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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilip | Structured version Visualization version GIF version |
Description: Inner product operation for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
Ref | Expression |
---|---|
hlhilip.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hlhilip.l | ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) |
hlhilip.v | ⊢ 𝑉 = (Base‘𝐿) |
hlhilip.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hlhilip.u | ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) |
hlhilip.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hlhilip.p | ⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥)) |
Ref | Expression |
---|---|
hlhilip | ⊢ (𝜑 → , = (·𝑖‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhilip.p | . . . 4 ⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥)) | |
2 | hlhilip.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝐿) | |
3 | 2 | fvexi 6936 | . . . . 5 ⊢ 𝑉 ∈ V |
4 | 3, 3 | mpoex 8122 | . . . 4 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥)) ∈ V |
5 | 1, 4 | eqeltri 2840 | . . 3 ⊢ , ∈ V |
6 | eqid 2740 | . . . 4 ⊢ ({〈(Base‘ndx), 𝑉〉, 〈(+g‘ndx), (+g‘𝐿)〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘𝐿)〉, 〈(·𝑖‘ndx), , 〉}) = ({〈(Base‘ndx), 𝑉〉, 〈(+g‘ndx), (+g‘𝐿)〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘𝐿)〉, 〈(·𝑖‘ndx), , 〉}) | |
7 | 6 | phlip 17412 | . . 3 ⊢ ( , ∈ V → , = (·𝑖‘({〈(Base‘ndx), 𝑉〉, 〈(+g‘ndx), (+g‘𝐿)〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘𝐿)〉, 〈(·𝑖‘ndx), , 〉}))) |
8 | 5, 7 | ax-mp 5 | . 2 ⊢ , = (·𝑖‘({〈(Base‘ndx), 𝑉〉, 〈(+g‘ndx), (+g‘𝐿)〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘𝐿)〉, 〈(·𝑖‘ndx), , 〉})) |
9 | hlhilip.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
10 | hlhilip.u | . . . 4 ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) | |
11 | hlhilip.l | . . . 4 ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) | |
12 | eqid 2740 | . . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿) | |
13 | eqid 2740 | . . . 4 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊) | |
14 | eqid 2740 | . . . 4 ⊢ ((HGMap‘𝐾)‘𝑊) = ((HGMap‘𝐾)‘𝑊) | |
15 | eqid 2740 | . . . 4 ⊢ (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) = (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) | |
16 | eqid 2740 | . . . 4 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿) | |
17 | hlhilip.s | . . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
18 | hlhilip.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
19 | 9, 10, 11, 2, 12, 13, 14, 15, 16, 17, 1, 18 | hlhilset 41893 | . . 3 ⊢ (𝜑 → 𝑈 = ({〈(Base‘ndx), 𝑉〉, 〈(+g‘ndx), (+g‘𝐿)〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘𝐿)〉, 〈(·𝑖‘ndx), , 〉})) |
20 | 19 | fveq2d 6926 | . 2 ⊢ (𝜑 → (·𝑖‘𝑈) = (·𝑖‘({〈(Base‘ndx), 𝑉〉, 〈(+g‘ndx), (+g‘𝐿)〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘𝐿)〉, 〈(·𝑖‘ndx), , 〉}))) |
21 | 8, 20 | eqtr4id 2799 | 1 ⊢ (𝜑 → , = (·𝑖‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 Vcvv 3488 ∪ cun 3974 {cpr 4650 {ctp 4652 〈cop 4654 ‘cfv 6575 (class class class)co 7450 ∈ cmpo 7452 sSet csts 17212 ndxcnx 17242 Basecbs 17260 +gcplusg 17313 *𝑟cstv 17315 Scalarcsca 17316 ·𝑠 cvsca 17317 ·𝑖cip 17318 HLchlt 39308 LHypclh 39943 EDRingcedring 40712 DVecHcdvh 41037 HDMapchdma 41751 HGMapchg 41842 HLHilchlh 41891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-7 12363 df-8 12364 df-n0 12556 df-z 12642 df-uz 12906 df-fz 13570 df-struct 17196 df-slot 17231 df-ndx 17243 df-base 17261 df-plusg 17326 df-sca 17329 df-vsca 17330 df-ip 17331 df-hlhil 41892 |
This theorem is referenced by: hlhilipval 41912 |
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