cdlemn11b - Mathbox for Norm Megill

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Theorem cdlemn11b 41197

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
cdlemn11b	⊢ (((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‘𝑄) ⊕ (𝐼‘𝑋)))

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 cdlemn11b**
Step	Hyp	Ref	Expression
1		simp3 1138	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋)))
2		cdlemn11a.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
3		cdlemn11a.l	. . 3 ⊢ ≤ = (le‘𝐾)
4		cdlemn11a.j	. . 3 ⊢ ∨ = (join‘𝐾)
5		cdlemn11a.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemn11a.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemn11a.p	. . 3 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
8		cdlemn11a.o	. . 3 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
9		cdlemn11a.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
10		cdlemn11a.r	. . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
11		cdlemn11a.e	. . 3 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
12		cdlemn11a.i	. . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
13		cdlemn11a.J	. . 3 ⊢ 𝐽 = ((DIsoC‘𝐾)‘𝑊)
14		cdlemn11a.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
15		cdlemn11a.d	. . 3 ⊢ + = (+_g‘𝑈)
16		cdlemn11a.s	. . 3 ⊢ ⊕ = (LSSum‘𝑈)
17		cdlemn11a.f	. . 3 ⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄)
18		cdlemn11a.g	. . 3 ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑁)
19	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	cdlemn11a 41196	. 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))
20	1, 19	sseldd 3936	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‘𝑄) ⊕ (𝐼‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ⊆ wss 3903 ⟨cop 4583 class class class wbr 5092 ↦ cmpt 5173 I cid 5513 ↾ cres 5621 ‘cfv 6482 ℩crio 7305 (class class class)co 7349 Basecbs 17120 +_gcplusg 17161 lecple 17168 occoc 17169 joincjn 18217 LSSumclsm 19513 Atomscatm 39252 HLchlt 39339 LHypclh 39973 LTrncltrn 40090 trLctrl 40147 TEndoctendo 40741 DVecHcdvh 41067 DIsoBcdib 41127 DIsoCcdic 41161

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-riotaBAD 38942

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-1st 7924 df-2nd 7925 df-undef 8206 df-map 8755 df-proset 18200 df-poset 18219 df-plt 18234 df-lub 18250 df-glb 18251 df-join 18252 df-meet 18253 df-p0 18329 df-p1 18330 df-lat 18338 df-clat 18405 df-oposet 39165 df-ol 39167 df-oml 39168 df-covers 39255 df-ats 39256 df-atl 39287 df-cvlat 39311 df-hlat 39340 df-llines 39487 df-lplanes 39488 df-lvols 39489 df-lines 39490 df-psubsp 39492 df-pmap 39493 df-padd 39785 df-lhyp 39977 df-laut 39978 df-ldil 40093 df-ltrn 40094 df-trl 40148 df-tendo 40744 df-dic 41162

This theorem is referenced by: cdlemn11c 41198

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