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Mirrors > Home > ILE Home > Th. List > ipsscad | GIF version |
Description: The set of scalars of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.) |
Ref | Expression |
---|---|
ipspart.a | ⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) |
ipsstrd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
ipsstrd.p | ⊢ (𝜑 → + ∈ 𝑊) |
ipsstrd.r | ⊢ (𝜑 → × ∈ 𝑋) |
ipsstrd.s | ⊢ (𝜑 → 𝑆 ∈ 𝑌) |
ipsstrd.x | ⊢ (𝜑 → · ∈ 𝑄) |
ipsstrd.i | ⊢ (𝜑 → 𝐼 ∈ 𝑍) |
Ref | Expression |
---|---|
ipsscad | ⊢ (𝜑 → 𝑆 = (Scalar‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scaslid 12093 | . 2 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ) | |
2 | ipspart.a | . . 3 ⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) | |
3 | ipsstrd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
4 | ipsstrd.p | . . 3 ⊢ (𝜑 → + ∈ 𝑊) | |
5 | ipsstrd.r | . . 3 ⊢ (𝜑 → × ∈ 𝑋) | |
6 | ipsstrd.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑌) | |
7 | ipsstrd.x | . . 3 ⊢ (𝜑 → · ∈ 𝑄) | |
8 | ipsstrd.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑍) | |
9 | 2, 3, 4, 5, 6, 7, 8 | ipsstrd 12105 | . 2 ⊢ (𝜑 → 𝐴 Struct 〈1, 8〉) |
10 | 1 | simpri 112 | . . . . 5 ⊢ (Scalar‘ndx) ∈ ℕ |
11 | opexg 4150 | . . . . 5 ⊢ (((Scalar‘ndx) ∈ ℕ ∧ 𝑆 ∈ 𝑌) → 〈(Scalar‘ndx), 𝑆〉 ∈ V) | |
12 | 10, 6, 11 | sylancr 410 | . . . 4 ⊢ (𝜑 → 〈(Scalar‘ndx), 𝑆〉 ∈ V) |
13 | tpid1g 3635 | . . . 4 ⊢ (〈(Scalar‘ndx), 𝑆〉 ∈ V → 〈(Scalar‘ndx), 𝑆〉 ∈ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) | |
14 | elun2 3244 | . . . 4 ⊢ (〈(Scalar‘ndx), 𝑆〉 ∈ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉} → 〈(Scalar‘ndx), 𝑆〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉})) | |
15 | 12, 13, 14 | 3syl 17 | . . 3 ⊢ (𝜑 → 〈(Scalar‘ndx), 𝑆〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉})) |
16 | 15, 2 | eleqtrrdi 2233 | . 2 ⊢ (𝜑 → 〈(Scalar‘ndx), 𝑆〉 ∈ 𝐴) |
17 | 1, 9, 6, 16 | opelstrsl 12060 | 1 ⊢ (𝜑 → 𝑆 = (Scalar‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∪ cun 3069 {ctp 3529 〈cop 3530 ‘cfv 5123 1c1 7626 ℕcn 8725 8c8 8782 ndxcnx 11961 Slot cslot 11963 Basecbs 11964 +gcplusg 12026 .rcmulr 12027 Scalarcsca 12029 ·𝑠 cvsca 12030 ·𝑖cip 12031 |
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 7716 ax-resscn 7717 ax-1cn 7718 ax-1re 7719 ax-icn 7720 ax-addcl 7721 ax-addrcl 7722 ax-mulcl 7723 ax-addcom 7725 ax-addass 7727 ax-distr 7729 ax-i2m1 7730 ax-0lt1 7731 ax-0id 7733 ax-rnegex 7734 ax-cnre 7736 ax-pre-ltirr 7737 ax-pre-ltwlin 7738 ax-pre-lttrn 7739 ax-pre-apti 7740 ax-pre-ltadd 7741 |
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 7807 df-mnf 7808 df-xr 7809 df-ltxr 7810 df-le 7811 df-sub 7940 df-neg 7941 df-inn 8726 df-2 8784 df-3 8785 df-4 8786 df-5 8787 df-6 8788 df-7 8789 df-8 8790 df-n0 8983 df-z 9060 df-uz 9332 df-fz 9796 df-struct 11966 df-ndx 11967 df-slot 11968 df-base 11970 df-plusg 12039 df-mulr 12040 df-sca 12042 df-vsca 12043 df-ip 12044 |
This theorem is referenced by: (None) |
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