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Mirrors > Home > MPE Home > Th. List > sraip | Structured version Visualization version Unicode version |
Description: The inner product operation of a subring algebra. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
Ref | Expression |
---|---|
srapart.a | subringAlg |
srapart.s |
Ref | Expression |
---|---|
sraip |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srapart.a | . . . . . 6 subringAlg | |
2 | 1 | adantl 482 | . . . . 5 subringAlg |
3 | srapart.s | . . . . . 6 | |
4 | sraval 19176 | . . . . . 6 subringAlg sSet Scalar ↾s sSet sSet | |
5 | 3, 4 | sylan2 491 | . . . . 5 subringAlg sSet Scalar ↾s sSet sSet |
6 | 2, 5 | eqtrd 2656 | . . . 4 sSet Scalar ↾s sSet sSet |
7 | 6 | fveq2d 6195 | . . 3 sSet Scalar ↾s sSet sSet |
8 | ovex 6678 | . . . 4 sSet Scalar ↾s sSet | |
9 | fvex 6201 | . . . 4 | |
10 | ipid 16023 | . . . . 5 Slot | |
11 | 10 | setsid 15914 | . . . 4 sSet Scalar ↾s sSet sSet Scalar ↾s sSet sSet |
12 | 8, 9, 11 | mp2an 708 | . . 3 sSet Scalar ↾s sSet sSet |
13 | 7, 12 | syl6reqr 2675 | . 2 |
14 | 10 | str0 15911 | . . 3 |
15 | fvprc 6185 | . . . 4 | |
16 | 15 | adantr 481 | . . 3 |
17 | fvprc 6185 | . . . . . . 7 subringAlg | |
18 | 17 | fveq1d 6193 | . . . . . 6 subringAlg |
19 | 0fv 6227 | . . . . . 6 | |
20 | 18, 19 | syl6eq 2672 | . . . . 5 subringAlg |
21 | 1, 20 | sylan9eqr 2678 | . . . 4 |
22 | 21 | fveq2d 6195 | . . 3 |
23 | 14, 16, 22 | 3eqtr4a 2682 | . 2 |
24 | 13, 23 | pm2.61ian 831 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 wceq 1483 wcel 1990 cvv 3200 wss 3574 c0 3915 cop 4183 cfv 5888 (class class class)co 6650 cnx 15854 sSet csts 15855 cbs 15857 ↾s cress 15858 cmulr 15942 Scalarcsca 15944 cvsca 15945 cip 15946 subringAlg csra 19168 |
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-i2m1 10004 ax-1ne0 10005 ax-rrecex 10008 ax-cnre 10009 |
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-ral 2917 df-rex 2918 df-reu 2919 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-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 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-sets 15864 df-ip 15959 df-sra 19172 |
This theorem is referenced by: frlmip 20117 rrxip 23178 |
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