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Mirrors > Home > ILE Home > Th. List > ipsstrd | GIF 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 | ⊢ 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 |
---|---|
ipsstrd | ⊢ (𝜑 → 𝐴 Struct 〈1, 8〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipspart.a | . 2 ⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) | |
2 | ipsstrd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
3 | ipsstrd.p | . . . 4 ⊢ (𝜑 → + ∈ 𝑊) | |
4 | ipsstrd.r | . . . 4 ⊢ (𝜑 → × ∈ 𝑋) | |
5 | eqid 2139 | . . . . 5 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} | |
6 | 5 | rngstrg 12074 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ × ∈ 𝑋) → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} Struct 〈1, 3〉) |
7 | 2, 3, 4, 6 | syl3anc 1216 | . . 3 ⊢ (𝜑 → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} Struct 〈1, 3〉) |
8 | ipsstrd.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑌) | |
9 | ipsstrd.x | . . . 4 ⊢ (𝜑 → · ∈ 𝑄) | |
10 | ipsstrd.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑍) | |
11 | 5nn 8884 | . . . . 5 ⊢ 5 ∈ ℕ | |
12 | scandx 12086 | . . . . 5 ⊢ (Scalar‘ndx) = 5 | |
13 | 5lt6 8899 | . . . . 5 ⊢ 5 < 6 | |
14 | 6nn 8885 | . . . . 5 ⊢ 6 ∈ ℕ | |
15 | vscandx 12089 | . . . . 5 ⊢ ( ·𝑠 ‘ndx) = 6 | |
16 | 6lt8 8911 | . . . . 5 ⊢ 6 < 8 | |
17 | 8nn 8887 | . . . . 5 ⊢ 8 ∈ ℕ | |
18 | ipndx 12097 | . . . . 5 ⊢ (·𝑖‘ndx) = 8 | |
19 | 11, 12, 13, 14, 15, 16, 17, 18 | strle3g 12051 | . . . 4 ⊢ ((𝑆 ∈ 𝑌 ∧ · ∈ 𝑄 ∧ 𝐼 ∈ 𝑍) → {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉} Struct 〈5, 8〉) |
20 | 8, 9, 10, 19 | syl3anc 1216 | . . 3 ⊢ (𝜑 → {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉} Struct 〈5, 8〉) |
21 | 3lt5 8896 | . . . 4 ⊢ 3 < 5 | |
22 | 21 | a1i 9 | . . 3 ⊢ (𝜑 → 3 < 5) |
23 | 7, 20, 22 | strleund 12047 | . 2 ⊢ (𝜑 → ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) Struct 〈1, 8〉) |
24 | 1, 23 | eqbrtrid 3963 | 1 ⊢ (𝜑 → 𝐴 Struct 〈1, 8〉) |
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 1c1 7621 < clt 7800 3c3 8772 5c5 8774 6c6 8775 8c8 8777 Struct cstr 11955 ndxcnx 11956 Basecbs 11959 +gcplusg 12021 .rcmulr 12022 Scalarcsca 12024 ·𝑠 cvsca 12025 ·𝑖cip 12026 |
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 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 |
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 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-5 8782 df-6 8783 df-7 8784 df-8 8785 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 df-struct 11961 df-ndx 11962 df-slot 11963 df-base 11965 df-plusg 12034 df-mulr 12035 df-sca 12037 df-vsca 12038 df-ip 12039 |
This theorem is referenced by: ipsbased 12101 ipsaddgd 12102 ipsmulrd 12103 ipsscad 12104 ipsvscad 12105 ipsipd 12106 |
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