Theorem dihjatcclem3 35727

Description: Lemma for dihjatcc 35729. (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 34323	. . . . . 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 34885	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
12	1, 7, 8, 11	syl3anc 1318	. . . 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 34885	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐷 ∈ 𝑇)
16	1, 7, 13, 15	syl3anc 1318	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑇)
17	4, 9	ltrncnv 34450	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ◡𝐷 ∈ 𝑇)
18	1, 16, 17	syl2anc 691	. . . 4 ⊢ (𝜑 → ◡𝐷 ∈ 𝑇)
19	4, 9	ltrnco 35025	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) → (𝐺 ∘ ◡𝐷) ∈ 𝑇)
20	1, 12, 18, 19	syl3anc 1318	. . 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 34468	. . 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 1318	. 2 ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊))
26	13	simpld 474	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
27	2, 3, 4, 9	ltrncoval 34449	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) ∧ 𝑄 ∈ 𝐴) → ((𝐺 ∘ ◡𝐷)‘𝑄) = (𝐺‘(◡𝐷‘𝑄)))
28	1, 12, 18, 26, 27	syl121anc 1323	. . . . . . 7 ⊢ (𝜑 → ((𝐺 ∘ ◡𝐷)‘𝑄) = (𝐺‘(◡𝐷‘𝑄)))
29	2, 3, 4, 9, 14	ltrniotacnvval 34888	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (◡𝐷‘𝑄) = 𝐶)
30	1, 7, 13, 29	syl3anc 1318	. . . . . . . . 9 ⊢ (𝜑 → (◡𝐷‘𝑄) = 𝐶)
31	30	fveq2d 6107	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘(◡𝐷‘𝑄)) = (𝐺‘𝐶))
32	2, 3, 4, 9, 10	ltrniotaval 34887	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺‘𝐶) = 𝑃)
33	1, 7, 8, 32	syl3anc 1318	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐶) = 𝑃)
34	31, 33	eqtrd 2644	. . . . . . 7 ⊢ (𝜑 → (𝐺‘(◡𝐷‘𝑄)) = 𝑃)
35	28, 34	eqtrd 2644	. . . . . 6 ⊢ (𝜑 → ((𝐺 ∘ ◡𝐷)‘𝑄) = 𝑃)
36	35	oveq2d 6565	. . . . 5 ⊢ (𝜑 → (𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) = (𝑄 ∨ 𝑃))
37	1	simpld 474	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL)
38	8	simpld 474	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
39	21, 3	hlatjcom 33672	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
40	37, 38, 26, 39	syl3anc 1318	. . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
41	36, 40	eqtr4d 2647	. . . 4 ⊢ (𝜑 → (𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) = (𝑃 ∨ 𝑄))
42	41	oveq1d 6564	. . 3 ⊢ (𝜑 → ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊) = ((𝑃 ∨ 𝑄) ∧ 𝑊))
43		dihjatcclem.v	. . 3 ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
44	42, 43	syl6eqr 2662	. 2 ⊢ (𝜑 → ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊) = 𝑉)
45	25, 44	eqtrd 2644	1 ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ^◡ccnv 5037   ∘ ccom 5042  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Basecbs 15695  lecple 15775  occoc 15776  joincjn 16767  meetcmee 16768  LSSumclsm 17872  Atomscatm 33568  HLchlt 33655  LHypclh 34288  LTrncltrn 34405  trLctrl 34463  TEndoctendo 35058  DVecHcdvh 35385  DIsoHcdih 35535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-riotaBAD 33257

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-undef 7286  df-map 7746  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804  df-lines 33805  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292  df-laut 34293  df-ldil 34408  df-ltrn 34409  df-trl 34464

This theorem is referenced by:  dihjatcclem4  35728