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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 6857 | . . 3 ⊢ · ∈ V |
3 | eqid 2737 | . . . 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 17232 | . . 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 2737 | . . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
10 | eqid 2737 | . . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿) | |
11 | eqid 2737 | . . . 4 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊) | |
12 | eqid 2737 | . . . 4 ⊢ ((HGMap‘𝐾)‘𝑊) = ((HGMap‘𝐾)‘𝑊) | |
13 | eqid 2737 | . . . 4 ⊢ (((EDRing‘𝐾)‘𝑊) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)⟩) = (((EDRing‘𝐾)‘𝑊) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)⟩) | |
14 | eqid 2737 | . . . 4 ⊢ ((HDMap‘𝐾)‘𝑊) = ((HDMap‘𝐾)‘𝑊) | |
15 | eqid 2737 | . . . 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 40400 | . . 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 6847 | . 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 2796 | 1 ⊢ (𝜑 → · = ( ·𝑠 ‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3446 ∪ cun 3909 {cpr 4589 {ctp 4591 ⟨cop 4593 ‘cfv 6497 (class class class)co 7358 ∈ cmpo 7360 sSet csts 17036 ndxcnx 17066 Basecbs 17084 +gcplusg 17134 *𝑟cstv 17136 Scalarcsca 17137 ·𝑠 cvsca 17138 ·𝑖cip 17139 HLchlt 37815 LHypclh 38450 EDRingcedring 39219 DVecHcdvh 39544 HDMapchdma 40258 HGMapchg 40349 HLHilchlh 40398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 df-struct 17020 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-sca 17150 df-vsca 17151 df-ip 17152 df-hlhil 40399 |
This theorem is referenced by: hlhillvec 40421 hlhilphllem 40429 |
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