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Mirrors > Home > MPE Home > Th. List > imasvalstr | Structured version Visualization version GIF version |
Description: Structure product value is a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
Ref | Expression |
---|---|
imasvalstr.u | ⊢ 𝑈 = (({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉}) ∪ {〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), 𝐿〉, 〈(dist‘ndx), 𝐷〉}) |
Ref | Expression |
---|---|
imasvalstr | ⊢ 𝑈 Struct 〈1, ;12〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasvalstr.u | . 2 ⊢ 𝑈 = (({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉}) ∪ {〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), 𝐿〉, 〈(dist‘ndx), 𝐷〉}) | |
2 | eqid 2825 | . . . 4 ⊢ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉}) = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉}) | |
3 | 2 | ipsstr 16390 | . . 3 ⊢ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉}) Struct 〈1, 8〉 |
4 | 9nn 11462 | . . . 4 ⊢ 9 ∈ ℕ | |
5 | tsetndx 16406 | . . . 4 ⊢ (TopSet‘ndx) = 9 | |
6 | 9lt10 11961 | . . . 4 ⊢ 9 < ;10 | |
7 | 10nn 11844 | . . . 4 ⊢ ;10 ∈ ℕ | |
8 | plendx 16413 | . . . 4 ⊢ (le‘ndx) = ;10 | |
9 | 1nn0 11643 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
10 | 0nn0 11642 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
11 | 2nn 11431 | . . . . 5 ⊢ 2 ∈ ℕ | |
12 | 2pos 11468 | . . . . 5 ⊢ 0 < 2 | |
13 | 9, 10, 11, 12 | declt 11857 | . . . 4 ⊢ ;10 < ;12 |
14 | 9, 11 | decnncl 11849 | . . . 4 ⊢ ;12 ∈ ℕ |
15 | dsndx 16422 | . . . 4 ⊢ (dist‘ndx) = ;12 | |
16 | 4, 5, 6, 7, 8, 13, 14, 15 | strle3 16341 | . . 3 ⊢ {〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), 𝐿〉, 〈(dist‘ndx), 𝐷〉} Struct 〈9, ;12〉 |
17 | 8lt9 11564 | . . 3 ⊢ 8 < 9 | |
18 | 3, 16, 17 | strleun 16338 | . 2 ⊢ (({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉}) ∪ {〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), 𝐿〉, 〈(dist‘ndx), 𝐷〉}) Struct 〈1, ;12〉 |
19 | 1, 18 | eqbrtri 4896 | 1 ⊢ 𝑈 Struct 〈1, ;12〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∪ cun 3796 {ctp 4403 〈cop 4405 class class class wbr 4875 ‘cfv 6127 0cc0 10259 1c1 10260 2c2 11413 8c8 11419 9c9 11420 ;cdc 11828 Struct cstr 16225 ndxcnx 16226 Basecbs 16229 +gcplusg 16312 .rcmulr 16313 Scalarcsca 16315 ·𝑠 cvsca 16316 ·𝑖cip 16317 TopSetcts 16318 lecple 16319 distcds 16321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 |
This theorem is referenced by: prdsvalstr 16473 imasbas 16532 imasds 16533 imasplusg 16537 imasmulr 16538 imassca 16539 imasvsca 16540 imasip 16541 imastset 16542 imasle 16543 |
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