cdlemn6 - Mathbox for Norm Megill

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Theorem cdlemn6 35509

Description: Part of proof of Lemma N of [Crawley] p. 121 line 35. (Contributed by NM, 26-Feb-2014.)

**Hypotheses**
Ref	Expression
cdlemn8.b	⊢ 𝐵 = (Base‘𝐾)
cdlemn8.l	⊢ ≤ = (le‘𝐾)
cdlemn8.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemn8.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemn8.p	⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
cdlemn8.o	⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
cdlemn8.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemn8.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
cdlemn8.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
cdlemn8.s	⊢ + = (+_g‘𝑈)
cdlemn8.f	⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄)

**Assertion**
Ref	Expression
cdlemn6	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩) = ⟨((𝑠‘𝐹) ∘ 𝑔), 𝑠⟩)

Distinct variable groups: ≤ ,ℎ 𝐴,ℎ 𝐵,ℎ ℎ,𝐻 ℎ,𝐾 𝑇,ℎ 𝑃,ℎ 𝑄,ℎ ℎ,𝑊 Allowed substitution hints: 𝐴(𝑔,𝑠) 𝐵(𝑔,𝑠) 𝑃(𝑔,𝑠) + (𝑔,ℎ,𝑠) 𝑄(𝑔,𝑠) 𝑅(𝑔,ℎ,𝑠) 𝑇(𝑔,𝑠) 𝑈(𝑔,ℎ,𝑠) 𝐸(𝑔,ℎ,𝑠) 𝐹(𝑔,ℎ,𝑠) 𝐻(𝑔,𝑠) 𝐾(𝑔,𝑠) ≤ (𝑔,𝑠) 𝑂(𝑔,ℎ,𝑠) 𝑊(𝑔,𝑠)

Proof of Theorem cdlemn6 Dummy variables 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1054	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp3l 1082	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → 𝑠 ∈ 𝐸)
3		cdlemn8.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
4		cdlemn8.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5		cdlemn8.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
6		cdlemn8.p	. . . . . . 7 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
7	3, 4, 5, 6	lhpocnel2 34323	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
8	1, 7	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
9		simp2l 1080	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
10		cdlemn8.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11		cdlemn8.f	. . . . . 6 ⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄)
12	3, 4, 5, 10, 11	ltrniotacl 34885	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
13	1, 8, 9, 12	syl3anc 1318	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → 𝐹 ∈ 𝑇)
14		cdlemn8.e	. . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
15	5, 10, 14	tendocl 35073	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑠‘𝐹) ∈ 𝑇)
16	1, 2, 13, 15	syl3anc 1318	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑠‘𝐹) ∈ 𝑇)
17		simp3r 1083	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → 𝑔 ∈ 𝑇)
18		cdlemn8.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
19		cdlemn8.o	. . . . 5 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
20	18, 5, 10, 14, 19	tendo0cl 35096	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸)
21	1, 20	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → 𝑂 ∈ 𝐸)
22		cdlemn8.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
23		eqid 2610	. . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
24		cdlemn8.s	. . . 4 ⊢ + = (+_g‘𝑈)
25		eqid 2610	. . . 4 ⊢ (+_g‘(Scalar‘𝑈)) = (+_g‘(Scalar‘𝑈))
26	5, 10, 14, 22, 23, 24, 25	dvhopvadd 35400	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑠‘𝐹) ∈ 𝑇 ∧ 𝑠 ∈ 𝐸) ∧ (𝑔 ∈ 𝑇 ∧ 𝑂 ∈ 𝐸)) → (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩) = ⟨((𝑠‘𝐹) ∘ 𝑔), (𝑠(+_g‘(Scalar‘𝑈))𝑂)⟩)
27	1, 16, 2, 17, 21, 26	syl122anc 1327	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩) = ⟨((𝑠‘𝐹) ∘ 𝑔), (𝑠(+_g‘(Scalar‘𝑈))𝑂)⟩)
28		eqid 2610	. . . . . . 7 ⊢ (𝑡 ∈ 𝐸, 𝑢 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑡‘ℎ) ∘ (𝑢‘ℎ)))) = (𝑡 ∈ 𝐸, 𝑢 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑡‘ℎ) ∘ (𝑢‘ℎ))))
29	5, 10, 14, 22, 23, 28, 25	dvhfplusr 35391	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+_g‘(Scalar‘𝑈)) = (𝑡 ∈ 𝐸, 𝑢 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑡‘ℎ) ∘ (𝑢‘ℎ)))))
30	1, 29	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (+_g‘(Scalar‘𝑈)) = (𝑡 ∈ 𝐸, 𝑢 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑡‘ℎ) ∘ (𝑢‘ℎ)))))
31	30	oveqd 6566	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑠(+_g‘(Scalar‘𝑈))𝑂) = (𝑠(𝑡 ∈ 𝐸, 𝑢 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑡‘ℎ) ∘ (𝑢‘ℎ))))𝑂))
32	18, 5, 10, 14, 19, 28	tendo0plr 35098	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝑠(𝑡 ∈ 𝐸, 𝑢 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑡‘ℎ) ∘ (𝑢‘ℎ))))𝑂) = 𝑠)
33	1, 2, 32	syl2anc 691	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑠(𝑡 ∈ 𝐸, 𝑢 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑡‘ℎ) ∘ (𝑢‘ℎ))))𝑂) = 𝑠)
34	31, 33	eqtrd 2644	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑠(+_g‘(Scalar‘𝑈))𝑂) = 𝑠)
35	34	opeq2d 4347	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → ⟨((𝑠‘𝐹) ∘ 𝑔), (𝑠(+_g‘(Scalar‘𝑈))𝑂)⟩ = ⟨((𝑠‘𝐹) ∘ 𝑔), 𝑠⟩)
36	27, 35	eqtrd 2644	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩) = ⟨((𝑠‘𝐹) ∘ 𝑔), 𝑠⟩)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⟨cop 4131 class class class wbr 4583 ↦ cmpt 4643 I cid 4948 ↾ cres 5040 ∘ ccom 5042 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 ↦ cmpt2 6551 Basecbs 15695 +_gcplusg 15768 Scalarcsca 15771 lecple 15775 occoc 15776 Atomscatm 33568 HLchlt 33655 LHypclh 34288 LTrncltrn 34405 TEndoctendo 35058 DVecHcdvh 35385

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-riotaBAD 33257

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-undef 7286 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-tendo 35061 df-edring 35063 df-dvech 35386

This theorem is referenced by: cdlemn7 35510 dihordlem6 35520

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