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Mirrors > Home > ILE Home > Th. List > ipsaddgd | Unicode version |
Description: The additive operation of a constructed inner product space. (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 |
---|---|
ipsaddgd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plusgslid 11981 | . 2 Slot | |
2 | ipspart.a | . . 3 Scalar | |
3 | ipsstrd.b | . . 3 | |
4 | ipsstrd.p | . . 3 | |
5 | ipsstrd.r | . . 3 | |
6 | ipsstrd.s | . . 3 | |
7 | ipsstrd.x | . . 3 | |
8 | ipsstrd.i | . . 3 | |
9 | 2, 3, 4, 5, 6, 7, 8 | ipsstrd 12027 | . 2 Struct |
10 | 1 | simpri 112 | . . . . 5 |
11 | opexg 4120 | . . . . 5 | |
12 | 10, 4, 11 | sylancr 410 | . . . 4 |
13 | tpid2g 3607 | . . . 4 | |
14 | elun1 3213 | . . . 4 Scalar | |
15 | 12, 13, 14 | 3syl 17 | . . 3 Scalar |
16 | 15, 2 | eleqtrrdi 2211 | . 2 |
17 | 1, 9, 4, 16 | opelstrsl 11982 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1316 wcel 1465 cvv 2660 cun 3039 ctp 3499 cop 3500 cfv 5093 c1 7589 cn 8688 c8 8745 cnx 11883 Slot cslot 11885 cbs 11886 cplusg 11948 cmulr 11949 Scalarcsca 11951 cvsca 11952 cip 11953 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-tp 3505 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-5 8750 df-6 8751 df-7 8752 df-8 8753 df-n0 8946 df-z 9023 df-uz 9295 df-fz 9759 df-struct 11888 df-ndx 11889 df-slot 11890 df-base 11892 df-plusg 11961 df-mulr 11962 df-sca 11964 df-vsca 11965 df-ip 11966 |
This theorem is referenced by: (None) |
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