cdlemn4a - Mathbox for Norm Megill

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Theorem cdlemn4a 42365

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 2760	. . . . 5 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cdlemn4 42364	. . . 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 4592	. . 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 6894	. 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 1154	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17	5, 8, 16	dvhlmod 42276	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝑈 ∈ LMod)
18	2, 3, 5, 4	lhpocnel2 41185	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
19	18	3ad2ant1 1151	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
20		simp2 1155	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
21	2, 3, 5, 6, 9	ltrniotacl 41745	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
22	16, 19, 20, 21	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
23		eqid 2760	. . . . . 6 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
24	5, 6, 23	tendoidcl 41935	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))
25	24	3ad2ant1 1151	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))
26		eqid 2760	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
27	5, 6, 23, 8, 26	dvhelvbasei 42254	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (Base‘𝑈))
28	16, 22, 25, 27	syl12anc 850	. . 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 41745	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐽 ∈ 𝑇)
30	1, 5, 6, 23, 7	tendo0cl 41956	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))
31	30	3ad2ant1 1151	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))
32	5, 6, 23, 8, 26	dvhelvbasei 42254	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐽 ∈ 𝑇 ∧ 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))) → ⟨𝐽, 𝑂⟩ ∈ (Base‘𝑈))
33	16, 29, 31, 32	syl12anc 850	. . 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 21381	. . 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 1398	. 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 3943	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ⊆ wss 3896 {csn 4584 ⟨cop 4834 class class class wbr 5102 ↦ cmpt 5185 I cid 5545 ↾ cres 5653 ‘cfv 6543 ℩crio 7380 (class class class)co 7424 Basecbs 17396 +_gcplusg 17437 lecple 17444 occoc 17445 LSSumclsm 19857 LModclmod 21144 LSpanclspn 21255 Atomscatm 40429 HLchlt 40516 LHypclh 41150 LTrncltrn 41267 TEndoctendo 41918 DVecHcdvh 42244

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-riotaBAD 40119

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7882 df-1st 8005 df-2nd 8006 df-tpos 8250 df-undef 8297 df-frecs 8306 df-wrecs 8337 df-recs 8386 df-rdg 8425 df-1o 8483 df-er 8724 df-map 8856 df-en 8981 df-dom 8982 df-sdom 8983 df-fin 8984 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-n0 12614 df-z 12701 df-uz 12973 df-fz 13647 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-sca 17453 df-vsca 17454 df-0g 17621 df-proset 18477 df-poset 18496 df-plt 18511 df-lub 18527 df-glb 18528 df-join 18529 df-meet 18530 df-p0 18606 df-p1 18607 df-lat 18615 df-clat 18682 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-submnd 18988 df-grp 19156 df-minusg 19157 df-sbg 19158 df-subg 19342 df-cntz 19540 df-lsm 19859 df-cmn 20005 df-abl 20006 df-mgp 20370 df-rng 20384 df-ur 20417 df-ring 20470 df-oppr 20576 df-dvdsr 20596 df-unit 20597 df-invr 20627 df-dvr 20640 df-drng 20991 df-lmod 21146 df-lss 21216 df-lsp 21256 df-lvec 21387 df-oposet 40342 df-ol 40344 df-oml 40345 df-covers 40432 df-ats 40433 df-atl 40464 df-cvlat 40488 df-hlat 40517 df-llines 40664 df-lplanes 40665 df-lvols 40666 df-lines 40667 df-psubsp 40669 df-pmap 40670 df-padd 40962 df-lhyp 41154 df-laut 41155 df-ldil 41270 df-ltrn 41271 df-trl 41325 df-tendo 41921 df-edring 41923 df-dvech 42245

This theorem is used by: cdlemn5pre 42366

W3C validator