cdleml9 - Mathbox for Norm Megill

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Theorem cdleml9 35093

Description: Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)

**Hypotheses**
Ref	Expression
cdleml6.b	⊢ 𝐵 = (Base‘𝐾)
cdleml6.j	⊢ ∨ = (join‘𝐾)
cdleml6.m	⊢ ∧ = (meet‘𝐾)
cdleml6.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleml6.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdleml6.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdleml6.p	⊢ 𝑄 = ((oc‘𝐾)‘𝑊)
cdleml6.z	⊢ 𝑍 = ((𝑄 ∨ (𝑅‘𝑏)) ∧ ((ℎ‘𝑄) ∨ (𝑅‘(𝑏 ∘ ^◡(𝑠‘ℎ)))))
cdleml6.y	⊢ 𝑌 = ((𝑄 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏))))
cdleml6.x	⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘(𝑠‘ℎ)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑄) = 𝑌))
cdleml6.u	⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if((𝑠‘ℎ) = ℎ, 𝑔, 𝑋))
cdleml6.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
cdleml6.o	⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))

**Assertion**
Ref	Expression
cdleml9	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → 𝑈 ≠ 0 )

Distinct variable groups: 𝑔,𝑏,𝑧, ∧ ∨ ,𝑏,𝑔,𝑧 𝐵,𝑏,𝑓,𝑔,𝑧 ℎ,𝑏,𝑔,𝑧 𝑠,𝑏,𝑔,𝑧 𝐻,𝑏,𝑔,𝑧 𝐾,𝑏,𝑔,𝑧 𝑄,𝑏,𝑔,𝑧 𝑅,𝑏,𝑔,𝑧 𝑇,𝑏,𝑓,𝑔,𝑧 𝑊,𝑏,𝑔,𝑧 𝑧,𝑌 𝑔,𝑍 Allowed substitution hints: 𝐵(ℎ,𝑠) 𝑄(𝑓,ℎ,𝑠) 𝑅(𝑓,ℎ,𝑠) 𝑇(ℎ,𝑠) 𝑈(𝑧,𝑓,𝑔,ℎ,𝑠,𝑏) 𝐸(𝑧,𝑓,𝑔,ℎ,𝑠,𝑏) 𝐻(𝑓,ℎ,𝑠) ∨ (𝑓,ℎ,𝑠) 𝐾(𝑓,ℎ,𝑠) ∧ (𝑓,ℎ,𝑠) 𝑊(𝑓,ℎ,𝑠) 𝑋(𝑧,𝑓,𝑔,ℎ,𝑠,𝑏) 𝑌(𝑓,𝑔,ℎ,𝑠,𝑏) 0 (𝑧,𝑓,𝑔,ℎ,𝑠,𝑏) 𝑍(𝑧,𝑓,ℎ,𝑠,𝑏)

**Proof of Theorem cdleml9**
Step	Hyp	Ref	Expression
1		cdleml6.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		cdleml6.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
3		cdleml6.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4		cdleml6.e	. . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
5		cdleml6.o	. . . 4 ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
6	1, 2, 3, 4, 5	tendo1ne0 34937	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ≠ 0 )
7	6	3ad2ant1 1074	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → ( I ↾ 𝑇) ≠ 0 )
8		cdleml6.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
9		cdleml6.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
10		cdleml6.r	. . . . . . 7 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
11		cdleml6.p	. . . . . . 7 ⊢ 𝑄 = ((oc‘𝐾)‘𝑊)
12		cdleml6.z	. . . . . . 7 ⊢ 𝑍 = ((𝑄 ∨ (𝑅‘𝑏)) ∧ ((ℎ‘𝑄) ∨ (𝑅‘(𝑏 ∘ ^◡(𝑠‘ℎ)))))
13		cdleml6.y	. . . . . . 7 ⊢ 𝑌 = ((𝑄 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏))))
14		cdleml6.x	. . . . . . 7 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘(𝑠‘ℎ)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑄) = 𝑌))
15		cdleml6.u	. . . . . . 7 ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if((𝑠‘ℎ) = ℎ, 𝑔, 𝑋))
16	1, 8, 9, 2, 3, 10, 11, 12, 13, 14, 15, 4, 5	cdleml8 35092	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (𝑈 ∘ 𝑠) = ( I ↾ 𝑇))
17	16	adantr 479	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) ∧ 𝑈 = 0 ) → (𝑈 ∘ 𝑠) = ( I ↾ 𝑇))
18		coeq1 5189	. . . . . 6 ⊢ (𝑈 = 0 → (𝑈 ∘ 𝑠) = ( 0 ∘ 𝑠))
19		simp1 1053	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
20		simp3l 1081	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → 𝑠 ∈ 𝐸)
21	1, 2, 3, 4, 5	tendo0mul 34935	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → ( 0 ∘ 𝑠) = 0 )
22	19, 20, 21	syl2anc 690	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → ( 0 ∘ 𝑠) = 0 )
23	18, 22	sylan9eqr 2665	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) ∧ 𝑈 = 0 ) → (𝑈 ∘ 𝑠) = 0 )
24	17, 23	eqtr3d 2645	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) ∧ 𝑈 = 0 ) → ( I ↾ 𝑇) = 0 )
25	24	ex 448	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (𝑈 = 0 → ( I ↾ 𝑇) = 0 ))
26	25	necon3d 2802	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (( I ↾ 𝑇) ≠ 0 → 𝑈 ≠ 0 ))
27	7, 26	mpd 15	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → 𝑈 ≠ 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 382 ∧ w3a 1030 = wceq 1474 ∈ wcel 1976 ≠ wne 2779 ∀wral 2895 ifcif 4035 ↦ cmpt 4637 I cid 4938 ^◡ccnv 5027 ↾ cres 5030 ∘ ccom 5032 ‘cfv 5790 ℩crio 6488 (class class class)co 6527 Basecbs 15641 occoc 15722 joincjn 16713 meetcmee 16714 HLchlt 33458 LHypclh 34091 LTrncltrn 34208 trLctrl 34266 TEndoctendo 34861

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1712 ax-4 1727 ax-5 1826 ax-6 1874 ax-7 1921 ax-8 1978 ax-9 1985 ax-10 2005 ax-11 2020 ax-12 2032 ax-13 2232 ax-ext 2589 ax-rep 4693 ax-sep 4703 ax-nul 4712 ax-pow 4764 ax-pr 4828 ax-un 6824 ax-riotaBAD 33060

This theorem depends on definitions: df-bi 195 df-or 383 df-an 384 df-3or 1031 df-3an 1032 df-tru 1477 df-fal 1480 df-ex 1695 df-nf 1700 df-sb 1867 df-eu 2461 df-mo 2462 df-clab 2596 df-cleq 2602 df-clel 2605 df-nfc 2739 df-ne 2781 df-nel 2782 df-ral 2900 df-rex 2901 df-reu 2902 df-rmo 2903 df-rab 2904 df-v 3174 df-sbc 3402 df-csb 3499 df-dif 3542 df-un 3544 df-in 3546 df-ss 3553 df-nul 3874 df-if 4036 df-pw 4109 df-sn 4125 df-pr 4127 df-op 4131 df-uni 4367 df-iun 4451 df-iin 4452 df-br 4578 df-opab 4638 df-mpt 4639 df-id 4943 df-xp 5034 df-rel 5035 df-cnv 5036 df-co 5037 df-dm 5038 df-rn 5039 df-res 5040 df-ima 5041 df-iota 5754 df-fun 5792 df-fn 5793 df-f 5794 df-f1 5795 df-fo 5796 df-f1o 5797 df-fv 5798 df-riota 6489 df-ov 6530 df-oprab 6531 df-mpt2 6532 df-1st 7036 df-2nd 7037 df-undef 7263 df-map 7723 df-preset 16697 df-poset 16715 df-plt 16727 df-lub 16743 df-glb 16744 df-join 16745 df-meet 16746 df-p0 16808 df-p1 16809 df-lat 16815 df-clat 16877 df-oposet 33284 df-ol 33286 df-oml 33287 df-covers 33374 df-ats 33375 df-atl 33406 df-cvlat 33430 df-hlat 33459 df-llines 33605 df-lplanes 33606 df-lvols 33607 df-lines 33608 df-psubsp 33610 df-pmap 33611 df-padd 33903 df-lhyp 34095 df-laut 34096 df-ldil 34211 df-ltrn 34212 df-trl 34267 df-tendo 34864

This theorem is referenced by: erngdvlem4 35100 erngdvlem4-rN 35108

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