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Mirrors > Home > ILE Home > Th. List > ipsbased | GIF version |
Description: The base set of a constructed inner product space. (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 |
---|---|
ipsbased | ⊢ (𝜑 → 𝐵 = (Base‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipspart.a | . . 3 ⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) | |
2 | ipsstrd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
3 | ipsstrd.p | . . 3 ⊢ (𝜑 → + ∈ 𝑊) | |
4 | ipsstrd.r | . . 3 ⊢ (𝜑 → × ∈ 𝑋) | |
5 | ipsstrd.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑌) | |
6 | ipsstrd.x | . . 3 ⊢ (𝜑 → · ∈ 𝑄) | |
7 | ipsstrd.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑍) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ipsstrd 11784 | . 2 ⊢ (𝜑 → 𝐴 Struct 〈1, 8〉) |
9 | basendxnn 11698 | . . . . 5 ⊢ (Base‘ndx) ∈ ℕ | |
10 | opexg 4079 | . . . . 5 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝐵 ∈ 𝑉) → 〈(Base‘ndx), 𝐵〉 ∈ V) | |
11 | 9, 2, 10 | sylancr 406 | . . . 4 ⊢ (𝜑 → 〈(Base‘ndx), 𝐵〉 ∈ V) |
12 | tpid1g 3574 | . . . 4 ⊢ (〈(Base‘ndx), 𝐵〉 ∈ V → 〈(Base‘ndx), 𝐵〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉}) | |
13 | elun1 3182 | . . . 4 ⊢ (〈(Base‘ndx), 𝐵〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} → 〈(Base‘ndx), 𝐵〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉})) | |
14 | 11, 12, 13 | 3syl 17 | . . 3 ⊢ (𝜑 → 〈(Base‘ndx), 𝐵〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉})) |
15 | 14, 1 | syl6eleqr 2188 | . 2 ⊢ (𝜑 → 〈(Base‘ndx), 𝐵〉 ∈ 𝐴) |
16 | 8, 2, 15 | opelstrbas 11740 | 1 ⊢ (𝜑 → 𝐵 = (Base‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1296 ∈ wcel 1445 Vcvv 2633 ∪ cun 3011 {ctp 3468 〈cop 3469 ‘cfv 5049 1c1 7448 ℕcn 8520 8c8 8577 ndxcnx 11640 Basecbs 11643 +gcplusg 11705 .rcmulr 11706 Scalarcsca 11708 ·𝑠 cvsca 11709 ·𝑖cip 11710 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-addcom 7542 ax-addass 7544 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-0id 7550 ax-rnegex 7551 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-tp 3474 df-op 3475 df-uni 3676 df-int 3711 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-inn 8521 df-2 8579 df-3 8580 df-4 8581 df-5 8582 df-6 8583 df-7 8584 df-8 8585 df-n0 8772 df-z 8849 df-uz 9119 df-fz 9574 df-struct 11645 df-ndx 11646 df-slot 11647 df-base 11649 df-plusg 11718 df-mulr 11719 df-sca 11721 df-vsca 11722 df-ip 11723 |
This theorem is referenced by: (None) |
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