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.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝐿) | |
2 | 1 | fvexi 6785 | . . 3 ⊢ · ∈ V |
3 | eqid 2740 | . . . 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))〉}) | |
4 | 3 | phlvsca 17058 | . . 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))〉}))) |
5 | 2, 4 | 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))〉})) |
6 | hlhilvsca.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | hlhilvsca.u | . . . 4 ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) | |
8 | hlhilvsca.l | . . . 4 ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) | |
9 | eqid 2740 | . . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
10 | eqid 2740 | . . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿) | |
11 | eqid 2740 | . . . 4 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊) | |
12 | eqid 2740 | . . . 4 ⊢ ((HGMap‘𝐾)‘𝑊) = ((HGMap‘𝐾)‘𝑊) | |
13 | eqid 2740 | . . . 4 ⊢ (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) = (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) | |
14 | eqid 2740 | . . . 4 ⊢ ((HDMap‘𝐾)‘𝑊) = ((HDMap‘𝐾)‘𝑊) | |
15 | eqid 2740 | . . . 4 ⊢ (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥)) = (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥)) | |
16 | hlhilvsca.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | 6, 7, 8, 9, 10, 11, 12, 13, 1, 14, 15, 16 | hlhilset 39944 | . . 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))〉})) |
18 | 17 | fveq2d 6775 | . 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))〉}))) |
19 | 5, 18 | eqtr4id 2799 | 1 ⊢ (𝜑 → · = ( ·𝑠 ‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ∪ cun 3890 {cpr 4569 {ctp 4571 〈cop 4573 ‘cfv 6432 (class class class)co 7271 ∈ cmpo 7273 sSet csts 16862 ndxcnx 16892 Basecbs 16910 +gcplusg 16960 *𝑟cstv 16962 Scalarcsca 16963 ·𝑠 cvsca 16964 ·𝑖cip 16965 HLchlt 37360 LHypclh 37994 EDRingcedring 38763 DVecHcdvh 39088 HDMapchdma 39802 HGMapchg 39893 HLHilchlh 39942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-struct 16846 df-slot 16881 df-ndx 16893 df-base 16911 df-plusg 16973 df-sca 16976 df-vsca 16977 df-ip 16978 df-hlhil 39943 |
This theorem is referenced by: hlhillvec 39965 hlhilphllem 39973 |
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