cdlemn4a - Mathbox for Norm Megill

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Theorem cdlemn4a 41858

Description: Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 24-Feb-2014.)

**Hypotheses**
Ref	Expression
cdlemn4.b	⊢ 𝐵 = (Base‘𝐾)
cdlemn4.l	⊢ ≤ = (le‘𝐾)
cdlemn4.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemn4.p	⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
cdlemn4.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemn4.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemn4.o	⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
cdlemn4.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
cdlemn4.f	⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄)
cdlemn4.g	⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅)
cdlemn4.j	⊢ 𝐽 = (℩ℎ ∈ 𝑇 (ℎ‘𝑄) = 𝑅)
cdlemn4a.n	⊢ 𝑁 = (LSpan‘𝑈)
cdlemn4a.s	⊢ ⊕ = (LSSum‘𝑈)

**Assertion**
Ref	Expression
cdlemn4a	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑁‘{⟨𝐺, ( I ↾ 𝑇)⟩}) ⊆ ((𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}) ⊕ (𝑁‘{⟨𝐽, 𝑂⟩})))

Distinct variable groups: ≤ ,ℎ 𝐴,ℎ 𝐵,ℎ ℎ,𝐻 ℎ,𝐾 𝑃,ℎ 𝑄,ℎ 𝑅,ℎ 𝑇,ℎ ℎ,𝑊 Allowed substitution hints: ⊕ (ℎ) 𝑈(ℎ) 𝐹(ℎ) 𝐺(ℎ) 𝐽(ℎ) 𝑁(ℎ) 𝑂(ℎ)

**Proof of Theorem cdlemn4a**
Step	Hyp	Ref	Expression
1		cdlemn4.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemn4.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		cdlemn4.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
4		cdlemn4.p	. . . . 5 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
5		cdlemn4.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
6		cdlemn4.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
7		cdlemn4.o	. . . . 5 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
8		cdlemn4.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
9		cdlemn4.f	. . . . 5 ⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄)
10		cdlemn4.g	. . . . 5 ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅)
11		cdlemn4.j	. . . . 5 ⊢ 𝐽 = (℩ℎ ∈ 𝑇 (ℎ‘𝑄) = 𝑅)
12		eqid 2769	. . . . 5 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cdlemn4 41857	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨𝐹, ( I ↾ 𝑇)⟩(+_g‘𝑈)⟨𝐽, 𝑂⟩))
14	13	sneqd 4603	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → {⟨𝐺, ( I ↾ 𝑇)⟩} = {(⟨𝐹, ( I ↾ 𝑇)⟩(+_g‘𝑈)⟨𝐽, 𝑂⟩)})
15	14	fveq2d 6883	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑁‘{⟨𝐺, ( I ↾ 𝑇)⟩}) = (𝑁‘{(⟨𝐹, ( I ↾ 𝑇)⟩(+_g‘𝑈)⟨𝐽, 𝑂⟩)}))
16		simp1 1152	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17	5, 8, 16	dvhlmod 41769	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝑈 ∈ LMod)
18	2, 3, 5, 4	lhpocnel2 40678	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
19	18	3ad2ant1 1149	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
20		simp2 1153	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
21	2, 3, 5, 6, 9	ltrniotacl 41238	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
22	16, 19, 20, 21	syl3anc 1396	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
23		eqid 2769	. . . . . 6 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
24	5, 6, 23	tendoidcl 41428	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))
25	24	3ad2ant1 1149	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))
26		eqid 2769	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
27	5, 6, 23, 8, 26	dvhelvbasei 41747	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (Base‘𝑈))
28	16, 22, 25, 27	syl12anc 849	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (Base‘𝑈))
29	2, 3, 5, 6, 11	ltrniotacl 41238	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐽 ∈ 𝑇)
30	1, 5, 6, 23, 7	tendo0cl 41449	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))
31	30	3ad2ant1 1149	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))
32	5, 6, 23, 8, 26	dvhelvbasei 41747	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐽 ∈ 𝑇 ∧ 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))) → ⟨𝐽, 𝑂⟩ ∈ (Base‘𝑈))
33	16, 29, 31, 32	syl12anc 849	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ⟨𝐽, 𝑂⟩ ∈ (Base‘𝑈))
34		cdlemn4a.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
35		cdlemn4a.s	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
36	26, 12, 34, 35	lspsntri 21192	. . 3 ⊢ ((𝑈 ∈ LMod ∧ ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (Base‘𝑈) ∧ ⟨𝐽, 𝑂⟩ ∈ (Base‘𝑈)) → (𝑁‘{(⟨𝐹, ( I ↾ 𝑇)⟩(+_g‘𝑈)⟨𝐽, 𝑂⟩)}) ⊆ ((𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}) ⊕ (𝑁‘{⟨𝐽, 𝑂⟩})))
37	17, 28, 33, 36	syl3anc 1396	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑁‘{(⟨𝐹, ( I ↾ 𝑇)⟩(+_g‘𝑈)⟨𝐽, 𝑂⟩)}) ⊆ ((𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}) ⊕ (𝑁‘{⟨𝐽, 𝑂⟩})))
38	15, 37	eqsstrd 3979	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑁‘{⟨𝐺, ( I ↾ 𝑇)⟩}) ⊆ ((𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}) ⊕ (𝑁‘{⟨𝐽, 𝑂⟩})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ⊆ wss 3913 {csn 4591 ⟨cop 4597 class class class wbr 5110 ↦ cmpt 5193 I cid 5553 ↾ cres 5661 ‘cfv 6533 ℩crio 7364 (class class class)co 7408 Basecbs 17265 +_gcplusg 17306 lecple 17313 occoc 17314 LSSumclsm 19700 LModclmod 20955 LSpanclspn 21066 Atomscatm 39922 HLchlt 40009 LHypclh 40643 LTrncltrn 40760 TEndoctendo 41411 DVecHcdvh 41737

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-riotaBAD 39612

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-1st 7982 df-2nd 7983 df-tpos 8218 df-undef 8265 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-n0 12501 df-z 12588 df-uz 12859 df-fz 13532 df-struct 17203 df-sets 17220 df-slot 17238 df-ndx 17250 df-base 17266 df-ress 17287 df-plusg 17319 df-mulr 17320 df-sca 17322 df-vsca 17323 df-0g 17490 df-proset 18346 df-poset 18365 df-plt 18380 df-lub 18396 df-glb 18397 df-join 18398 df-meet 18399 df-p0 18475 df-p1 18476 df-lat 18484 df-clat 18551 df-mgm 18694 df-sgrp 18773 df-mnd 18789 df-submnd 18838 df-grp 18999 df-minusg 19000 df-sbg 19001 df-subg 19185 df-cntz 19383 df-lsm 19702 df-cmn 19848 df-abl 19849 df-mgp 20213 df-rng 20227 df-ur 20260 df-ring 20313 df-oppr 20415 df-dvdsr 20435 df-unit 20436 df-invr 20466 df-dvr 20479 df-drng 20811 df-lmod 20957 df-lss 21027 df-lsp 21067 df-lvec 21198 df-oposet 39835 df-ol 39837 df-oml 39838 df-covers 39925 df-ats 39926 df-atl 39957 df-cvlat 39981 df-hlat 40010 df-llines 40157 df-lplanes 40158 df-lvols 40159 df-lines 40160 df-psubsp 40162 df-pmap 40163 df-padd 40455 df-lhyp 40647 df-laut 40648 df-ldil 40763 df-ltrn 40764 df-trl 40818 df-tendo 41414 df-edring 41416 df-dvech 41738

This theorem is referenced by: cdlemn5pre 41859

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