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Mirrors > Home > ILE Home > Th. List > ipsstrd | Unicode version |
Description: A constructed inner product space is a structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.) |
Ref | Expression |
---|---|
ipspart.a | Scalar |
ipsstrd.b | |
ipsstrd.p | |
ipsstrd.r | |
ipsstrd.s | |
ipsstrd.x | |
ipsstrd.i |
Ref | Expression |
---|---|
ipsstrd | Struct |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipspart.a | . 2 Scalar | |
2 | ipsstrd.b | . . . 4 | |
3 | ipsstrd.p | . . . 4 | |
4 | ipsstrd.r | . . . 4 | |
5 | eqid 2139 | . . . . 5 | |
6 | 5 | rngstrg 12077 | . . . 4 Struct |
7 | 2, 3, 4, 6 | syl3anc 1216 | . . 3 Struct |
8 | ipsstrd.s | . . . 4 | |
9 | ipsstrd.x | . . . 4 | |
10 | ipsstrd.i | . . . 4 | |
11 | 5nn 8887 | . . . . 5 | |
12 | scandx 12089 | . . . . 5 Scalar | |
13 | 5lt6 8902 | . . . . 5 | |
14 | 6nn 8888 | . . . . 5 | |
15 | vscandx 12092 | . . . . 5 | |
16 | 6lt8 8914 | . . . . 5 | |
17 | 8nn 8890 | . . . . 5 | |
18 | ipndx 12100 | . . . . 5 | |
19 | 11, 12, 13, 14, 15, 16, 17, 18 | strle3g 12054 | . . . 4 Scalar Struct |
20 | 8, 9, 10, 19 | syl3anc 1216 | . . 3 Scalar Struct |
21 | 3lt5 8899 | . . . 4 | |
22 | 21 | a1i 9 | . . 3 |
23 | 7, 20, 22 | strleund 12050 | . 2 Scalar Struct |
24 | 1, 23 | eqbrtrid 3963 | 1 Struct |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wcel 1480 cun 3069 ctp 3529 cop 3530 class class class wbr 3929 cfv 5123 c1 7624 clt 7803 c3 8775 c5 8777 c6 8778 c8 8780 Struct cstr 11958 cnx 11959 cbs 11962 cplusg 12024 cmulr 12025 Scalarcsca 12027 cvsca 12028 cip 12029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-addcom 7723 ax-addass 7725 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-0id 7731 ax-rnegex 7732 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-tp 3535 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-inn 8724 df-2 8782 df-3 8783 df-4 8784 df-5 8785 df-6 8786 df-7 8787 df-8 8788 df-n0 8981 df-z 9058 df-uz 9330 df-fz 9794 df-struct 11964 df-ndx 11965 df-slot 11966 df-base 11968 df-plusg 12037 df-mulr 12038 df-sca 12040 df-vsca 12041 df-ip 12042 |
This theorem is referenced by: ipsbased 12104 ipsaddgd 12105 ipsmulrd 12106 ipsscad 12107 ipsvscad 12108 ipsipd 12109 |
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