Theorem dihjatcclem3 41403

Description: Lemma for dihjatcc 41405. (Contributed by NM, 28-Sep-2014.)

Hypotheses

Ref	Expression
dihjatcclem.b	⊢ 𝐵 = (Base‘𝐾)
dihjatcclem.l	⊢ ≤ = (le‘𝐾)
dihjatcclem.h	⊢ 𝐻 = (LHyp‘𝐾)
dihjatcclem.j	⊢ ∨ = (join‘𝐾)
dihjatcclem.m	⊢ ∧ = (meet‘𝐾)
dihjatcclem.a	⊢ 𝐴 = (Atoms‘𝐾)
dihjatcclem.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihjatcclem.s	⊢ ⊕ = (LSSum‘𝑈)
dihjatcclem.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihjatcclem.v	⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
dihjatcclem.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dihjatcclem.p	⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
dihjatcclem.q	⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
dihjatcc.w	⊢ 𝐶 = ((oc‘𝐾)‘𝑊)
dihjatcc.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dihjatcc.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dihjatcc.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dihjatcc.g	⊢ 𝐺 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑃)
dihjatcc.dd	⊢ 𝐷 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑄)

Assertion

Ref	Expression
dihjatcclem3	⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = 𝑉)

Distinct variable groups:   ≤ ,𝑑   𝐴,𝑑   𝐵,𝑑   𝐶,𝑑   𝐻,𝑑   𝑃,𝑑   𝐾,𝑑   𝑄,𝑑   𝑇,𝑑   𝑊,𝑑

Allowed substitution hints:   𝜑(𝑑)   𝐷(𝑑)   ⊕ (𝑑)   𝑅(𝑑)   𝑈(𝑑)   𝐸(𝑑)   𝐺(𝑑)   𝐼(𝑑)   ∨ (𝑑)   ∧ (𝑑)   𝑉(𝑑)

Proof of Theorem dihjatcclem3

Step	Hyp	Ref	Expression
1		dihjatcclem.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		dihjatcclem.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
3		dihjatcclem.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
4		dihjatcclem.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
5		dihjatcc.w	. . . . . . 7 ⊢ 𝐶 = ((oc‘𝐾)‘𝑊)
6	2, 3, 4, 5	lhpocnel2 40002	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊))
7	1, 6	syl 17	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊))
8		dihjatcclem.p	. . . . 5 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
9		dihjatcc.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
10		dihjatcc.g	. . . . . 6 ⊢ 𝐺 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑃)
11	2, 3, 4, 9, 10	ltrniotacl 40562	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
12	1, 7, 8, 11	syl3anc 1370	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑇)
13		dihjatcclem.q	. . . . . 6 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
14		dihjatcc.dd	. . . . . . 7 ⊢ 𝐷 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑄)
15	2, 3, 4, 9, 14	ltrniotacl 40562	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐷 ∈ 𝑇)
16	1, 7, 13, 15	syl3anc 1370	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑇)
17	4, 9	ltrncnv 40129	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ◡𝐷 ∈ 𝑇)
18	1, 16, 17	syl2anc 584	. . . 4 ⊢ (𝜑 → ◡𝐷 ∈ 𝑇)
19	4, 9	ltrnco 40702	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) → (𝐺 ∘ ◡𝐷) ∈ 𝑇)
20	1, 12, 18, 19	syl3anc 1370	. . 3 ⊢ (𝜑 → (𝐺 ∘ ◡𝐷) ∈ 𝑇)
21		dihjatcclem.j	. . . 4 ⊢ ∨ = (join‘𝐾)
22		dihjatcclem.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
23		dihjatcc.r	. . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
24	2, 21, 22, 3, 4, 9, 23	trlval2 40146	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∘ ◡𝐷) ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑅‘(𝐺 ∘ ◡𝐷)) = ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊))
25	1, 20, 13, 24	syl3anc 1370	. 2 ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊))
26	13	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
27	2, 3, 4, 9	ltrncoval 40128	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) ∧ 𝑄 ∈ 𝐴) → ((𝐺 ∘ ◡𝐷)‘𝑄) = (𝐺‘(◡𝐷‘𝑄)))
28	1, 12, 18, 26, 27	syl121anc 1374	. . . . . . 7 ⊢ (𝜑 → ((𝐺 ∘ ◡𝐷)‘𝑄) = (𝐺‘(◡𝐷‘𝑄)))
29	2, 3, 4, 9, 14	ltrniotacnvval 40565	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (◡𝐷‘𝑄) = 𝐶)
30	1, 7, 13, 29	syl3anc 1370	. . . . . . . . 9 ⊢ (𝜑 → (◡𝐷‘𝑄) = 𝐶)
31	30	fveq2d 6911	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘(◡𝐷‘𝑄)) = (𝐺‘𝐶))
32	2, 3, 4, 9, 10	ltrniotaval 40564	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺‘𝐶) = 𝑃)
33	1, 7, 8, 32	syl3anc 1370	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐶) = 𝑃)
34	31, 33	eqtrd 2775	. . . . . . 7 ⊢ (𝜑 → (𝐺‘(◡𝐷‘𝑄)) = 𝑃)
35	28, 34	eqtrd 2775	. . . . . 6 ⊢ (𝜑 → ((𝐺 ∘ ◡𝐷)‘𝑄) = 𝑃)
36	35	oveq2d 7447	. . . . 5 ⊢ (𝜑 → (𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) = (𝑄 ∨ 𝑃))
37	1	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL)
38	8	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
39	21, 3	hlatjcom 39350	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
40	37, 38, 26, 39	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
41	36, 40	eqtr4d 2778	. . . 4 ⊢ (𝜑 → (𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) = (𝑃 ∨ 𝑄))
42	41	oveq1d 7446	. . 3 ⊢ (𝜑 → ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊) = ((𝑃 ∨ 𝑄) ∧ 𝑊))
43		dihjatcclem.v	. . 3 ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
44	42, 43	eqtr4di 2793	. 2 ⊢ (𝜑 → ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊) = 𝑉)
45	25, 44	eqtrd 2775	1 ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   class class class wbr 5148  ^◡ccnv 5688   ∘ ccom 5693  ‘cfv 6563  ℩crio 7387  (class class class)co 7431  Basecbs 17245  lecple 17305  occoc 17306  joincjn 18369  meetcmee 18370  LSSumclsm 19667  Atomscatm 39245  HLchlt 39332  LHypclh 39967  LTrncltrn 40084  trLctrl 40141  TEndoctendo 40735  DVecHcdvh 41061  DIsoHcdih 41211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-riotaBAD 38935

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-undef 8297  df-map 8867  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lplanes 39482  df-lvols 39483  df-lines 39484  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088  df-trl 40142

This theorem is referenced by:  dihjatcclem4  41404