Theorem dihjatcclem1 39359

Description: Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-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))

Assertion

Ref	Expression
dihjatcclem1	⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)))

Proof of Theorem dihjatcclem1

Step	Hyp	Ref	Expression
1		dihjatcclem.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		dihjatcclem.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		dihjatcclem.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	dvhlmod 39051	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		lmodabl 20085	. . . 4 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Abel)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝑈 ∈ Abel)
7		eqid 2738	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8	7	lsssssubg 20135	. . . . 5 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
9	4, 8	syl 17	. . . 4 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
10		dihjatcclem.p	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
11	10	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
12		dihjatcclem.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
13		dihjatcclem.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
14	12, 13	atbase 37230	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
15	11, 14	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
16		dihjatcclem.i	. . . . . 6 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
17	12, 1, 16, 2, 7	dihlss 39191	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐵) → (𝐼‘𝑃) ∈ (LSubSp‘𝑈))
18	3, 15, 17	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (LSubSp‘𝑈))
19	9, 18	sseldd 3918	. . 3 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (SubGrp‘𝑈))
20		dihjatcclem.v	. . . . . 6 ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
21	3	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ HL)
22	21	hllatd 37305	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Lat)
23		dihjatcclem.q	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
24	23	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
25		dihjatcclem.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
26	12, 25, 13	hlatjcl 37308	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵)
27	21, 11, 24, 26	syl3anc 1369	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ 𝐵)
28	3	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
29	12, 1	lhpbase 37939	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
30	28, 29	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
31		dihjatcclem.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
32	12, 31	latmcl 18073	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
33	22, 27, 30, 32	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
34	20, 33	eqeltrid 2843	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
35	12, 1, 16, 2, 7	dihlss 39191	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ 𝐵) → (𝐼‘𝑉) ∈ (LSubSp‘𝑈))
36	3, 34, 35	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝐼‘𝑉) ∈ (LSubSp‘𝑈))
37	9, 36	sseldd 3918	. . 3 ⊢ (𝜑 → (𝐼‘𝑉) ∈ (SubGrp‘𝑈))
38	12, 13	atbase 37230	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
39	24, 38	syl 17	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
40	12, 1, 16, 2, 7	dihlss 39191	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐵) → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
41	3, 39, 40	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
42	9, 41	sseldd 3918	. . 3 ⊢ (𝜑 → (𝐼‘𝑄) ∈ (SubGrp‘𝑈))
43		dihjatcclem.s	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
44	43	lsm4 19376	. . 3 ⊢ ((𝑈 ∈ Abel ∧ ((𝐼‘𝑃) ∈ (SubGrp‘𝑈) ∧ (𝐼‘𝑉) ∈ (SubGrp‘𝑈)) ∧ ((𝐼‘𝑄) ∈ (SubGrp‘𝑈) ∧ (𝐼‘𝑉) ∈ (SubGrp‘𝑈))) → (((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) ⊕ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉))) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ ((𝐼‘𝑉) ⊕ (𝐼‘𝑉))))
45	6, 19, 37, 42, 37, 44	syl122anc 1377	. 2 ⊢ (𝜑 → (((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) ⊕ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉))) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ ((𝐼‘𝑉) ⊕ (𝐼‘𝑉))))
46	23	simprd 495	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑄 ≤ 𝑊)
47	46	intnand 488	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊))
48		dihjatcclem.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
49	12, 48, 25	latjle12 18083	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊) ↔ (𝑃 ∨ 𝑄) ≤ 𝑊))
50	22, 15, 39, 30, 49	syl13anc 1370	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊) ↔ (𝑃 ∨ 𝑄) ≤ 𝑊))
51	47, 50	mtbid 323	. . . . . 6 ⊢ (𝜑 → ¬ (𝑃 ∨ 𝑄) ≤ 𝑊)
52	48, 25, 13	hlatlej1 37316	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑄))
53	21, 11, 24, 52	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → 𝑃 ≤ (𝑃 ∨ 𝑄))
54	12, 48, 25, 31, 13, 1, 16, 2, 43	dihvalcq2 39188	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∨ 𝑄) ∈ 𝐵 ∧ ¬ (𝑃 ∨ 𝑄) ≤ 𝑊) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑃 ≤ (𝑃 ∨ 𝑄))) → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))))
55	3, 27, 51, 10, 53, 54	syl122anc 1377	. . . . 5 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))))
56	20	fveq2i 6759	. . . . . 6 ⊢ (𝐼‘𝑉) = (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))
57	56	oveq2i 7266	. . . . 5 ⊢ ((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) = ((𝐼‘𝑃) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊)))
58	55, 57	eqtr4di 2797	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑉)))
59	48, 25, 13	hlatlej2 37317	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑃 ∨ 𝑄))
60	21, 11, 24, 59	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → 𝑄 ≤ (𝑃 ∨ 𝑄))
61	12, 48, 25, 31, 13, 1, 16, 2, 43	dihvalcq2 39188	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∨ 𝑄) ∈ 𝐵 ∧ ¬ (𝑃 ∨ 𝑄) ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ 𝑄))) → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑄) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))))
62	3, 27, 51, 23, 60, 61	syl122anc 1377	. . . . 5 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑄) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))))
63	56	oveq2i 7266	. . . . 5 ⊢ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉)) = ((𝐼‘𝑄) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊)))
64	62, 63	eqtr4di 2797	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑄) ⊕ (𝐼‘𝑉)))
65	58, 64	oveq12d 7273	. . 3 ⊢ (𝜑 → ((𝐼‘(𝑃 ∨ 𝑄)) ⊕ (𝐼‘(𝑃 ∨ 𝑄))) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) ⊕ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉))))
66	12, 1, 16, 2, 7	dihlss 39191	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝐼‘(𝑃 ∨ 𝑄)) ∈ (LSubSp‘𝑈))
67	3, 27, 66	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) ∈ (LSubSp‘𝑈))
68	9, 67	sseldd 3918	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) ∈ (SubGrp‘𝑈))
69	43	lsmidm 19183	. . . 4 ⊢ ((𝐼‘(𝑃 ∨ 𝑄)) ∈ (SubGrp‘𝑈) → ((𝐼‘(𝑃 ∨ 𝑄)) ⊕ (𝐼‘(𝑃 ∨ 𝑄))) = (𝐼‘(𝑃 ∨ 𝑄)))
70	68, 69	syl 17	. . 3 ⊢ (𝜑 → ((𝐼‘(𝑃 ∨ 𝑄)) ⊕ (𝐼‘(𝑃 ∨ 𝑄))) = (𝐼‘(𝑃 ∨ 𝑄)))
71	65, 70	eqtr3d 2780	. 2 ⊢ (𝜑 → (((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) ⊕ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉))) = (𝐼‘(𝑃 ∨ 𝑄)))
72	43	lsmidm 19183	. . . 4 ⊢ ((𝐼‘𝑉) ∈ (SubGrp‘𝑈) → ((𝐼‘𝑉) ⊕ (𝐼‘𝑉)) = (𝐼‘𝑉))
73	37, 72	syl 17	. . 3 ⊢ (𝜑 → ((𝐼‘𝑉) ⊕ (𝐼‘𝑉)) = (𝐼‘𝑉))
74	73	oveq2d 7271	. 2 ⊢ (𝜑 → (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ ((𝐼‘𝑉) ⊕ (𝐼‘𝑉))) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)))
75	45, 71, 74	3eqtr3d 2786	1 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  lecple 16895  joincjn 17944  meetcmee 17945  Latclat 18064  SubGrpcsubg 18664  LSSumclsm 19154  Abelcabl 19302  LModclmod 20038  LSubSpclss 20108  Atomscatm 37204  HLchlt 37291  LHypclh 37925  DVecHcdvh 39019  DIsoHcdih 39169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-riotaBAD 36894

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-tpos 8013  df-undef 8060  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-0g 17069  df-proset 17928  df-poset 17946  df-plt 17963  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-p0 18058  df-p1 18059  df-lat 18065  df-clat 18132  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-subg 18667  df-cntz 18838  df-lsm 19156  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-oppr 19777  df-dvdsr 19798  df-unit 19799  df-invr 19829  df-dvr 19840  df-drng 19908  df-lmod 20040  df-lss 20109  df-lsp 20149  df-lvec 20280  df-oposet 37117  df-ol 37119  df-oml 37120  df-covers 37207  df-ats 37208  df-atl 37239  df-cvlat 37263  df-hlat 37292  df-llines 37439  df-lplanes 37440  df-lvols 37441  df-lines 37442  df-psubsp 37444  df-pmap 37445  df-padd 37737  df-lhyp 37929  df-laut 37930  df-ldil 38045  df-ltrn 38046  df-trl 38100  df-tendo 38696  df-edring 38698  df-disoa 38970  df-dvech 39020  df-dib 39080  df-dic 39114  df-dih 39170

This theorem is referenced by:  dihjatcclem2  39360