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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 2189 | . . . . 5 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} | |
6 | 5 | rngstrg 12649 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ × ∈ 𝑋) → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} Struct 〈1, 3〉) |
7 | 2, 3, 4, 6 | syl3anc 1249 | . . 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 9114 | . . . . 5 ⊢ 5 ∈ ℕ | |
12 | scandx 12665 | . . . . 5 ⊢ (Scalar‘ndx) = 5 | |
13 | 5lt6 9129 | . . . . 5 ⊢ 5 < 6 | |
14 | 6nn 9115 | . . . . 5 ⊢ 6 ∈ ℕ | |
15 | vscandx 12671 | . . . . 5 ⊢ ( ·𝑠 ‘ndx) = 6 | |
16 | 6lt8 9141 | . . . . 5 ⊢ 6 < 8 | |
17 | 8nn 9117 | . . . . 5 ⊢ 8 ∈ ℕ | |
18 | ipndx 12683 | . . . . 5 ⊢ (·𝑖‘ndx) = 8 | |
19 | 11, 12, 13, 14, 15, 16, 17, 18 | strle3g 12623 | . . . 4 ⊢ ((𝑆 ∈ 𝑌 ∧ · ∈ 𝑄 ∧ 𝐼 ∈ 𝑍) → {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉} Struct 〈5, 8〉) |
20 | 8, 9, 10, 19 | syl3anc 1249 | . . 3 ⊢ (𝜑 → {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉} Struct 〈5, 8〉) |
21 | 3lt5 9126 | . . . 4 ⊢ 3 < 5 | |
22 | 21 | a1i 9 | . . 3 ⊢ (𝜑 → 3 < 5) |
23 | 7, 20, 22 | strleund 12618 | . 2 ⊢ (𝜑 → ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) Struct 〈1, 8〉) |
24 | 1, 23 | eqbrtrid 4053 | 1 ⊢ (𝜑 → 𝐴 Struct 〈1, 8〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 ∪ cun 3142 {ctp 3609 〈cop 3610 class class class wbr 4018 ‘cfv 5235 1c1 7843 < clt 8023 3c3 9002 5c5 9004 6c6 9005 8c8 9007 Struct cstr 12511 ndxcnx 12512 Basecbs 12515 +gcplusg 12592 .rcmulr 12593 Scalarcsca 12595 ·𝑠 cvsca 12596 ·𝑖cip 12597 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-addass 7944 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-tp 3615 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-inn 8951 df-2 9009 df-3 9010 df-4 9011 df-5 9012 df-6 9013 df-7 9014 df-8 9015 df-n0 9208 df-z 9285 df-uz 9560 df-fz 10041 df-struct 12517 df-ndx 12518 df-slot 12519 df-base 12521 df-plusg 12605 df-mulr 12606 df-sca 12608 df-vsca 12609 df-ip 12610 |
This theorem is referenced by: ipsbased 12691 ipsaddgd 12692 ipsmulrd 12693 ipsscad 12694 ipsvscad 12695 ipsipd 12696 |
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