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| Mirrors > Home > MPE Home > Th. List > imasvalstr | Structured version Visualization version GIF version | ||
| Description: An image structure 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 2737 | . . . 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 17380 | . . 3 ⊢ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉}) Struct 〈1, 8〉 |
| 4 | 9nn 12364 | . . . 4 ⊢ 9 ∈ ℕ | |
| 5 | tsetndx 17396 | . . . 4 ⊢ (TopSet‘ndx) = 9 | |
| 6 | 9lt10 12864 | . . . 4 ⊢ 9 < ;10 | |
| 7 | 10nn 12749 | . . . 4 ⊢ ;10 ∈ ℕ | |
| 8 | plendx 17410 | . . . 4 ⊢ (le‘ndx) = ;10 | |
| 9 | 1nn0 12542 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
| 10 | 0nn0 12541 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
| 11 | 2nn 12339 | . . . . 5 ⊢ 2 ∈ ℕ | |
| 12 | 2pos 12369 | . . . . 5 ⊢ 0 < 2 | |
| 13 | 9, 10, 11, 12 | declt 12761 | . . . 4 ⊢ ;10 < ;12 |
| 14 | 9, 11 | decnncl 12753 | . . . 4 ⊢ ;12 ∈ ℕ |
| 15 | dsndx 17429 | . . . 4 ⊢ (dist‘ndx) = ;12 | |
| 16 | 4, 5, 6, 7, 8, 13, 14, 15 | strle3 17197 | . . 3 ⊢ {〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), 𝐿〉, 〈(dist‘ndx), 𝐷〉} Struct 〈9, ;12〉 |
| 17 | 8lt9 12465 | . . 3 ⊢ 8 < 9 | |
| 18 | 3, 16, 17 | strleun 17194 | . 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 5164 | 1 ⊢ 𝑈 Struct 〈1, ;12〉 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∪ cun 3949 {ctp 4630 〈cop 4632 class class class wbr 5143 ‘cfv 6561 0cc0 11155 1c1 11156 2c2 12321 8c8 12327 9c9 12328 ;cdc 12733 Struct cstr 17183 ndxcnx 17230 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 ·𝑖cip 17302 TopSetcts 17303 lecple 17304 distcds 17306 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 |
| This theorem is referenced by: prdsvalstr 17497 imasbas 17557 imasds 17558 imasplusg 17562 imasmulr 17563 imassca 17564 imasvsca 17565 imasip 17566 imastset 17567 imasle 17568 rlocbas 33271 rlocaddval 33272 rlocmulval 33273 |
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