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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 16482 (which has 4 members), we chain strleun 16437 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 4438 | . . . 4 ⊢ {〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉} = ({〈( ·𝑠 ‘ndx), · 〉} ∪ {〈(·𝑖‘ndx), , 〉}) | |
2 | 1 | uneq2i 4021 | . . 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 4027 | . . 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 2806 | . 2 ⊢ 𝐻 = (({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) ∪ {〈(·𝑖‘ndx), , 〉}) |
6 | eqid 2772 | . . . 4 ⊢ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) | |
7 | 6 | lmodstr 16482 | . . 3 ⊢ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) Struct 〈1, 6〉 |
8 | 8nn 11533 | . . . 4 ⊢ 8 ∈ ℕ | |
9 | ipndx 16487 | . . . 4 ⊢ (·𝑖‘ndx) = 8 | |
10 | 8, 9 | strle1 16438 | . . 3 ⊢ {〈(·𝑖‘ndx), , 〉} Struct 〈8, 8〉 |
11 | 6lt8 11633 | . . 3 ⊢ 6 < 8 | |
12 | 7, 10, 11 | strleun 16437 | . 2 ⊢ (({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) ∪ {〈(·𝑖‘ndx), , 〉}) Struct 〈1, 8〉 |
13 | 5, 12 | eqbrtri 4944 | 1 ⊢ 𝐻 Struct 〈1, 8〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∪ cun 3823 {csn 4435 {cpr 4437 {ctp 4439 〈cop 4441 class class class wbr 4923 ‘cfv 6182 1c1 10328 6c6 11492 8c8 11494 Struct cstr 16325 ndxcnx 16326 Basecbs 16329 +gcplusg 16411 Scalarcsca 16414 ·𝑠 cvsca 16415 ·𝑖cip 16416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-n0 11701 df-z 11787 df-uz 12052 df-fz 12702 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-plusg 16424 df-sca 16427 df-vsca 16428 df-ip 16429 |
This theorem is referenced by: phlbase 16500 phlplusg 16501 phlsca 16502 phlvsca 16503 phlip 16504 |
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