cdlemn11a - Mathbox for Norm Megill

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Theorem cdlemn11a 39221

Description: Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

**Hypotheses**
Ref	Expression
cdlemn11a.b	⊢ 𝐵 = (Base‘𝐾)
cdlemn11a.l	⊢ ≤ = (le‘𝐾)
cdlemn11a.j	⊢ ∨ = (join‘𝐾)
cdlemn11a.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemn11a.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemn11a.p	⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
cdlemn11a.o	⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
cdlemn11a.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemn11a.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemn11a.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
cdlemn11a.i	⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
cdlemn11a.J	⊢ 𝐽 = ((DIsoC‘𝐾)‘𝑊)
cdlemn11a.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
cdlemn11a.d	⊢ + = (+_g‘𝑈)
cdlemn11a.s	⊢ ⊕ = (LSSum‘𝑈)
cdlemn11a.f	⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄)
cdlemn11a.g	⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑁)

**Assertion**
Ref	Expression
cdlemn11a	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → ⟨𝐺, ( I ↾ 𝑇)⟩ ∈ (𝐽‘𝑁))

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

**Proof of Theorem cdlemn11a**
Step	Hyp	Ref	Expression
1		simp1 1135	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		cdlemn11a.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
3		cdlemn11a.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
4		cdlemn11a.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
5		cdlemn11a.p	. . . . . . 7 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
6	2, 3, 4, 5	lhpocnel2 38033	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
7	6	3ad2ant1 1132	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
8		simp22 1206	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊))
9		cdlemn11a.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
10		cdlemn11a.g	. . . . . 6 ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑁)
11	2, 3, 4, 9, 10	ltrniotacl 38593	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
12	1, 7, 8, 11	syl3anc 1370	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → 𝐺 ∈ 𝑇)
13		fvresi 7045	. . . 4 ⊢ (𝐺 ∈ 𝑇 → (( I ↾ 𝑇)‘𝐺) = 𝐺)
14	12, 13	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → (( I ↾ 𝑇)‘𝐺) = 𝐺)
15	14	eqcomd 2744	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → 𝐺 = (( I ↾ 𝑇)‘𝐺))
16		cdlemn11a.e	. . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
17	4, 9, 16	tendoidcl 38783	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸)
18	17	3ad2ant1 1132	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → ( I ↾ 𝑇) ∈ 𝐸)
19		cdlemn11a.J	. . . 4 ⊢ 𝐽 = ((DIsoC‘𝐾)‘𝑊)
20		riotaex 7236	. . . . 5 ⊢ (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑁) ∈ V
21	10, 20	eqeltri 2835	. . . 4 ⊢ 𝐺 ∈ V
22	9	fvexi 6788	. . . . 5 ⊢ 𝑇 ∈ V
23		resiexg 7761	. . . . 5 ⊢ (𝑇 ∈ V → ( I ↾ 𝑇) ∈ V)
24	22, 23	ax-mp 5	. . . 4 ⊢ ( I ↾ 𝑇) ∈ V
25	2, 3, 4, 5, 9, 16, 19, 10, 21, 24	dicopelval2 39195	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) → (⟨𝐺, ( I ↾ 𝑇)⟩ ∈ (𝐽‘𝑁) ↔ (𝐺 = (( I ↾ 𝑇)‘𝐺) ∧ ( I ↾ 𝑇) ∈ 𝐸)))
26	1, 8, 25	syl2anc 584	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → (⟨𝐺, ( I ↾ 𝑇)⟩ ∈ (𝐽‘𝑁) ↔ (𝐺 = (( I ↾ 𝑇)‘𝐺) ∧ ( I ↾ 𝑇) ∈ 𝐸)))
27	15, 18, 26	mpbir2and 710	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → ⟨𝐺, ( I ↾ 𝑇)⟩ ∈ (𝐽‘𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ⊆ wss 3887 ⟨cop 4567 class class class wbr 5074 ↦ cmpt 5157 I cid 5488 ↾ cres 5591 ‘cfv 6433 ℩crio 7231 (class class class)co 7275 Basecbs 16912 +_gcplusg 16962 lecple 16969 occoc 16970 joincjn 18029 LSSumclsm 19239 Atomscatm 37277 HLchlt 37364 LHypclh 37998 LTrncltrn 38115 trLctrl 38172 TEndoctendo 38766 DVecHcdvh 39092 DIsoBcdib 39152 DIsoCcdic 39186

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-undef 8089 df-map 8617 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tendo 38769 df-dic 39187

This theorem is referenced by: cdlemn11b 39222

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