Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilvsca | Structured version Visualization version GIF version |
Description: The scalar product for the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
Ref | Expression |
---|---|
hlhilvsca.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hlhilvsca.l | ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) |
hlhilvsca.t | ⊢ · = ( ·𝑠 ‘𝐿) |
hlhilvsca.u | ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) |
hlhilvsca.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
hlhilvsca | ⊢ (𝜑 → · = ( ·𝑠 ‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhilvsca.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hlhilvsca.u | . . . 4 ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) | |
3 | hlhilvsca.l | . . . 4 ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) | |
4 | eqid 2610 | . . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
5 | eqid 2610 | . . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿) | |
6 | eqid 2610 | . . . 4 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊) | |
7 | eqid 2610 | . . . 4 ⊢ ((HGMap‘𝐾)‘𝑊) = ((HGMap‘𝐾)‘𝑊) | |
8 | eqid 2610 | . . . 4 ⊢ (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) = (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) | |
9 | hlhilvsca.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝐿) | |
10 | eqid 2610 | . . . 4 ⊢ ((HDMap‘𝐾)‘𝑊) = ((HDMap‘𝐾)‘𝑊) | |
11 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥)) = (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥)) | |
12 | hlhilvsca.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | hlhilset 36244 | . . 3 ⊢ (𝜑 → 𝑈 = ({〈(Base‘ndx), (Base‘𝐿)〉, 〈(+g‘ndx), (+g‘𝐿)〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉})) |
14 | 13 | fveq2d 6107 | . 2 ⊢ (𝜑 → ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘({〈(Base‘ndx), (Base‘𝐿)〉, 〈(+g‘ndx), (+g‘𝐿)〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉}))) |
15 | fvex 6113 | . . . 4 ⊢ ( ·𝑠 ‘𝐿) ∈ V | |
16 | 9, 15 | eqeltri 2684 | . . 3 ⊢ · ∈ V |
17 | eqid 2610 | . . . 4 ⊢ ({〈(Base‘ndx), (Base‘𝐿)〉, 〈(+g‘ndx), (+g‘𝐿)〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉}) = ({〈(Base‘ndx), (Base‘𝐿)〉, 〈(+g‘ndx), (+g‘𝐿)〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉}) | |
18 | 17 | phlvsca 15861 | . . 3 ⊢ ( · ∈ V → · = ( ·𝑠 ‘({〈(Base‘ndx), (Base‘𝐿)〉, 〈(+g‘ndx), (+g‘𝐿)〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉}))) |
19 | 16, 18 | ax-mp 5 | . 2 ⊢ · = ( ·𝑠 ‘({〈(Base‘ndx), (Base‘𝐿)〉, 〈(+g‘ndx), (+g‘𝐿)〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉})) |
20 | 14, 19 | syl6reqr 2663 | 1 ⊢ (𝜑 → · = ( ·𝑠 ‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 {cpr 4127 {ctp 4129 〈cop 4131 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ndxcnx 15692 sSet csts 15693 Basecbs 15695 +gcplusg 15768 *𝑟cstv 15770 Scalarcsca 15771 ·𝑠 cvsca 15772 ·𝑖cip 15773 HLchlt 33655 LHypclh 34288 EDRingcedring 35059 DVecHcdvh 35385 HDMapchdma 36100 HGMapchg 36193 HLHilchlh 36242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-sca 15784 df-vsca 15785 df-ip 15786 df-hlhil 36243 |
This theorem is referenced by: hlhillvec 36261 hlhilphllem 36269 |
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