . 6 | |
27 | 15, 26 | lmodvsubcl 18908 | . . . . 5 |
28 | 6, 13, 18, 27 | syl3anc 1326 | . . . 4 |
29 | 7, 14, 15, 26, 3 | ipsubdir 19987 | . . . 4 |
30 | 4, 12, 21, 28, 29 | syl13anc 1328 | . . 3 |
31 | 7, 14, 15, 26, 3 | ipsubdi 19988 | . . . . 5 |
32 | 4, 12, 13, 18, 31 | syl13anc 1328 | . . . 4 |
33 | 7, 14, 15, 26, 3 | ipsubdi 19988 | . . . . 5 |
34 | 4, 21, 13, 18, 33 | syl13anc 1328 | . . . 4 |
35 | 32, 34 | oveq12d 6668 | . . 3 |
36 | ringgrp 18552 | . . . . . 6 | |
37 | 9, 36 | syl 17 | . . . . 5 |
38 | 1, 3 | grpsubcl 17495 | . . . . 5 |
39 | 37, 17, 20, 38 | syl3anc 1326 | . . . 4 |
40 | 7, 14, 15, 1 | ipcl 19978 | . . . . 5 |
41 | 4, 21, 18, 40 | syl3anc 1326 | . . . 4 |
42 | 1, 2, 3, 11, 39, 23, 41 | ablsubsub 18223 | . . 3 |
43 | 30, 35, 42 | 3eqtrd 2660 | . 2 |
44 | 1, 2 | ringacl 18578 | . . . 4 |
45 | 9, 20, 23, 44 | syl3anc 1326 | . . 3 |
46 | 1, 2, 3 | abladdsub 18220 | . . 3 |
47 | 11, 17, 41, 45, 46 | syl13anc 1328 | . 2 |
48 | 25, 43, 47 | 3eqtr4d 2666 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 Scalarcsca 15944 cip 15946 cgrp 17422 csg 17424 cabl 18194 crg 18547 clmod 18863 cphl 19969 |
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-map 7859 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-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-grp 17425 df-minusg 17426 df-sbg 17427 df-ghm 17658 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-rnghom 18715 df-staf 18845 df-srng 18846 df-lmod 18865 df-lmhm 19022 df-lvec 19103 df-sra 19172 df-rgmod 19173 df-phl 19971 |
This theorem is referenced by: cph2subdi 23010 ipcau2 23033 tchcphlem1 23034 |
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