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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 2725 | . . . 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 17311 | . . 3 ⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) Struct ⟨1, 8⟩ |
4 | 9nn 12335 | . . . 4 ⊢ 9 ∈ ℕ | |
5 | tsetndx 17327 | . . . 4 ⊢ (TopSet‘ndx) = 9 | |
6 | 9lt10 12833 | . . . 4 ⊢ 9 < ;10 | |
7 | 10nn 12718 | . . . 4 ⊢ ;10 ∈ ℕ | |
8 | plendx 17341 | . . . 4 ⊢ (le‘ndx) = ;10 | |
9 | 1nn0 12513 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
10 | 0nn0 12512 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
11 | 2nn 12310 | . . . . 5 ⊢ 2 ∈ ℕ | |
12 | 2pos 12340 | . . . . 5 ⊢ 0 < 2 | |
13 | 9, 10, 11, 12 | declt 12730 | . . . 4 ⊢ ;10 < ;12 |
14 | 9, 11 | decnncl 12722 | . . . 4 ⊢ ;12 ∈ ℕ |
15 | dsndx 17360 | . . . 4 ⊢ (dist‘ndx) = ;12 | |
16 | 4, 5, 6, 7, 8, 13, 14, 15 | strle3 17123 | . . 3 ⊢ {⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), 𝐿⟩, ⟨(dist‘ndx), 𝐷⟩} Struct ⟨9, ;12⟩ |
17 | 8lt9 12436 | . . 3 ⊢ 8 < 9 | |
18 | 3, 16, 17 | strleun 17120 | . 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 5165 | 1 ⊢ 𝑈 Struct ⟨1, ;12⟩ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∪ cun 3939 {ctp 4629 ⟨cop 4631 class class class wbr 5144 ‘cfv 6543 0cc0 11133 1c1 11134 2c2 12292 8c8 12298 9c9 12299 ;cdc 12702 Struct cstr 17109 ndxcnx 17156 Basecbs 17174 +gcplusg 17227 .rcmulr 17228 Scalarcsca 17230 ·𝑠 cvsca 17231 ·𝑖cip 17232 TopSetcts 17233 lecple 17234 distcds 17236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-struct 17110 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 |
This theorem is referenced by: prdsvalstr 17428 imasbas 17488 imasds 17489 imasplusg 17493 imasmulr 17494 imassca 17495 imasvsca 17496 imasip 17497 imastset 17498 imasle 17499 rlocbas 33004 rlocaddval 33005 rlocmulval 33006 |
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