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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilsca | Structured version Visualization version GIF version |
Description: The scalar of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
Ref | Expression |
---|---|
hlhilbase.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hlhilbase.u | ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) |
hlhilbase.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hlhilsca.e | ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) |
hlhilsca.g | ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) |
hlhilsca.r | ⊢ 𝑅 = (𝐸 sSet 〈(*𝑟‘ndx), 𝐺〉) |
Ref | Expression |
---|---|
hlhilsca | ⊢ (𝜑 → 𝑅 = (Scalar‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhilsca.r | . . . 4 ⊢ 𝑅 = (𝐸 sSet 〈(*𝑟‘ndx), 𝐺〉) | |
2 | ovex 7168 | . . . 4 ⊢ (𝐸 sSet 〈(*𝑟‘ndx), 𝐺〉) ∈ V | |
3 | 1, 2 | eqeltri 2886 | . . 3 ⊢ 𝑅 ∈ V |
4 | eqid 2798 | . . . 4 ⊢ ({〈(Base‘ndx), (Base‘((DVecH‘𝐾)‘𝑊))〉, 〈(+g‘ndx), (+g‘((DVecH‘𝐾)‘𝑊))〉, 〈(Scalar‘ndx), 𝑅〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘((DVecH‘𝐾)‘𝑊))〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘((DVecH‘𝐾)‘𝑊)), 𝑦 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉}) = ({〈(Base‘ndx), (Base‘((DVecH‘𝐾)‘𝑊))〉, 〈(+g‘ndx), (+g‘((DVecH‘𝐾)‘𝑊))〉, 〈(Scalar‘ndx), 𝑅〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘((DVecH‘𝐾)‘𝑊))〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘((DVecH‘𝐾)‘𝑊)), 𝑦 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉}) | |
5 | 4 | phlsca 16648 | . . 3 ⊢ (𝑅 ∈ V → 𝑅 = (Scalar‘({〈(Base‘ndx), (Base‘((DVecH‘𝐾)‘𝑊))〉, 〈(+g‘ndx), (+g‘((DVecH‘𝐾)‘𝑊))〉, 〈(Scalar‘ndx), 𝑅〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘((DVecH‘𝐾)‘𝑊))〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘((DVecH‘𝐾)‘𝑊)), 𝑦 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉}))) |
6 | 3, 5 | ax-mp 5 | . 2 ⊢ 𝑅 = (Scalar‘({〈(Base‘ndx), (Base‘((DVecH‘𝐾)‘𝑊))〉, 〈(+g‘ndx), (+g‘((DVecH‘𝐾)‘𝑊))〉, 〈(Scalar‘ndx), 𝑅〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘((DVecH‘𝐾)‘𝑊))〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘((DVecH‘𝐾)‘𝑊)), 𝑦 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉})) |
7 | hlhilbase.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
8 | hlhilbase.u | . . . 4 ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) | |
9 | eqid 2798 | . . . 4 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
10 | eqid 2798 | . . . 4 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊)) | |
11 | eqid 2798 | . . . 4 ⊢ (+g‘((DVecH‘𝐾)‘𝑊)) = (+g‘((DVecH‘𝐾)‘𝑊)) | |
12 | hlhilsca.e | . . . 4 ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) | |
13 | hlhilsca.g | . . . 4 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) | |
14 | eqid 2798 | . . . 4 ⊢ ( ·𝑠 ‘((DVecH‘𝐾)‘𝑊)) = ( ·𝑠 ‘((DVecH‘𝐾)‘𝑊)) | |
15 | eqid 2798 | . . . 4 ⊢ ((HDMap‘𝐾)‘𝑊) = ((HDMap‘𝐾)‘𝑊) | |
16 | eqid 2798 | . . . 4 ⊢ (𝑥 ∈ (Base‘((DVecH‘𝐾)‘𝑊)), 𝑦 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥)) = (𝑥 ∈ (Base‘((DVecH‘𝐾)‘𝑊)), 𝑦 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥)) | |
17 | hlhilbase.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
18 | 7, 8, 9, 10, 11, 12, 13, 1, 14, 15, 16, 17 | hlhilset 39230 | . . 3 ⊢ (𝜑 → 𝑈 = ({〈(Base‘ndx), (Base‘((DVecH‘𝐾)‘𝑊))〉, 〈(+g‘ndx), (+g‘((DVecH‘𝐾)‘𝑊))〉, 〈(Scalar‘ndx), 𝑅〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘((DVecH‘𝐾)‘𝑊))〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘((DVecH‘𝐾)‘𝑊)), 𝑦 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉})) |
19 | 18 | fveq2d 6649 | . 2 ⊢ (𝜑 → (Scalar‘𝑈) = (Scalar‘({〈(Base‘ndx), (Base‘((DVecH‘𝐾)‘𝑊))〉, 〈(+g‘ndx), (+g‘((DVecH‘𝐾)‘𝑊))〉, 〈(Scalar‘ndx), 𝑅〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘((DVecH‘𝐾)‘𝑊))〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘((DVecH‘𝐾)‘𝑊)), 𝑦 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉}))) |
20 | 6, 19 | eqtr4id 2852 | 1 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∪ cun 3879 {cpr 4527 {ctp 4529 〈cop 4531 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ndxcnx 16472 sSet csts 16473 Basecbs 16475 +gcplusg 16557 *𝑟cstv 16559 Scalarcsca 16560 ·𝑠 cvsca 16561 ·𝑖cip 16562 HLchlt 36646 LHypclh 37280 EDRingcedring 38049 DVecHcdvh 38374 HDMapchdma 39088 HGMapchg 39179 HLHilchlh 39228 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-sca 16573 df-vsca 16574 df-ip 16575 df-hlhil 39229 |
This theorem is referenced by: hlhilslem 39234 hlhilnvl 39246 |
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