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 7206 | . . . 4 ⊢ (𝐸 sSet 〈(*𝑟‘ndx), 𝐺〉) ∈ V | |
3 | 1, 2 | eqeltri 2830 | . . 3 ⊢ 𝑅 ∈ V |
4 | eqid 2739 | . . . 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 16762 | . . 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 2739 | . . . 4 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
10 | eqid 2739 | . . . 4 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊)) | |
11 | eqid 2739 | . . . 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 2739 | . . . 4 ⊢ ( ·𝑠 ‘((DVecH‘𝐾)‘𝑊)) = ( ·𝑠 ‘((DVecH‘𝐾)‘𝑊)) | |
15 | eqid 2739 | . . . 4 ⊢ ((HDMap‘𝐾)‘𝑊) = ((HDMap‘𝐾)‘𝑊) | |
16 | eqid 2739 | . . . 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 39594 | . . 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 6681 | . 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 2793 | 1 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3399 ∪ cun 3842 {cpr 4519 {ctp 4521 〈cop 4523 ‘cfv 6340 (class class class)co 7173 ∈ cmpo 7175 ndxcnx 16586 sSet csts 16587 Basecbs 16589 +gcplusg 16671 *𝑟cstv 16673 Scalarcsca 16674 ·𝑠 cvsca 16675 ·𝑖cip 16676 HLchlt 37010 LHypclh 37644 EDRingcedring 38413 DVecHcdvh 38738 HDMapchdma 39452 HGMapchg 39543 HLHilchlh 39592 |
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 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-1st 7717 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-7 11787 df-8 11788 df-n0 11980 df-z 12066 df-uz 12328 df-fz 12985 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-plusg 16684 df-sca 16687 df-vsca 16688 df-ip 16689 df-hlhil 39593 |
This theorem is referenced by: hlhilslem 39598 hlhilnvl 39610 |
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