Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilplus | Structured version Visualization version GIF version |
Description: The vector addition for the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.) |
Ref | Expression |
---|---|
hlhilbase.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hlhilbase.u | ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) |
hlhilbase.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hlhilbase.l | ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) |
hlhilplus.a | ⊢ + = (+g‘𝐿) |
Ref | Expression |
---|---|
hlhilplus | ⊢ (𝜑 → + = (+g‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhilplus.a | . . . 4 ⊢ + = (+g‘𝐿) | |
2 | 1 | fvexi 6782 | . . 3 ⊢ + ∈ V |
3 | eqid 2739 | . . . 4 ⊢ ({〈(Base‘ndx), (Base‘𝐿)〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘𝐿)〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉}) = ({〈(Base‘ndx), (Base‘𝐿)〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘𝐿)〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉}) | |
4 | 3 | phlplusg 17039 | . . 3 ⊢ ( + ∈ V → + = (+g‘({〈(Base‘ndx), (Base‘𝐿)〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘𝐿)〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉}))) |
5 | 2, 4 | ax-mp 5 | . 2 ⊢ + = (+g‘({〈(Base‘ndx), (Base‘𝐿)〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘𝐿)〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉})) |
6 | hlhilbase.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | hlhilbase.u | . . . 4 ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) | |
8 | hlhilbase.l | . . . 4 ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) | |
9 | eqid 2739 | . . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
10 | eqid 2739 | . . . 4 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊) | |
11 | eqid 2739 | . . . 4 ⊢ ((HGMap‘𝐾)‘𝑊) = ((HGMap‘𝐾)‘𝑊) | |
12 | eqid 2739 | . . . 4 ⊢ (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) = (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) | |
13 | eqid 2739 | . . . 4 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿) | |
14 | eqid 2739 | . . . 4 ⊢ ((HDMap‘𝐾)‘𝑊) = ((HDMap‘𝐾)‘𝑊) | |
15 | eqid 2739 | . . . 4 ⊢ (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥)) = (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥)) | |
16 | hlhilbase.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | 6, 7, 8, 9, 1, 10, 11, 12, 13, 14, 15, 16 | hlhilset 39927 | . . 3 ⊢ (𝜑 → 𝑈 = ({〈(Base‘ndx), (Base‘𝐿)〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘𝐿)〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉})) |
18 | 17 | fveq2d 6772 | . 2 ⊢ (𝜑 → (+g‘𝑈) = (+g‘({〈(Base‘ndx), (Base‘𝐿)〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), (((EDRing‘𝐾)‘𝑊) sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘𝐿)〉, 〈(·𝑖‘ndx), (𝑥 ∈ (Base‘𝐿), 𝑦 ∈ (Base‘𝐿) ↦ ((((HDMap‘𝐾)‘𝑊)‘𝑦)‘𝑥))〉}))) |
19 | 5, 18 | eqtr4id 2798 | 1 ⊢ (𝜑 → + = (+g‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 Vcvv 3430 ∪ cun 3889 {cpr 4568 {ctp 4570 〈cop 4572 ‘cfv 6430 (class class class)co 7268 ∈ cmpo 7270 sSet csts 16845 ndxcnx 16875 Basecbs 16893 +gcplusg 16943 *𝑟cstv 16945 Scalarcsca 16946 ·𝑠 cvsca 16947 ·𝑖cip 16948 HLchlt 37343 LHypclh 37977 EDRingcedring 38746 DVecHcdvh 39071 HDMapchdma 39785 HGMapchg 39876 HLHilchlh 39925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-n0 12217 df-z 12303 df-uz 12565 df-fz 13222 df-struct 16829 df-slot 16864 df-ndx 16876 df-base 16894 df-plusg 16956 df-sca 16959 df-vsca 16960 df-ip 16961 df-hlhil 39926 |
This theorem is referenced by: hlhillvec 39948 hlhil0 39952 hlhillsm 39953 hlhilphllem 39956 |
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