cdlemn2a - Mathbox for Norm Megill

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Theorem cdlemn2a 37270

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

**Hypotheses**
Ref	Expression
cdlemn2a.b	⊢ 𝐵 = (Base‘𝐾)
cdlemn2a.l	⊢ ≤ = (le‘𝐾)
cdlemn2a.j	⊢ ∨ = (join‘𝐾)
cdlemn2a.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemn2a.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemn2a.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemn2a.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemn2a.o	⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
cdlemn2a.i	⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
cdlemn2a.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
cdlemn2a.n	⊢ 𝑁 = (LSpan‘𝑈)
cdlemn2a.f	⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑄) = 𝑆)

**Assertion**
Ref	Expression
cdlemn2a	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑁‘{⟨𝐹, 𝑂⟩}) ⊆ (𝐼‘𝑋))

Distinct variable groups: ≤ ,ℎ 𝐴,ℎ 𝐵,𝑓 ℎ,𝐻 𝑓,𝐾 ℎ,𝐾 𝑄,ℎ 𝑆,ℎ 𝑇,𝑓 𝑇,ℎ 𝑓,𝑊 ℎ,𝑊 Allowed substitution hints: 𝐴(𝑓) 𝐵(ℎ) 𝑄(𝑓) 𝑅(𝑓,ℎ) 𝑆(𝑓) 𝑈(𝑓,ℎ) 𝐹(𝑓,ℎ) 𝐻(𝑓) 𝐼(𝑓,ℎ) ∨ (𝑓,ℎ) ≤ (𝑓) 𝑁(𝑓,ℎ) 𝑂(𝑓,ℎ) 𝑋(𝑓,ℎ)

**Proof of Theorem cdlemn2a**
Step	Hyp	Ref	Expression
1		simp1 1170	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp21 1267	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
3		simp22 1268	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))
4		cdlemn2a.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
5		cdlemn2a.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemn2a.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemn2a.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8		cdlemn2a.f	. . . . 5 ⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑄) = 𝑆)
9	4, 5, 6, 7, 8	ltrniotacl 36653	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
10	1, 2, 3, 9	syl3anc 1494	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝐹 ∈ 𝑇)
11		cdlemn2a.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
12		cdlemn2a.r	. . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
13		cdlemn2a.o	. . . 4 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
14		cdlemn2a.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
15		cdlemn2a.i	. . . 4 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
16		cdlemn2a.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
17	11, 6, 7, 12, 13, 14, 15, 16	dib1dim2 37242	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐼‘(𝑅‘𝐹)) = (𝑁‘{⟨𝐹, 𝑂⟩}))
18	1, 10, 17	syl2anc 579	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝐼‘(𝑅‘𝐹)) = (𝑁‘{⟨𝐹, 𝑂⟩}))
19		cdlemn2a.j	. . . 4 ⊢ ∨ = (join‘𝐾)
20	11, 4, 19, 5, 6, 7, 12, 8	cdlemn2 37269	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑅‘𝐹) ≤ 𝑋)
21	11, 6, 7, 12	trlcl 36238	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ 𝐵)
22	1, 10, 21	syl2anc 579	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑅‘𝐹) ∈ 𝐵)
23	4, 6, 7, 12	trlle 36258	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ≤ 𝑊)
24	1, 10, 23	syl2anc 579	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑅‘𝐹) ≤ 𝑊)
25		simp23 1269	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
26	11, 4, 6, 15	dibord 37233	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅‘𝐹) ∈ 𝐵 ∧ (𝑅‘𝐹) ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ((𝐼‘(𝑅‘𝐹)) ⊆ (𝐼‘𝑋) ↔ (𝑅‘𝐹) ≤ 𝑋))
27	1, 22, 24, 25, 26	syl121anc 1498	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → ((𝐼‘(𝑅‘𝐹)) ⊆ (𝐼‘𝑋) ↔ (𝑅‘𝐹) ≤ 𝑋))
28	20, 27	mpbird 249	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝐼‘(𝑅‘𝐹)) ⊆ (𝐼‘𝑋))
29	18, 28	eqsstr3d 3865	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑁‘{⟨𝐹, 𝑂⟩}) ⊆ (𝐼‘𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 {csn 4399 ⟨cop 4405 class class class wbr 4875 ↦ cmpt 4954 I cid 5251 ↾ cres 5348 ‘cfv 6127 ℩crio 6870 (class class class)co 6910 Basecbs 16229 lecple 16319 joincjn 17304 LSpanclspn 19337 Atomscatm 35337 HLchlt 35424 LHypclh 36058 LTrncltrn 36175 trLctrl 36232 DVecHcdvh 37152 DIsoBcdib 37212

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-riotaBAD 35027

This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-tpos 7622 df-undef 7669 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-0g 16462 df-proset 17288 df-poset 17306 df-plt 17318 df-lub 17334 df-glb 17335 df-join 17336 df-meet 17337 df-p0 17399 df-p1 17400 df-lat 17406 df-clat 17468 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-minusg 17787 df-sbg 17788 df-mgp 18851 df-ur 18863 df-ring 18910 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-drng 19112 df-lmod 19228 df-lss 19296 df-lsp 19338 df-lvec 19469 df-oposet 35250 df-ol 35252 df-oml 35253 df-covers 35340 df-ats 35341 df-atl 35372 df-cvlat 35396 df-hlat 35425 df-llines 35572 df-lplanes 35573 df-lvols 35574 df-lines 35575 df-psubsp 35577 df-pmap 35578 df-padd 35870 df-lhyp 36062 df-laut 36063 df-ldil 36178 df-ltrn 36179 df-trl 36233 df-tendo 36829 df-edring 36831 df-disoa 37103 df-dvech 37153 df-dib 37213

This theorem is referenced by: cdlemn5pre 37274

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