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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.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 2825 | . . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿) | |
6 | eqid 2825 | . . . 4 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊) | |
7 | eqid 2825 | . . . 4 ⊢ ((HGMap‘𝐾)‘𝑊) = ((HGMap‘𝐾)‘𝑊) | |
8 | eqid 2825 | . . . 4 ⊢ (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) = (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) | |
9 | eqid 2825 | . . . 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 38002 | . . 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 6437 | . 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 6447 | . . . . 5 ⊢ 𝑉 ∈ V |
16 | 15, 15 | mpt2ex 7510 | . . . 4 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥)) ∈ V |
17 | 11, 16 | eqeltri 2902 | . . 3 ⊢ , ∈ V |
18 | eqid 2825 | . . . 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 16398 | . . 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 2880 | 1 ⊢ (𝜑 → , = (·𝑖‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∪ cun 3796 {cpr 4399 {ctp 4401 〈cop 4403 ‘cfv 6123 (class class class)co 6905 ↦ cmpt2 6907 ndxcnx 16219 sSet csts 16220 Basecbs 16222 +gcplusg 16305 *𝑟cstv 16307 Scalarcsca 16308 ·𝑠 cvsca 16309 ·𝑖cip 16310 HLchlt 35418 LHypclh 36052 EDRingcedring 36821 DVecHcdvh 37146 HDMapchdma 37860 HGMapchg 37951 HLHilchlh 38000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-plusg 16318 df-sca 16321 df-vsca 16322 df-ip 16323 df-hlhil 38001 |
This theorem is referenced by: hlhilipval 38017 |
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