diclss - Mathbox for Norm Megill

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Theorem diclss 35948

Description: The value of partial isomorphism C is a subspace of partial vector space H. (Contributed by NM, 16-Feb-2014.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
diclss	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ∈ 𝑆)

Proof of Theorem diclss Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2627	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (Scalar‘𝑈) = (Scalar‘𝑈))
2		diclss.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2626	. . . . 5 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
4		diclss.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		eqid 2626	. . . . 5 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
6		eqid 2626	. . . . 5 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
7	2, 3, 4, 5, 6	dvhbase 35838	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘(Scalar‘𝑈)) = ((TEndo‘𝐾)‘𝑊))
8	7	eqcomd 2632	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((TEndo‘𝐾)‘𝑊) = (Base‘(Scalar‘𝑈)))
9	8	adantr 481	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((TEndo‘𝐾)‘𝑊) = (Base‘(Scalar‘𝑈)))
10		eqid 2626	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
11		eqid 2626	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
12	2, 10, 3, 4, 11	dvhvbase 35842	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
13	12	eqcomd 2632	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
14	13	adantr 481	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) = (Base‘𝑈))
15		eqidd 2627	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (+_g‘𝑈) = (+_g‘𝑈))
16		eqidd 2627	. 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 35941	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ⊆ (Base‘𝑈))
23	22, 14	sseqtr4d 3626	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
24	19, 20, 2, 21	dicn0 35947	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ≠ ∅)
25		simpll 789	. . 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 791	. . 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 1065	. . . 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 1066	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑄) ∧ 𝑏 ∈ (𝐼‘𝑄))) → 𝑎 ∈ (𝐼‘𝑄))
29		eqid 2626	. . . . 5 ⊢ ( ·_𝑠 ‘𝑈) = ( ·_𝑠 ‘𝑈)
30	19, 20, 2, 3, 4, 21, 29	dicvscacl 35946	. . . 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 1327	. . 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 1067	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑄) ∧ 𝑏 ∈ (𝐼‘𝑄))) → 𝑏 ∈ (𝐼‘𝑄))
33		eqid 2626	. . . 4 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
34	19, 20, 2, 4, 21, 33	dicvaddcl 35945	. . 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 1327	. 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 18850	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 384 ∧ w3a 1036 = wceq 1480 ∈ wcel 1992 class class class wbr 4618 × cxp 5077 ‘cfv 5850 (class class class)co 6605 Basecbs 15776 +_gcplusg 15857 Scalarcsca 15860 ·_𝑠 cvsca 15861 lecple 15864 LSubSpclss 18846 Atomscatm 34016 HLchlt 34103 LHypclh 34736 LTrncltrn 34853 TEndoctendo 35506 DVecHcdvh 35833 DIsoCcdic 35927

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1719 ax-4 1734 ax-5 1841 ax-6 1890 ax-7 1937 ax-8 1994 ax-9 2001 ax-10 2021 ax-11 2036 ax-12 2049 ax-13 2250 ax-ext 2606 ax-rep 4736 ax-sep 4746 ax-nul 4754 ax-pow 4808 ax-pr 4872 ax-un 6903 ax-cnex 9937 ax-resscn 9938 ax-1cn 9939 ax-icn 9940 ax-addcl 9941 ax-addrcl 9942 ax-mulcl 9943 ax-mulrcl 9944 ax-mulcom 9945 ax-addass 9946 ax-mulass 9947 ax-distr 9948 ax-i2m1 9949 ax-1ne0 9950 ax-1rid 9951 ax-rnegex 9952 ax-rrecex 9953 ax-cnre 9954 ax-pre-lttri 9955 ax-pre-lttrn 9956 ax-pre-ltadd 9957 ax-pre-mulgt0 9958 ax-riotaBAD 33705

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1037 df-3an 1038 df-tru 1483 df-ex 1702 df-nf 1707 df-sb 1883 df-eu 2478 df-mo 2479 df-clab 2613 df-cleq 2619 df-clel 2622 df-nfc 2756 df-ne 2797 df-nel 2900 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3193 df-sbc 3423 df-csb 3520 df-dif 3563 df-un 3565 df-in 3567 df-ss 3574 df-pss 3576 df-nul 3897 df-if 4064 df-pw 4137 df-sn 4154 df-pr 4156 df-tp 4158 df-op 4160 df-uni 4408 df-int 4446 df-iun 4492 df-iin 4493 df-br 4619 df-opab 4679 df-mpt 4680 df-tr 4718 df-eprel 4990 df-id 4994 df-po 5000 df-so 5001 df-fr 5038 df-we 5040 df-xp 5085 df-rel 5086 df-cnv 5087 df-co 5088 df-dm 5089 df-rn 5090 df-res 5091 df-ima 5092 df-pred 5642 df-ord 5688 df-on 5689 df-lim 5690 df-suc 5691 df-iota 5813 df-fun 5852 df-fn 5853 df-f 5854 df-f1 5855 df-fo 5856 df-f1o 5857 df-fv 5858 df-riota 6566 df-ov 6608 df-oprab 6609 df-mpt2 6610 df-om 7014 df-1st 7116 df-2nd 7117 df-undef 7345 df-wrecs 7353 df-recs 7414 df-rdg 7452 df-1o 7506 df-oadd 7510 df-er 7688 df-map 7805 df-en 7901 df-dom 7902 df-sdom 7903 df-fin 7904 df-pnf 10021 df-mnf 10022 df-xr 10023 df-ltxr 10024 df-le 10025 df-sub 10213 df-neg 10214 df-nn 10966 df-2 11024 df-3 11025 df-4 11026 df-5 11027 df-6 11028 df-n0 11238 df-z 11323 df-uz 11632 df-fz 12266 df-struct 15778 df-ndx 15779 df-slot 15780 df-base 15781 df-plusg 15870 df-mulr 15871 df-sca 15873 df-vsca 15874 df-preset 16844 df-poset 16862 df-plt 16874 df-lub 16890 df-glb 16891 df-join 16892 df-meet 16893 df-p0 16955 df-p1 16956 df-lat 16962 df-clat 17024 df-lss 18847 df-oposet 33929 df-ol 33931 df-oml 33932 df-covers 34019 df-ats 34020 df-atl 34051 df-cvlat 34075 df-hlat 34104 df-llines 34250 df-lplanes 34251 df-lvols 34252 df-lines 34253 df-psubsp 34255 df-pmap 34256 df-padd 34548 df-lhyp 34740 df-laut 34741 df-ldil 34856 df-ltrn 34857 df-trl 34912 df-tendo 35509 df-edring 35511 df-dvech 35834 df-dic 35928

This theorem is referenced by: cdlemn5pre 35955 cdlemn11c 35964 dihjustlem 35971 dihord1 35973 dihord2a 35974 dihord2b 35975 dihord11c 35979 dihlsscpre 35989 dihvalcqat 35994 dihopelvalcpre 36003 dihord6apre 36011 dihord5b 36014 dihord5apre 36017

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