cdlemn11a - Mathbox for Norm Megill

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Theorem cdlemn11a 41962

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 1154	. . . . 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 40774	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
7	6	3ad2ant1 1151	. . . . 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 1226	. . . . 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 41334	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
12	1, 7, 8, 11	syl3anc 1398	. . . 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 7173	. . . 4 ⊢ (𝐺 ∈ 𝑇 → (( I ↾ 𝑇)‘𝐺) = 𝐺)
14	12, 13	syl 18	. . 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 2769	. 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 41524	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸)
18	17	3ad2ant1 1151	. 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 7373	. . . . 5 ⊢ (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑁) ∈ V
21	10, 20	eqeltri 2859	. . . 4 ⊢ 𝐺 ∈ V
22	9	fvexi 6897	. . . . 5 ⊢ 𝑇 ∈ V
23		resiexg 7910	. . . . 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 41936	. . 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 595	. 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 725	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 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 Vcvv 3455 ⊆ wss 3906 ⟨cop 4596 class class class wbr 5110 ↦ cmpt 5193 I cid 5557 ↾ cres 5665 ‘cfv 6538 ℩crio 7368 (class class class)co 7412 Basecbs 17270 +_gcplusg 17311 lecple 17318 occoc 17319 joincjn 18368 LSSumclsm 19705 Atomscatm 40018 HLchlt 40105 LHypclh 40739 LTrncltrn 40856 trLctrl 40913 TEndoctendo 41507 DVecHcdvh 41833 DIsoBcdib 41893 DIsoCcdic 41927

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-riotaBAD 39708

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7987 df-2nd 7988 df-undef 8270 df-map 8827 df-proset 18351 df-poset 18370 df-plt 18385 df-lub 18401 df-glb 18402 df-join 18403 df-meet 18404 df-p0 18480 df-p1 18481 df-lat 18489 df-clat 18556 df-oposet 39931 df-ol 39933 df-oml 39934 df-covers 40021 df-ats 40022 df-atl 40053 df-cvlat 40077 df-hlat 40106 df-llines 40253 df-lplanes 40254 df-lvols 40255 df-lines 40256 df-psubsp 40258 df-pmap 40259 df-padd 40551 df-lhyp 40743 df-laut 40744 df-ldil 40859 df-ltrn 40860 df-trl 40914 df-tendo 41510 df-dic 41928

This theorem is referenced by: cdlemn11b 41963

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