Theorem dihjatcclem3 42668

Description: Lemma for dihjatcc 42670. (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 41267	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊))
7	1, 6	syl 18	. . . . 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 41827	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
12	1, 7, 8, 11	syl3anc 1399	. . . 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 41827	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐷 ∈ 𝑇)
16	1, 7, 13, 15	syl3anc 1399	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑇)
17	4, 9	ltrncnv 41394	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ◡𝐷 ∈ 𝑇)
18	1, 16, 17	syl2anc 596	. . . 4 ⊢ (𝜑 → ◡𝐷 ∈ 𝑇)
19	4, 9	ltrnco 41967	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) → (𝐺 ∘ ◡𝐷) ∈ 𝑇)
20	1, 12, 18, 19	syl3anc 1399	. . 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 41411	. . 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 1399	. 2 ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊))
26	13	simpld 500	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
27	2, 3, 4, 9	ltrncoval 41393	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) ∧ 𝑄 ∈ 𝐴) → ((𝐺 ∘ ◡𝐷)‘𝑄) = (𝐺‘(◡𝐷‘𝑄)))
28	1, 12, 18, 26, 27	syl121anc 1403	. . . . . . 7 ⊢ (𝜑 → ((𝐺 ∘ ◡𝐷)‘𝑄) = (𝐺‘(◡𝐷‘𝑄)))
29	2, 3, 4, 9, 14	ltrniotacnvval 41830	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (◡𝐷‘𝑄) = 𝐶)
30	1, 7, 13, 29	syl3anc 1399	. . . . . . . . 9 ⊢ (𝜑 → (◡𝐷‘𝑄) = 𝐶)
31	30	fveq2d 6909	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘(◡𝐷‘𝑄)) = (𝐺‘𝐶))
32	2, 3, 4, 9, 10	ltrniotaval 41829	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺‘𝐶) = 𝑃)
33	1, 7, 8, 32	syl3anc 1399	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐶) = 𝑃)
34	31, 33	eqtrd 2798	. . . . . . 7 ⊢ (𝜑 → (𝐺‘(◡𝐷‘𝑄)) = 𝑃)
35	28, 34	eqtrd 2798	. . . . . 6 ⊢ (𝜑 → ((𝐺 ∘ ◡𝐷)‘𝑄) = 𝑃)
36	35	oveq2d 7456	. . . . 5 ⊢ (𝜑 → (𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) = (𝑄 ∨ 𝑃))
37	1	simpld 500	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL)
38	8	simpld 500	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
39	21, 3	hlatjcom 40616	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
40	37, 38, 26, 39	syl3anc 1399	. . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
41	36, 40	eqtr4d 2801	. . . 4 ⊢ (𝜑 → (𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) = (𝑃 ∨ 𝑄))
42	41	oveq1d 7455	. . 3 ⊢ (𝜑 → ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊) = ((𝑃 ∨ 𝑄) ∧ 𝑊))
43		dihjatcclem.v	. . 3 ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
44	42, 43	eqtr4di 2816	. 2 ⊢ (𝜑 → ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊) = 𝑉)
45	25, 44	eqtrd 2798	1 ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = 𝑉)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   class class class wbr 5106  ^◡ccnv 5654   ∘ ccom 5659  ‘cfv 6555  ℩crio 7396  (class class class)co 7440  Basecbs 17413  lecple 17461  occoc 17462  joincjn 18511  meetcmee 18512  LSSumclsm 19875  Atomscatm 40511  HLchlt 40598  LHypclh 41232  LTrncltrn 41349  trLctrl 41406  TEndoctendo 42000  DVecHcdvh 42326  DIsoHcdih 42476

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-riotaBAD 40201

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  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-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-1st 8022  df-2nd 8023  df-undef 8315  df-map 8874  df-proset 18494  df-poset 18513  df-plt 18528  df-lub 18544  df-glb 18545  df-join 18546  df-meet 18547  df-p0 18623  df-p1 18624  df-lat 18632  df-clat 18699  df-oposet 40424  df-ol 40426  df-oml 40427  df-covers 40514  df-ats 40515  df-atl 40546  df-cvlat 40570  df-hlat 40599  df-llines 40746  df-lplanes 40747  df-lvols 40748  df-lines 40749  df-psubsp 40751  df-pmap 40752  df-padd 41044  df-lhyp 41236  df-laut 41237  df-ldil 41352  df-ltrn 41353  df-trl 41407

This theorem is used by:  dihjatcclem4  42669