cdlemn11a - Mathbox for Norm Megill

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Theorem cdlemn11a 41580

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 1137	. . . . 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 40392	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
7	6	3ad2ant1 1134	. . . . 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 1209	. . . . 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 40952	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
12	1, 7, 8, 11	syl3anc 1374	. . . 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 7129	. . . 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 2743	. 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 41142	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸)
18	17	3ad2ant1 1134	. 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 7329	. . . . 5 ⊢ (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑁) ∈ V
21	10, 20	eqeltri 2833	. . . 4 ⊢ 𝐺 ∈ V
22	9	fvexi 6856	. . . . 5 ⊢ 𝑇 ∈ V
23		resiexg 7864	. . . . 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 41554	. . 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 585	. 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 714	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 Vcvv 3442 ⊆ wss 3903 ⟨cop 4588 class class class wbr 5100 ↦ cmpt 5181 I cid 5526 ↾ cres 5634 ‘cfv 6500 ℩crio 7324 (class class class)co 7368 Basecbs 17148 +_gcplusg 17189 lecple 17196 occoc 17197 joincjn 18246 LSSumclsm 19575 Atomscatm 39636 HLchlt 39723 LHypclh 40357 LTrncltrn 40474 trLctrl 40531 TEndoctendo 41125 DVecHcdvh 41451 DIsoBcdib 41511 DIsoCcdic 41545

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-riotaBAD 39326

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-1st 7943 df-2nd 7944 df-undef 8225 df-map 8777 df-proset 18229 df-poset 18248 df-plt 18263 df-lub 18279 df-glb 18280 df-join 18281 df-meet 18282 df-p0 18358 df-p1 18359 df-lat 18367 df-clat 18434 df-oposet 39549 df-ol 39551 df-oml 39552 df-covers 39639 df-ats 39640 df-atl 39671 df-cvlat 39695 df-hlat 39724 df-llines 39871 df-lplanes 39872 df-lvols 39873 df-lines 39874 df-psubsp 39876 df-pmap 39877 df-padd 40169 df-lhyp 40361 df-laut 40362 df-ldil 40477 df-ltrn 40478 df-trl 40532 df-tendo 41128 df-dic 41546

This theorem is referenced by: cdlemn11b 41581

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