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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > diclss | Structured version Visualization version GIF version |
Description: The value of partial isomorphism C is a subspace of partial vector space H. (Contributed by NM, 16-Feb-2014.) |
Ref | Expression |
---|---|
diclss.l | ⊢ ≤ = (le‘𝐾) |
diclss.a | ⊢ 𝐴 = (Atoms‘𝐾) |
diclss.h | ⊢ 𝐻 = (LHyp‘𝐾) |
diclss.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
diclss.i | ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊) |
diclss.s | ⊢ 𝑆 = (LSubSp‘𝑈) |
Ref | Expression |
---|---|
diclss | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2652 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (Scalar‘𝑈) = (Scalar‘𝑈)) | |
2 | diclss.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | eqid 2651 | . . . . 5 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊) | |
4 | diclss.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | eqid 2651 | . . . . 5 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
6 | eqid 2651 | . . . . 5 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) | |
7 | 2, 3, 4, 5, 6 | dvhbase 36689 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘(Scalar‘𝑈)) = ((TEndo‘𝐾)‘𝑊)) |
8 | 7 | eqcomd 2657 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((TEndo‘𝐾)‘𝑊) = (Base‘(Scalar‘𝑈))) |
9 | 8 | adantr 480 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((TEndo‘𝐾)‘𝑊) = (Base‘(Scalar‘𝑈))) |
10 | eqid 2651 | . . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
11 | eqid 2651 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
12 | 2, 10, 3, 4, 11 | dvhvbase 36693 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊))) |
13 | 12 | eqcomd 2657 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈)) |
14 | 13 | adantr 480 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈)) |
15 | eqidd 2652 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (+g‘𝑈) = (+g‘𝑈)) | |
16 | eqidd 2652 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈)) | |
17 | diclss.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑈) | |
18 | 17 | a1i 11 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝑆 = (LSubSp‘𝑈)) |
19 | diclss.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
20 | diclss.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
21 | diclss.i | . . . 4 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊) | |
22 | 19, 20, 2, 21, 4, 11 | dicssdvh 36792 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ⊆ (Base‘𝑈)) |
23 | 22, 14 | sseqtr4d 3675 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊))) |
24 | 19, 20, 2, 21 | dicn0 36798 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ≠ ∅) |
25 | simpll 805 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑄) ∧ 𝑏 ∈ (𝐼‘𝑄))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
26 | simplr 807 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑄) ∧ 𝑏 ∈ (𝐼‘𝑄))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) | |
27 | simpr1 1087 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑄) ∧ 𝑏 ∈ (𝐼‘𝑄))) → 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) | |
28 | simpr2 1088 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑄) ∧ 𝑏 ∈ (𝐼‘𝑄))) → 𝑎 ∈ (𝐼‘𝑄)) | |
29 | eqid 2651 | . . . . 5 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
30 | 19, 20, 2, 3, 4, 21, 29 | dicvscacl 36797 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑄))) → (𝑥( ·𝑠 ‘𝑈)𝑎) ∈ (𝐼‘𝑄)) |
31 | 25, 26, 27, 28, 30 | syl112anc 1370 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑄) ∧ 𝑏 ∈ (𝐼‘𝑄))) → (𝑥( ·𝑠 ‘𝑈)𝑎) ∈ (𝐼‘𝑄)) |
32 | simpr3 1089 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑄) ∧ 𝑏 ∈ (𝐼‘𝑄))) → 𝑏 ∈ (𝐼‘𝑄)) | |
33 | eqid 2651 | . . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
34 | 19, 20, 2, 4, 21, 33 | dicvaddcl 36796 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ((𝑥( ·𝑠 ‘𝑈)𝑎) ∈ (𝐼‘𝑄) ∧ 𝑏 ∈ (𝐼‘𝑄))) → ((𝑥( ·𝑠 ‘𝑈)𝑎)(+g‘𝑈)𝑏) ∈ (𝐼‘𝑄)) |
35 | 25, 26, 31, 32, 34 | syl112anc 1370 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑄) ∧ 𝑏 ∈ (𝐼‘𝑄))) → ((𝑥( ·𝑠 ‘𝑈)𝑎)(+g‘𝑈)𝑏) ∈ (𝐼‘𝑄)) |
36 | 1, 9, 14, 15, 16, 18, 23, 24, 35 | islssd 18984 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 class class class wbr 4685 × cxp 5141 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 +gcplusg 15988 Scalarcsca 15991 ·𝑠 cvsca 15992 lecple 15995 LSubSpclss 18980 Atomscatm 34868 HLchlt 34955 LHypclh 35588 LTrncltrn 35705 TEndoctendo 36357 DVecHcdvh 36684 DIsoCcdic 36778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-riotaBAD 34557 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-undef 7444 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-preset 16975 df-poset 16993 df-plt 17005 df-lub 17021 df-glb 17022 df-join 17023 df-meet 17024 df-p0 17086 df-p1 17087 df-lat 17093 df-clat 17155 df-lss 18981 df-oposet 34781 df-ol 34783 df-oml 34784 df-covers 34871 df-ats 34872 df-atl 34903 df-cvlat 34927 df-hlat 34956 df-llines 35102 df-lplanes 35103 df-lvols 35104 df-lines 35105 df-psubsp 35107 df-pmap 35108 df-padd 35400 df-lhyp 35592 df-laut 35593 df-ldil 35708 df-ltrn 35709 df-trl 35764 df-tendo 36360 df-edring 36362 df-dvech 36685 df-dic 36779 |
This theorem is referenced by: cdlemn5pre 36806 cdlemn11c 36815 dihjustlem 36822 dihord1 36824 dihord2a 36825 dihord2b 36826 dihord11c 36830 dihlsscpre 36840 dihvalcqat 36845 dihopelvalcpre 36854 dihord6apre 36862 dihord5b 36865 dihord5apre 36868 |
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