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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 2777 | . . . 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 16416 | . . 3 ⊢ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉}) Struct 〈1, 8〉 |
4 | 9nn 11479 | . . . 4 ⊢ 9 ∈ ℕ | |
5 | tsetndx 16432 | . . . 4 ⊢ (TopSet‘ndx) = 9 | |
6 | 9lt10 11978 | . . . 4 ⊢ 9 < ;10 | |
7 | 10nn 11861 | . . . 4 ⊢ ;10 ∈ ℕ | |
8 | plendx 16439 | . . . 4 ⊢ (le‘ndx) = ;10 | |
9 | 1nn0 11660 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
10 | 0nn0 11659 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
11 | 2nn 11448 | . . . . 5 ⊢ 2 ∈ ℕ | |
12 | 2pos 11485 | . . . . 5 ⊢ 0 < 2 | |
13 | 9, 10, 11, 12 | declt 11874 | . . . 4 ⊢ ;10 < ;12 |
14 | 9, 11 | decnncl 11866 | . . . 4 ⊢ ;12 ∈ ℕ |
15 | dsndx 16448 | . . . 4 ⊢ (dist‘ndx) = ;12 | |
16 | 4, 5, 6, 7, 8, 13, 14, 15 | strle3 16367 | . . 3 ⊢ {〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), 𝐿〉, 〈(dist‘ndx), 𝐷〉} Struct 〈9, ;12〉 |
17 | 8lt9 11581 | . . 3 ⊢ 8 < 9 | |
18 | 3, 16, 17 | strleun 16364 | . 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 4907 | 1 ⊢ 𝑈 Struct 〈1, ;12〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∪ cun 3789 {ctp 4401 〈cop 4403 class class class wbr 4886 ‘cfv 6135 0cc0 10272 1c1 10273 2c2 11430 8c8 11436 9c9 11437 ;cdc 11845 Struct cstr 16251 ndxcnx 16252 Basecbs 16255 +gcplusg 16338 .rcmulr 16339 Scalarcsca 16341 ·𝑠 cvsca 16342 ·𝑖cip 16343 TopSetcts 16344 lecple 16345 distcds 16347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 |
This theorem is referenced by: prdsvalstr 16499 imasbas 16558 imasds 16559 imasplusg 16563 imasmulr 16564 imassca 16565 imasvsca 16566 imasip 16567 imastset 16568 imasle 16569 |
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