diclss - Mathbox for Norm Megill

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Theorem diclss 41365

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 2734	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (Scalar‘𝑈) = (Scalar‘𝑈))
2		diclss.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2733	. . . . 5 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
4		diclss.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		eqid 2733	. . . . 5 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
6		eqid 2733	. . . . 5 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
7	2, 3, 4, 5, 6	dvhbase 41255	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘(Scalar‘𝑈)) = ((TEndo‘𝐾)‘𝑊))
8	7	eqcomd 2739	. . 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 2733	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
11		eqid 2733	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
12	2, 10, 3, 4, 11	dvhvbase 41259	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
13	12	eqcomd 2739	. . 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 2734	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (+_g‘𝑈) = (+_g‘𝑈))
16		eqidd 2734	. 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 41358	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ⊆ (Base‘𝑈))
23	22, 14	sseqtrrd 3968	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ⊆ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)))
24	19, 20, 2, 21	dicn0 41364	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ≠ ∅)
25		simpll 766	. . 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 768	. . 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 1195	. . . 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 1196	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑄) ∧ 𝑏 ∈ (𝐼‘𝑄))) → 𝑎 ∈ (𝐼‘𝑄))
29		eqid 2733	. . . . 5 ⊢ ( ·_𝑠 ‘𝑈) = ( ·_𝑠 ‘𝑈)
30	19, 20, 2, 3, 4, 21, 29	dicvscacl 41363	. . . 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 1376	. . 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 1197	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑎 ∈ (𝐼‘𝑄) ∧ 𝑏 ∈ (𝐼‘𝑄))) → 𝑏 ∈ (𝐼‘𝑄))
33		eqid 2733	. . . 4 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
34	19, 20, 2, 4, 21, 33	dicvaddcl 41362	. . 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 1376	. 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 20877	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 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 class class class wbr 5095 × cxp 5619 ‘cfv 6489 (class class class)co 7355 Basecbs 17127 +_gcplusg 17168 Scalarcsca 17171 ·_𝑠 cvsca 17172 lecple 17175 LSubSpclss 20873 Atomscatm 39435 HLchlt 39522 LHypclh 40156 LTrncltrn 40273 TEndoctendo 40924 DVecHcdvh 41250 DIsoCcdic 41344

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-riotaBAD 39125

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-undef 8212 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8631 df-map 8761 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-n0 12393 df-z 12480 df-uz 12743 df-fz 13415 df-struct 17065 df-slot 17100 df-ndx 17112 df-base 17128 df-plusg 17181 df-mulr 17182 df-sca 17184 df-vsca 17185 df-proset 18208 df-poset 18227 df-plt 18242 df-lub 18258 df-glb 18259 df-join 18260 df-meet 18261 df-p0 18337 df-p1 18338 df-lat 18346 df-clat 18413 df-lss 20874 df-oposet 39348 df-ol 39350 df-oml 39351 df-covers 39438 df-ats 39439 df-atl 39470 df-cvlat 39494 df-hlat 39523 df-llines 39670 df-lplanes 39671 df-lvols 39672 df-lines 39673 df-psubsp 39675 df-pmap 39676 df-padd 39968 df-lhyp 40160 df-laut 40161 df-ldil 40276 df-ltrn 40277 df-trl 40331 df-tendo 40927 df-edring 40929 df-dvech 41251 df-dic 41345

This theorem is referenced by: cdlemn5pre 41372 cdlemn11c 41381 dihjustlem 41388 dihord1 41390 dihord2a 41391 dihord2b 41392 dihord11c 41396 dihlsscpre 41406 dihvalcqat 41411 dihopelvalcpre 41420 dihord6apre 41428 dihord5b 41431 dihord5apre 41434

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