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Mirrors > Home > ILE Home > Th. List > ipsipd | GIF version |
Description: The multiplicative operation 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 |
---|---|
ipsipd | ⊢ (𝜑 → 𝐼 = (·𝑖‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipslid 12545 | . 2 ⊢ (·𝑖 = Slot (·𝑖‘ndx) ∧ (·𝑖‘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 12546 | . 2 ⊢ (𝜑 → 𝐴 Struct 〈1, 8〉) |
10 | 1 | simpri 112 | . . . . 5 ⊢ (·𝑖‘ndx) ∈ ℕ |
11 | opexg 4211 | . . . . 5 ⊢ (((·𝑖‘ndx) ∈ ℕ ∧ 𝐼 ∈ 𝑍) → 〈(·𝑖‘ndx), 𝐼〉 ∈ V) | |
12 | 10, 8, 11 | sylancr 412 | . . . 4 ⊢ (𝜑 → 〈(·𝑖‘ndx), 𝐼〉 ∈ V) |
13 | tpid3g 3696 | . . . 4 ⊢ (〈(·𝑖‘ndx), 𝐼〉 ∈ V → 〈(·𝑖‘ndx), 𝐼〉 ∈ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) | |
14 | elun2 3295 | . . . 4 ⊢ (〈(·𝑖‘ndx), 𝐼〉 ∈ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉} → 〈(·𝑖‘ndx), 𝐼〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉})) | |
15 | 12, 13, 14 | 3syl 17 | . . 3 ⊢ (𝜑 → 〈(·𝑖‘ndx), 𝐼〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉})) |
16 | 15, 2 | eleqtrrdi 2264 | . 2 ⊢ (𝜑 → 〈(·𝑖‘ndx), 𝐼〉 ∈ 𝐴) |
17 | 1, 9, 8, 16 | opelstrsl 12501 | 1 ⊢ (𝜑 → 𝐼 = (·𝑖‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 Vcvv 2730 ∪ cun 3119 {ctp 3583 〈cop 3584 ‘cfv 5196 1c1 7762 ℕcn 8865 8c8 8922 ndxcnx 12400 Slot cslot 12402 Basecbs 12403 +gcplusg 12467 .rcmulr 12468 Scalarcsca 12470 ·𝑠 cvsca 12471 ·𝑖cip 12472 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-addass 7863 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-tp 3589 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-inn 8866 df-2 8924 df-3 8925 df-4 8926 df-5 8927 df-6 8928 df-7 8929 df-8 8930 df-n0 9123 df-z 9200 df-uz 9475 df-fz 9953 df-struct 12405 df-ndx 12406 df-slot 12407 df-base 12409 df-plusg 12480 df-mulr 12481 df-sca 12483 df-vsca 12484 df-ip 12485 |
This theorem is referenced by: (None) |
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