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Mirrors > Home > MPE Home > Th. List > phlstr | Structured version Visualization version GIF version |
Description: A constructed pre-Hilbert space is a structure. Starting from lmodstr 16064 (which has 4 members), we chain strleun 16019 once more, adding an ordered pair to the function, to get all 5 members. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.) |
Ref | Expression |
---|---|
phlfn.h | ⊢ 𝐻 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉}) |
Ref | Expression |
---|---|
phlstr | ⊢ 𝐻 Struct 〈1, 8〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4213 | . . . 4 ⊢ {〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉} = ({〈( ·𝑠 ‘ndx), · 〉} ∪ {〈(·𝑖‘ndx), , 〉}) | |
2 | 1 | uneq2i 3797 | . . 3 ⊢ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉}) = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ ({〈( ·𝑠 ‘ndx), · 〉} ∪ {〈(·𝑖‘ndx), , 〉})) |
3 | phlfn.h | . . 3 ⊢ 𝐻 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉}) | |
4 | unass 3803 | . . 3 ⊢ (({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) ∪ {〈(·𝑖‘ndx), , 〉}) = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ ({〈( ·𝑠 ‘ndx), · 〉} ∪ {〈(·𝑖‘ndx), , 〉})) | |
5 | 2, 3, 4 | 3eqtr4i 2683 | . 2 ⊢ 𝐻 = (({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) ∪ {〈(·𝑖‘ndx), , 〉}) |
6 | eqid 2651 | . . . 4 ⊢ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) | |
7 | 6 | lmodstr 16064 | . . 3 ⊢ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) Struct 〈1, 6〉 |
8 | 8nn 11229 | . . . 4 ⊢ 8 ∈ ℕ | |
9 | ipndx 16069 | . . . 4 ⊢ (·𝑖‘ndx) = 8 | |
10 | 8, 9 | strle1 16020 | . . 3 ⊢ {〈(·𝑖‘ndx), , 〉} Struct 〈8, 8〉 |
11 | 6lt8 11254 | . . 3 ⊢ 6 < 8 | |
12 | 7, 10, 11 | strleun 16019 | . 2 ⊢ (({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) ∪ {〈(·𝑖‘ndx), , 〉}) Struct 〈1, 8〉 |
13 | 5, 12 | eqbrtri 4706 | 1 ⊢ 𝐻 Struct 〈1, 8〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 ∪ cun 3605 {csn 4210 {cpr 4212 {ctp 4214 〈cop 4216 class class class wbr 4685 ‘cfv 5926 1c1 9975 6c6 11112 8c8 11114 Struct cstr 15900 ndxcnx 15901 Basecbs 15904 +gcplusg 15988 Scalarcsca 15991 ·𝑠 cvsca 15992 ·𝑖cip 15993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-plusg 16001 df-sca 16004 df-vsca 16005 df-ip 16006 |
This theorem is referenced by: phlbase 16082 phlplusg 16083 phlsca 16084 phlvsca 16085 phlip 16086 |
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