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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 6853 | . . . . 5 ⊢ 𝑉 ∈ V |
4 | 3, 3 | mpoex 8008 | . . . 4 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥)) ∈ V |
5 | 1, 4 | eqeltri 2834 | . . 3 ⊢ , ∈ V |
6 | eqid 2736 | . . . 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 17224 | . . 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 2736 | . . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿) | |
13 | eqid 2736 | . . . 4 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊) | |
14 | eqid 2736 | . . . 4 ⊢ ((HGMap‘𝐾)‘𝑊) = ((HGMap‘𝐾)‘𝑊) | |
15 | eqid 2736 | . . . 4 ⊢ (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) = (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) | |
16 | eqid 2736 | . . . 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 40364 | . . 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 6843 | . 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 2795 | 1 ⊢ (𝜑 → , = (·𝑖‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∪ cun 3906 {cpr 4586 {ctp 4588 〈cop 4590 ‘cfv 6493 (class class class)co 7353 ∈ cmpo 7355 sSet csts 17027 ndxcnx 17057 Basecbs 17075 +gcplusg 17125 *𝑟cstv 17127 Scalarcsca 17128 ·𝑠 cvsca 17129 ·𝑖cip 17130 HLchlt 37779 LHypclh 38414 EDRingcedring 39183 DVecHcdvh 39508 HDMapchdma 40222 HGMapchg 40313 HLHilchlh 40362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-n0 12410 df-z 12496 df-uz 12760 df-fz 13417 df-struct 17011 df-slot 17046 df-ndx 17058 df-base 17076 df-plusg 17138 df-sca 17141 df-vsca 17142 df-ip 17143 df-hlhil 40363 |
This theorem is referenced by: hlhilipval 40383 |
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