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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 2140 | . . . . 5 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} | |
6 | 5 | rngstrg 12113 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ × ∈ 𝑋) → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} Struct 〈1, 3〉) |
7 | 2, 3, 4, 6 | syl3anc 1217 | . . 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 8908 | . . . . 5 ⊢ 5 ∈ ℕ | |
12 | scandx 12125 | . . . . 5 ⊢ (Scalar‘ndx) = 5 | |
13 | 5lt6 8923 | . . . . 5 ⊢ 5 < 6 | |
14 | 6nn 8909 | . . . . 5 ⊢ 6 ∈ ℕ | |
15 | vscandx 12128 | . . . . 5 ⊢ ( ·𝑠 ‘ndx) = 6 | |
16 | 6lt8 8935 | . . . . 5 ⊢ 6 < 8 | |
17 | 8nn 8911 | . . . . 5 ⊢ 8 ∈ ℕ | |
18 | ipndx 12136 | . . . . 5 ⊢ (·𝑖‘ndx) = 8 | |
19 | 11, 12, 13, 14, 15, 16, 17, 18 | strle3g 12090 | . . . 4 ⊢ ((𝑆 ∈ 𝑌 ∧ · ∈ 𝑄 ∧ 𝐼 ∈ 𝑍) → {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉} Struct 〈5, 8〉) |
20 | 8, 9, 10, 19 | syl3anc 1217 | . . 3 ⊢ (𝜑 → {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉} Struct 〈5, 8〉) |
21 | 3lt5 8920 | . . . 4 ⊢ 3 < 5 | |
22 | 21 | a1i 9 | . . 3 ⊢ (𝜑 → 3 < 5) |
23 | 7, 20, 22 | strleund 12086 | . 2 ⊢ (𝜑 → ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) Struct 〈1, 8〉) |
24 | 1, 23 | eqbrtrid 3971 | 1 ⊢ (𝜑 → 𝐴 Struct 〈1, 8〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 ∪ cun 3074 {ctp 3534 〈cop 3535 class class class wbr 3937 ‘cfv 5131 1c1 7645 < clt 7824 3c3 8796 5c5 8798 6c6 8799 8c8 8801 Struct cstr 11994 ndxcnx 11995 Basecbs 11998 +gcplusg 12060 .rcmulr 12061 Scalarcsca 12063 ·𝑠 cvsca 12064 ·𝑖cip 12065 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-tp 3540 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-7 8808 df-8 8809 df-n0 9002 df-z 9079 df-uz 9351 df-fz 9822 df-struct 12000 df-ndx 12001 df-slot 12002 df-base 12004 df-plusg 12073 df-mulr 12074 df-sca 12076 df-vsca 12077 df-ip 12078 |
This theorem is referenced by: ipsbased 12140 ipsaddgd 12141 ipsmulrd 12142 ipsscad 12143 ipsvscad 12144 ipsipd 12145 |
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