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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibss | Structured version Visualization version GIF version |
Description: The partial isomorphism B maps to a set of vectors in full vector space H. (Contributed by NM, 1-Jan-2014.) |
Ref | Expression |
---|---|
dibss.b | β’ π΅ = (BaseβπΎ) |
dibss.l | β’ β€ = (leβπΎ) |
dibss.h | β’ π» = (LHypβπΎ) |
dibss.i | β’ πΌ = ((DIsoBβπΎ)βπ) |
dibss.u | β’ π = ((DVecHβπΎ)βπ) |
dibss.v | β’ π = (Baseβπ) |
Ref | Expression |
---|---|
dibss | β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΌβπ) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dibss.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | dibss.l | . . . 4 β’ β€ = (leβπΎ) | |
3 | dibss.h | . . . 4 β’ π» = (LHypβπΎ) | |
4 | eqid 2724 | . . . 4 β’ ((LTrnβπΎ)βπ) = ((LTrnβπΎ)βπ) | |
5 | eqid 2724 | . . . 4 β’ ((DIsoAβπΎ)βπ) = ((DIsoAβπΎ)βπ) | |
6 | 1, 2, 3, 4, 5 | diass 40407 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (((DIsoAβπΎ)βπ)βπ) β ((LTrnβπΎ)βπ)) |
7 | eqid 2724 | . . . . . 6 β’ ((TEndoβπΎ)βπ) = ((TEndoβπΎ)βπ) | |
8 | eqid 2724 | . . . . . 6 β’ (π β ((LTrnβπΎ)βπ) β¦ ( I βΎ π΅)) = (π β ((LTrnβπΎ)βπ) β¦ ( I βΎ π΅)) | |
9 | 1, 3, 4, 7, 8 | tendo0cl 40155 | . . . . 5 β’ ((πΎ β HL β§ π β π») β (π β ((LTrnβπΎ)βπ) β¦ ( I βΎ π΅)) β ((TEndoβπΎ)βπ)) |
10 | 9 | snssd 4805 | . . . 4 β’ ((πΎ β HL β§ π β π») β {(π β ((LTrnβπΎ)βπ) β¦ ( I βΎ π΅))} β ((TEndoβπΎ)βπ)) |
11 | 10 | adantr 480 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β {(π β ((LTrnβπΎ)βπ) β¦ ( I βΎ π΅))} β ((TEndoβπΎ)βπ)) |
12 | xpss12 5682 | . . 3 β’ (((((DIsoAβπΎ)βπ)βπ) β ((LTrnβπΎ)βπ) β§ {(π β ((LTrnβπΎ)βπ) β¦ ( I βΎ π΅))} β ((TEndoβπΎ)βπ)) β ((((DIsoAβπΎ)βπ)βπ) Γ {(π β ((LTrnβπΎ)βπ) β¦ ( I βΎ π΅))}) β (((LTrnβπΎ)βπ) Γ ((TEndoβπΎ)βπ))) | |
13 | 6, 11, 12 | syl2anc 583 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β ((((DIsoAβπΎ)βπ)βπ) Γ {(π β ((LTrnβπΎ)βπ) β¦ ( I βΎ π΅))}) β (((LTrnβπΎ)βπ) Γ ((TEndoβπΎ)βπ))) |
14 | dibss.i | . . 3 β’ πΌ = ((DIsoBβπΎ)βπ) | |
15 | 1, 2, 3, 4, 8, 5, 14 | dibval2 40509 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΌβπ) = ((((DIsoAβπΎ)βπ)βπ) Γ {(π β ((LTrnβπΎ)βπ) β¦ ( I βΎ π΅))})) |
16 | dibss.u | . . . 4 β’ π = ((DVecHβπΎ)βπ) | |
17 | dibss.v | . . . 4 β’ π = (Baseβπ) | |
18 | 3, 4, 7, 16, 17 | dvhvbase 40452 | . . 3 β’ ((πΎ β HL β§ π β π») β π = (((LTrnβπΎ)βπ) Γ ((TEndoβπΎ)βπ))) |
19 | 18 | adantr 480 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β π = (((LTrnβπΎ)βπ) Γ ((TEndoβπΎ)βπ))) |
20 | 13, 15, 19 | 3sstr4d 4022 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΌβπ) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wss 3941 {csn 4621 class class class wbr 5139 β¦ cmpt 5222 I cid 5564 Γ cxp 5665 βΎ cres 5669 βcfv 6534 Basecbs 17145 lecple 17205 HLchlt 38714 LHypclh 39349 LTrncltrn 39466 TEndoctendo 40117 DIsoAcdia 40393 DVecHcdvh 40443 DIsoBcdib 40503 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-riotaBAD 38317 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-n0 12471 df-z 12557 df-uz 12821 df-fz 13483 df-struct 17081 df-slot 17116 df-ndx 17128 df-base 17146 df-plusg 17211 df-sca 17214 df-vsca 17215 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-oposet 38540 df-ol 38542 df-oml 38543 df-covers 38630 df-ats 38631 df-atl 38662 df-cvlat 38686 df-hlat 38715 df-llines 38863 df-lplanes 38864 df-lvols 38865 df-lines 38866 df-psubsp 38868 df-pmap 38869 df-padd 39161 df-lhyp 39353 df-laut 39354 df-ldil 39469 df-ltrn 39470 df-trl 39524 df-tendo 40120 df-disoa 40394 df-dvech 40444 df-dib 40504 |
This theorem is referenced by: diblss 40535 |
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