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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 12610 | . 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 12633 | . 2 ⊢ (𝜑 → 𝐴 Struct 〈1, 8〉) |
10 | 1 | simpri 113 | . . . . 5 ⊢ (Scalar‘ndx) ∈ ℕ |
11 | opexg 4228 | . . . . 5 ⊢ (((Scalar‘ndx) ∈ ℕ ∧ 𝑆 ∈ 𝑌) → 〈(Scalar‘ndx), 𝑆〉 ∈ V) | |
12 | 10, 6, 11 | sylancr 414 | . . . 4 ⊢ (𝜑 → 〈(Scalar‘ndx), 𝑆〉 ∈ V) |
13 | tpid1g 3704 | . . . 4 ⊢ (〈(Scalar‘ndx), 𝑆〉 ∈ V → 〈(Scalar‘ndx), 𝑆〉 ∈ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) | |
14 | elun2 3303 | . . . 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 2271 | . 2 ⊢ (𝜑 → 〈(Scalar‘ndx), 𝑆〉 ∈ 𝐴) |
17 | 1, 9, 6, 16 | opelstrsl 12572 | 1 ⊢ (𝜑 → 𝑆 = (Scalar‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∪ cun 3127 {ctp 3594 〈cop 3595 ‘cfv 5216 1c1 7811 ℕcn 8918 8c8 8975 ndxcnx 12458 Slot cslot 12460 Basecbs 12461 +gcplusg 12535 .rcmulr 12536 Scalarcsca 12538 ·𝑠 cvsca 12539 ·𝑖cip 12540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-tp 3600 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-inn 8919 df-2 8977 df-3 8978 df-4 8979 df-5 8980 df-6 8981 df-7 8982 df-8 8983 df-n0 9176 df-z 9253 df-uz 9528 df-fz 10008 df-struct 12463 df-ndx 12464 df-slot 12465 df-base 12467 df-plusg 12548 df-mulr 12549 df-sca 12551 df-vsca 12552 df-ip 12553 |
This theorem is referenced by: (None) |
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