cdlemn6 - Mathbox for Norm Megill

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Theorem cdlemn6 42075

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 1154	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp3l 1220	. . . 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 40892	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
8	1, 7	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
9		simp2l 1218	. . . . 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 41452	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
13	1, 8, 9, 12	syl3anc 1398	. . . 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 41640	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑠‘𝐹) ∈ 𝑇)
16	1, 2, 13, 15	syl3anc 1398	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑠‘𝐹) ∈ 𝑇)
17		simp3r 1221	. . 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 41663	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸)
21	1, 20	syl 18	. . 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 2760	. . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
24		cdlemn8.s	. . . 4 ⊢ + = (+_g‘𝑈)
25		eqid 2760	. . . 4 ⊢ (+_g‘(Scalar‘𝑈)) = (+_g‘(Scalar‘𝑈))
26	5, 10, 14, 22, 23, 24, 25	dvhopvadd 41966	. . 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 1406	. 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 2760	. . . . . . 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 41957	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+_g‘(Scalar‘𝑈)) = (𝑡 ∈ 𝐸, 𝑢 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑡‘ℎ) ∘ (𝑢‘ℎ)))))
30	1, 29	syl 18	. . . . 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 7430	. . . 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 41665	. . . . 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 596	. . . 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 2795	. . 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 4840	. 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 2795	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ⟨cop 4590 class class class wbr 5103 ↦ cmpt 5186 I cid 5549 ↾ cres 5657 ∘ ccom 5659 ‘cfv 6533 ℩crio 7369 (class class class)co 7413 ∈ cmpo 7415 Basecbs 17301 +_gcplusg 17342 Scalarcsca 17345 lecple 17349 occoc 17350 Atomscatm 40136 HLchlt 40223 LHypclh 40857 LTrncltrn 40974 TEndoctendo 41625 DVecHcdvh 41951

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-riotaBAD 39826

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-undef 8271 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-map 8828 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12529 df-z 12616 df-uz 12888 df-fz 13562 df-struct 17239 df-slot 17274 df-ndx 17286 df-base 17302 df-plusg 17355 df-mulr 17356 df-sca 17358 df-vsca 17359 df-proset 18382 df-poset 18401 df-plt 18416 df-lub 18432 df-glb 18433 df-join 18434 df-meet 18435 df-p0 18511 df-p1 18512 df-lat 18520 df-clat 18587 df-oposet 40049 df-ol 40051 df-oml 40052 df-covers 40139 df-ats 40140 df-atl 40171 df-cvlat 40195 df-hlat 40224 df-llines 40371 df-lplanes 40372 df-lvols 40373 df-lines 40374 df-psubsp 40376 df-pmap 40377 df-padd 40669 df-lhyp 40861 df-laut 40862 df-ldil 40977 df-ltrn 40978 df-trl 41032 df-tendo 41628 df-edring 41630 df-dvech 41952

This theorem is used by: cdlemn7 42076 dihordlem6 42086

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