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.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hlhilip.u | . . . 4 ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) | |
3 | hlhilip.l | . . . 4 ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) | |
4 | hlhilip.v | . . . 4 ⊢ 𝑉 = (Base‘𝐿) | |
5 | eqid 2821 | . . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿) | |
6 | eqid 2821 | . . . 4 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊) | |
7 | eqid 2821 | . . . 4 ⊢ ((HGMap‘𝐾)‘𝑊) = ((HGMap‘𝐾)‘𝑊) | |
8 | eqid 2821 | . . . 4 ⊢ (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) = (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) | |
9 | eqid 2821 | . . . 4 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿) | |
10 | hlhilip.s | . . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
11 | hlhilip.p | . . . 4 ⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥)) | |
12 | hlhilip.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | hlhilset 39064 | . . 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), , 〉})) |
14 | 13 | fveq2d 6669 | . 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), , 〉}))) |
15 | 4 | fvexi 6679 | . . . . 5 ⊢ 𝑉 ∈ V |
16 | 15, 15 | mpoex 7771 | . . . 4 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥)) ∈ V |
17 | 11, 16 | eqeltri 2909 | . . 3 ⊢ , ∈ V |
18 | eqid 2821 | . . . 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), , 〉}) | |
19 | 18 | phlip 16652 | . . 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), , 〉}))) |
20 | 17, 19 | 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), , 〉})) |
21 | 14, 20 | syl6reqr 2875 | 1 ⊢ (𝜑 → , = (·𝑖‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∪ cun 3934 {cpr 4563 {ctp 4565 〈cop 4567 ‘cfv 6350 (class class class)co 7150 ∈ cmpo 7152 ndxcnx 16474 sSet csts 16475 Basecbs 16477 +gcplusg 16559 *𝑟cstv 16561 Scalarcsca 16562 ·𝑠 cvsca 16563 ·𝑖cip 16564 HLchlt 36480 LHypclh 37114 EDRingcedring 37883 DVecHcdvh 38208 HDMapchdma 38922 HGMapchg 39013 HLHilchlh 39062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-sca 16575 df-vsca 16576 df-ip 16577 df-hlhil 39063 |
This theorem is referenced by: hlhilipval 39079 |
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