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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 2798 | . . . 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 16635 | . . 3 ⊢ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉}) Struct 〈1, 8〉 |
4 | 9nn 11723 | . . . 4 ⊢ 9 ∈ ℕ | |
5 | tsetndx 16651 | . . . 4 ⊢ (TopSet‘ndx) = 9 | |
6 | 9lt10 12217 | . . . 4 ⊢ 9 < ;10 | |
7 | 10nn 12102 | . . . 4 ⊢ ;10 ∈ ℕ | |
8 | plendx 16658 | . . . 4 ⊢ (le‘ndx) = ;10 | |
9 | 1nn0 11901 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
10 | 0nn0 11900 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
11 | 2nn 11698 | . . . . 5 ⊢ 2 ∈ ℕ | |
12 | 2pos 11728 | . . . . 5 ⊢ 0 < 2 | |
13 | 9, 10, 11, 12 | declt 12114 | . . . 4 ⊢ ;10 < ;12 |
14 | 9, 11 | decnncl 12106 | . . . 4 ⊢ ;12 ∈ ℕ |
15 | dsndx 16667 | . . . 4 ⊢ (dist‘ndx) = ;12 | |
16 | 4, 5, 6, 7, 8, 13, 14, 15 | strle3 16586 | . . 3 ⊢ {〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), 𝐿〉, 〈(dist‘ndx), 𝐷〉} Struct 〈9, ;12〉 |
17 | 8lt9 11824 | . . 3 ⊢ 8 < 9 | |
18 | 3, 16, 17 | strleun 16583 | . 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 5051 | 1 ⊢ 𝑈 Struct 〈1, ;12〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∪ cun 3879 {ctp 4529 〈cop 4531 class class class wbr 5030 ‘cfv 6324 0cc0 10526 1c1 10527 2c2 11680 8c8 11686 9c9 11687 ;cdc 12086 Struct cstr 16471 ndxcnx 16472 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 Scalarcsca 16560 ·𝑠 cvsca 16561 ·𝑖cip 16562 TopSetcts 16563 lecple 16564 distcds 16566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 |
This theorem is referenced by: prdsvalstr 16718 imasbas 16777 imasds 16778 imasplusg 16782 imasmulr 16783 imassca 16784 imasvsca 16785 imasip 16786 imastset 16787 imasle 16788 |
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