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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 17275 (which has 4 members), we chain strleun 17095 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 4624 | . . . 4 ⊢ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩} = ({⟨( ·𝑠 ‘ndx), · ⟩} ∪ {⟨(·𝑖‘ndx), , ⟩}) | |
2 | 1 | uneq2i 4153 | . . 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 4159 | . . 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 2762 | . 2 ⊢ 𝐻 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩}) ∪ {⟨(·𝑖‘ndx), , ⟩}) |
6 | eqid 2724 | . . . 4 ⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩}) | |
7 | 6 | lmodstr 17275 | . . 3 ⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩}) Struct ⟨1, 6⟩ |
8 | 8nn 12306 | . . . 4 ⊢ 8 ∈ ℕ | |
9 | ipndx 17280 | . . . 4 ⊢ (·𝑖‘ndx) = 8 | |
10 | 8, 9 | strle1 17096 | . . 3 ⊢ {⟨(·𝑖‘ndx), , ⟩} Struct ⟨8, 8⟩ |
11 | 6lt8 12404 | . . 3 ⊢ 6 < 8 | |
12 | 7, 10, 11 | strleun 17095 | . 2 ⊢ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩}) ∪ {⟨(·𝑖‘ndx), , ⟩}) Struct ⟨1, 8⟩ |
13 | 5, 12 | eqbrtri 5160 | 1 ⊢ 𝐻 Struct ⟨1, 8⟩ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∪ cun 3939 {csn 4621 {cpr 4623 {ctp 4625 ⟨cop 4627 class class class wbr 5139 ‘cfv 6534 1c1 11108 6c6 12270 8c8 12272 Struct cstr 17084 ndxcnx 17131 Basecbs 17149 +gcplusg 17202 Scalarcsca 17205 ·𝑠 cvsca 17206 ·𝑖cip 17207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13486 df-struct 17085 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-sca 17218 df-vsca 17219 df-ip 17220 |
This theorem is referenced by: phlbase 17297 phlplusg 17298 phlsca 17299 phlvsca 17300 phlip 17301 |
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