Theorem dihjatcclem3 41422

Description: Lemma for dihjatcc 41424. (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 40021	. . . . . 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 40581	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
12	1, 7, 8, 11	syl3anc 1373	. . . 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 40581	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐷 ∈ 𝑇)
16	1, 7, 13, 15	syl3anc 1373	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑇)
17	4, 9	ltrncnv 40148	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ◡𝐷 ∈ 𝑇)
18	1, 16, 17	syl2anc 584	. . . 4 ⊢ (𝜑 → ◡𝐷 ∈ 𝑇)
19	4, 9	ltrnco 40721	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) → (𝐺 ∘ ◡𝐷) ∈ 𝑇)
20	1, 12, 18, 19	syl3anc 1373	. . 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 40165	. . 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 1373	. 2 ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊))
26	13	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
27	2, 3, 4, 9	ltrncoval 40147	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) ∧ 𝑄 ∈ 𝐴) → ((𝐺 ∘ ◡𝐷)‘𝑄) = (𝐺‘(◡𝐷‘𝑄)))
28	1, 12, 18, 26, 27	syl121anc 1377	. . . . . . 7 ⊢ (𝜑 → ((𝐺 ∘ ◡𝐷)‘𝑄) = (𝐺‘(◡𝐷‘𝑄)))
29	2, 3, 4, 9, 14	ltrniotacnvval 40584	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (◡𝐷‘𝑄) = 𝐶)
30	1, 7, 13, 29	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (◡𝐷‘𝑄) = 𝐶)
31	30	fveq2d 6910	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘(◡𝐷‘𝑄)) = (𝐺‘𝐶))
32	2, 3, 4, 9, 10	ltrniotaval 40583	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺‘𝐶) = 𝑃)
33	1, 7, 8, 32	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐶) = 𝑃)
34	31, 33	eqtrd 2777	. . . . . . 7 ⊢ (𝜑 → (𝐺‘(◡𝐷‘𝑄)) = 𝑃)
35	28, 34	eqtrd 2777	. . . . . 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 39369	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
40	37, 38, 26, 39	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
41	36, 40	eqtr4d 2780	. . . 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 2795	. 2 ⊢ (𝜑 → ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊) = 𝑉)
45	25, 44	eqtrd 2777	1 ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   class class class wbr 5143  ^◡ccnv 5684   ∘ ccom 5689  ‘cfv 6561  ℩crio 7387  (class class class)co 7431  Basecbs 17247  lecple 17304  occoc 17305  joincjn 18357  meetcmee 18358  LSSumclsm 19652  Atomscatm 39264  HLchlt 39351  LHypclh 39986  LTrncltrn 40103  trLctrl 40160  TEndoctendo 40754  DVecHcdvh 41080  DIsoHcdih 41230

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-riotaBAD 38954

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-undef 8298  df-map 8868  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-p1 18471  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501  df-lvols 39502  df-lines 39503  df-psubsp 39505  df-pmap 39506  df-padd 39798  df-lhyp 39990  df-laut 39991  df-ldil 40106  df-ltrn 40107  df-trl 40161

This theorem is referenced by:  dihjatcclem4  41423