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| Description: A constructed pre-Hilbert space is a structure. Starting from lmodstr 17370 (which has 4 members), we chain strleun 17195 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 4628 | . . . 4 ⊢ {〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉} = ({〈( ·𝑠 ‘ndx), · 〉} ∪ {〈(·𝑖‘ndx), , 〉}) | |
| 2 | 1 | uneq2i 4164 | . . 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 4171 | . . 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 2774 | . 2 ⊢ 𝐻 = (({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) ∪ {〈(·𝑖‘ndx), , 〉}) | 
| 6 | eqid 2736 | . . . 4 ⊢ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) | |
| 7 | 6 | lmodstr 17370 | . . 3 ⊢ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) Struct 〈1, 6〉 | 
| 8 | 8nn 12362 | . . . 4 ⊢ 8 ∈ ℕ | |
| 9 | ipndx 17375 | . . . 4 ⊢ (·𝑖‘ndx) = 8 | |
| 10 | 8, 9 | strle1 17196 | . . 3 ⊢ {〈(·𝑖‘ndx), , 〉} Struct 〈8, 8〉 | 
| 11 | 6lt8 12460 | . . 3 ⊢ 6 < 8 | |
| 12 | 7, 10, 11 | strleun 17195 | . 2 ⊢ (({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑇〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) ∪ {〈(·𝑖‘ndx), , 〉}) Struct 〈1, 8〉 | 
| 13 | 5, 12 | eqbrtri 5163 | 1 ⊢ 𝐻 Struct 〈1, 8〉 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∪ cun 3948 {csn 4625 {cpr 4627 {ctp 4629 〈cop 4631 class class class wbr 5142 ‘cfv 6560 1c1 11157 6c6 12326 8c8 12328 Struct cstr 17184 ndxcnx 17231 Basecbs 17248 +gcplusg 17298 Scalarcsca 17301 ·𝑠 cvsca 17302 ·𝑖cip 17303 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-struct 17185 df-slot 17220 df-ndx 17232 df-base 17249 df-plusg 17311 df-sca 17314 df-vsca 17315 df-ip 17316 | 
| This theorem is referenced by: phlbase 17392 phlplusg 17393 phlsca 17394 phlvsca 17395 phlip 17396 | 
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