cdlemn11a - Mathbox for Norm Megill

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Theorem cdlemn11a 38817

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 1133	. . . . 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 37629	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
7	6	3ad2ant1 1130	. . . . 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 1204	. . . . 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 38189	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
12	1, 7, 8, 11	syl3anc 1368	. . . 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 6932	. . . 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 2764	. 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 38379	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸)
18	17	3ad2ant1 1130	. 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 7118	. . . . 5 ⊢ (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑁) ∈ V
21	10, 20	eqeltri 2848	. . . 4 ⊢ 𝐺 ∈ V
22	9	fvexi 6677	. . . . 5 ⊢ 𝑇 ∈ V
23		resiexg 7630	. . . . 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 38791	. . 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 587	. 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 712	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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3409 ⊆ wss 3860 ⟨cop 4531 class class class wbr 5036 ↦ cmpt 5116 I cid 5433 ↾ cres 5530 ‘cfv 6340 ℩crio 7113 (class class class)co 7156 Basecbs 16554 +_gcplusg 16636 lecple 16643 occoc 16644 joincjn 17633 LSSumclsm 18839 Atomscatm 36873 HLchlt 36960 LHypclh 37594 LTrncltrn 37711 trLctrl 37768 TEndoctendo 38362 DVecHcdvh 38688 DIsoBcdib 38748 DIsoCcdic 38782

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-riotaBAD 36563

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-iun 4888 df-iin 4889 df-br 5037 df-opab 5099 df-mpt 5117 df-id 5434 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7699 df-2nd 7700 df-undef 7955 df-map 8424 df-proset 17617 df-poset 17635 df-plt 17647 df-lub 17663 df-glb 17664 df-join 17665 df-meet 17666 df-p0 17728 df-p1 17729 df-lat 17735 df-clat 17797 df-oposet 36786 df-ol 36788 df-oml 36789 df-covers 36876 df-ats 36877 df-atl 36908 df-cvlat 36932 df-hlat 36961 df-llines 37108 df-lplanes 37109 df-lvols 37110 df-lines 37111 df-psubsp 37113 df-pmap 37114 df-padd 37406 df-lhyp 37598 df-laut 37599 df-ldil 37714 df-ltrn 37715 df-trl 37769 df-tendo 38365 df-dic 38783

This theorem is referenced by: cdlemn11b 38818

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