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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 12466 | . 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 12478 | . 2 ⊢ (𝜑 → 𝐴 Struct 〈1, 8〉) |
10 | 1 | simpri 112 | . . . . 5 ⊢ (Scalar‘ndx) ∈ ℕ |
11 | opexg 4200 | . . . . 5 ⊢ (((Scalar‘ndx) ∈ ℕ ∧ 𝑆 ∈ 𝑌) → 〈(Scalar‘ndx), 𝑆〉 ∈ V) | |
12 | 10, 6, 11 | sylancr 411 | . . . 4 ⊢ (𝜑 → 〈(Scalar‘ndx), 𝑆〉 ∈ V) |
13 | tpid1g 3682 | . . . 4 ⊢ (〈(Scalar‘ndx), 𝑆〉 ∈ V → 〈(Scalar‘ndx), 𝑆〉 ∈ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) | |
14 | elun2 3285 | . . . 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 2258 | . 2 ⊢ (𝜑 → 〈(Scalar‘ndx), 𝑆〉 ∈ 𝐴) |
17 | 1, 9, 6, 16 | opelstrsl 12433 | 1 ⊢ (𝜑 → 𝑆 = (Scalar‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ∪ cun 3109 {ctp 3572 〈cop 3573 ‘cfv 5182 1c1 7745 ℕcn 8848 8c8 8905 ndxcnx 12334 Slot cslot 12336 Basecbs 12337 +gcplusg 12399 .rcmulr 12400 Scalarcsca 12402 ·𝑠 cvsca 12403 ·𝑖cip 12404 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-tp 3578 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-n0 9106 df-z 9183 df-uz 9458 df-fz 9936 df-struct 12339 df-ndx 12340 df-slot 12341 df-base 12343 df-plusg 12412 df-mulr 12413 df-sca 12415 df-vsca 12416 df-ip 12417 |
This theorem is referenced by: (None) |
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