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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 17211 (which has 4 members), we chain strleun 17034 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 4590 | . . . 4 ⊢ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩} = ({⟨( ·𝑠 ‘ndx), · ⟩} ∪ {⟨(·𝑖‘ndx), , ⟩}) | |
2 | 1 | uneq2i 4121 | . . 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 4127 | . . 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 2771 | . 2 ⊢ 𝐻 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩}) ∪ {⟨(·𝑖‘ndx), , ⟩}) |
6 | eqid 2733 | . . . 4 ⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩}) | |
7 | 6 | lmodstr 17211 | . . 3 ⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩}) Struct ⟨1, 6⟩ |
8 | 8nn 12253 | . . . 4 ⊢ 8 ∈ ℕ | |
9 | ipndx 17216 | . . . 4 ⊢ (·𝑖‘ndx) = 8 | |
10 | 8, 9 | strle1 17035 | . . 3 ⊢ {⟨(·𝑖‘ndx), , ⟩} Struct ⟨8, 8⟩ |
11 | 6lt8 12351 | . . 3 ⊢ 6 < 8 | |
12 | 7, 10, 11 | strleun 17034 | . 2 ⊢ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩}) ∪ {⟨(·𝑖‘ndx), , ⟩}) Struct ⟨1, 8⟩ |
13 | 5, 12 | eqbrtri 5127 | 1 ⊢ 𝐻 Struct ⟨1, 8⟩ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∪ cun 3909 {csn 4587 {cpr 4589 {ctp 4591 ⟨cop 4593 class class class wbr 5106 ‘cfv 6497 1c1 11057 6c6 12217 8c8 12219 Struct cstr 17023 ndxcnx 17070 Basecbs 17088 +gcplusg 17138 Scalarcsca 17141 ·𝑠 cvsca 17142 ·𝑖cip 17143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-struct 17024 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-sca 17154 df-vsca 17155 df-ip 17156 |
This theorem is referenced by: phlbase 17233 phlplusg 17234 phlsca 17235 phlvsca 17236 phlip 17237 |
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