Theorem dia2dimlem4 38237

Description: Lemma for dia2dim 38247. Show that the composition (sum) of translations (vectors) 𝐺 and 𝐷 equals 𝐹. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)

Hypotheses

Ref	Expression
dia2dimlem4.l	⊢ ≤ = (le‘𝐾)
dia2dimlem4.a	⊢ 𝐴 = (Atoms‘𝐾)
dia2dimlem4.h	⊢ 𝐻 = (LHyp‘𝐾)
dia2dimlem4.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dia2dimlem4.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dia2dimlem4.p	⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
dia2dimlem4.f	⊢ (𝜑 → 𝐹 ∈ 𝑇)
dia2dimlem4.g	⊢ (𝜑 → 𝐺 ∈ 𝑇)
dia2dimlem4.gv	⊢ (𝜑 → (𝐺‘𝑃) = 𝑄)
dia2dimlem4.d	⊢ (𝜑 → 𝐷 ∈ 𝑇)
dia2dimlem4.dv	⊢ (𝜑 → (𝐷‘𝑄) = (𝐹‘𝑃))

Assertion

Ref	Expression
dia2dimlem4	⊢ (𝜑 → (𝐷 ∘ 𝐺) = 𝐹)

Proof of Theorem dia2dimlem4

Step	Hyp	Ref	Expression
1		dia2dimlem4.k	. 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		dia2dimlem4.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑇)
3		dia2dimlem4.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑇)
4		dia2dimlem4.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
5		dia2dimlem4.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
6	4, 5	ltrnco 37889	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐷 ∘ 𝐺) ∈ 𝑇)
7	1, 2, 3, 6	syl3anc 1366	. 2 ⊢ (𝜑 → (𝐷 ∘ 𝐺) ∈ 𝑇)
8		dia2dimlem4.f	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝑇)
9		dia2dimlem4.p	. 2 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
10	9	simpld 497	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
11		dia2dimlem4.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
12		dia2dimlem4.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
13	11, 12, 4, 5	ltrncoval 37315	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ∈ 𝐴) → ((𝐷 ∘ 𝐺)‘𝑃) = (𝐷‘(𝐺‘𝑃)))
14	1, 2, 3, 10, 13	syl121anc 1370	. . 3 ⊢ (𝜑 → ((𝐷 ∘ 𝐺)‘𝑃) = (𝐷‘(𝐺‘𝑃)))
15		dia2dimlem4.gv	. . . 4 ⊢ (𝜑 → (𝐺‘𝑃) = 𝑄)
16	15	fveq2d 6667	. . 3 ⊢ (𝜑 → (𝐷‘(𝐺‘𝑃)) = (𝐷‘𝑄))
17		dia2dimlem4.dv	. . 3 ⊢ (𝜑 → (𝐷‘𝑄) = (𝐹‘𝑃))
18	14, 16, 17	3eqtrd 2859	. 2 ⊢ (𝜑 → ((𝐷 ∘ 𝐺)‘𝑃) = (𝐹‘𝑃))
19	11, 12, 4, 5	cdlemd 37377	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐷 ∘ 𝐺) ∈ 𝑇 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ ((𝐷 ∘ 𝐺)‘𝑃) = (𝐹‘𝑃)) → (𝐷 ∘ 𝐺) = 𝐹)
20	1, 7, 8, 9, 18, 19	syl311anc 1379	1 ⊢ (𝜑 → (𝐷 ∘ 𝐺) = 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113   class class class wbr 5059   ∘ ccom 5552  ‘cfv 6348  lecple 16565  Atomscatm 36433  HLchlt 36520  LHypclh 37154  LTrncltrn 37271

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-riotaBAD 36123

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-reu 3144  df-rmo 3145  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4914  df-iin 4915  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7682  df-2nd 7683  df-undef 7932  df-map 8401  df-proset 17531  df-poset 17549  df-plt 17561  df-lub 17577  df-glb 17578  df-join 17579  df-meet 17580  df-p0 17642  df-p1 17643  df-lat 17649  df-clat 17711  df-oposet 36346  df-ol 36348  df-oml 36349  df-covers 36436  df-ats 36437  df-atl 36468  df-cvlat 36492  df-hlat 36521  df-llines 36668  df-lplanes 36669  df-lvols 36670  df-lines 36671  df-psubsp 36673  df-pmap 36674  df-padd 36966  df-lhyp 37158  df-laut 37159  df-ldil 37274  df-ltrn 37275  df-trl 37329

This theorem is referenced by:  dia2dimlem5  38238