Theorem dihjatcclem1 39044

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 38736	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		lmodabl 19793	. . . 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 19842	. . . . 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 498	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
12		dihjatcclem.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
13		dihjatcclem.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
14	12, 13	atbase 36915	. . . . . 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 38876	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐵) → (𝐼‘𝑃) ∈ (LSubSp‘𝑈))
18	3, 15, 17	syl2anc 587	. . . 4 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (LSubSp‘𝑈))
19	9, 18	sseldd 3876	. . 3 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (SubGrp‘𝑈))
20		dihjatcclem.v	. . . . . 6 ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
21	3	simpld 498	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ HL)
22	21	hllatd 36990	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Lat)
23		dihjatcclem.q	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
24	23	simpld 498	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
25		dihjatcclem.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
26	12, 25, 13	hlatjcl 36993	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵)
27	21, 11, 24, 26	syl3anc 1372	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ 𝐵)
28	3	simprd 499	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
29	12, 1	lhpbase 37624	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
30	28, 29	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
31		dihjatcclem.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
32	12, 31	latmcl 17771	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
33	22, 27, 30, 32	syl3anc 1372	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
34	20, 33	eqeltrid 2837	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
35	12, 1, 16, 2, 7	dihlss 38876	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ 𝐵) → (𝐼‘𝑉) ∈ (LSubSp‘𝑈))
36	3, 34, 35	syl2anc 587	. . . 4 ⊢ (𝜑 → (𝐼‘𝑉) ∈ (LSubSp‘𝑈))
37	9, 36	sseldd 3876	. . 3 ⊢ (𝜑 → (𝐼‘𝑉) ∈ (SubGrp‘𝑈))
38	12, 13	atbase 36915	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
39	24, 38	syl 17	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
40	12, 1, 16, 2, 7	dihlss 38876	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐵) → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
41	3, 39, 40	syl2anc 587	. . . 4 ⊢ (𝜑 → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
42	9, 41	sseldd 3876	. . 3 ⊢ (𝜑 → (𝐼‘𝑄) ∈ (SubGrp‘𝑈))
43		dihjatcclem.s	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
44	43	lsm4 19092	. . 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 1380	. 2 ⊢ (𝜑 → (((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) ⊕ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉))) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ ((𝐼‘𝑉) ⊕ (𝐼‘𝑉))))
46	23	simprd 499	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑄 ≤ 𝑊)
47	46	intnand 492	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊))
48		dihjatcclem.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
49	12, 48, 25	latjle12 17781	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊) ↔ (𝑃 ∨ 𝑄) ≤ 𝑊))
50	22, 15, 39, 30, 49	syl13anc 1373	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊) ↔ (𝑃 ∨ 𝑄) ≤ 𝑊))
51	47, 50	mtbid 327	. . . . . 6 ⊢ (𝜑 → ¬ (𝑃 ∨ 𝑄) ≤ 𝑊)
52	48, 25, 13	hlatlej1 37001	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑄))
53	21, 11, 24, 52	syl3anc 1372	. . . . . 6 ⊢ (𝜑 → 𝑃 ≤ (𝑃 ∨ 𝑄))
54	12, 48, 25, 31, 13, 1, 16, 2, 43	dihvalcq2 38873	. . . . . 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 1380	. . . . 5 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))))
56	20	fveq2i 6671	. . . . . 6 ⊢ (𝐼‘𝑉) = (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))
57	56	oveq2i 7175	. . . . 5 ⊢ ((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) = ((𝐼‘𝑃) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊)))
58	55, 57	eqtr4di 2791	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑉)))
59	48, 25, 13	hlatlej2 37002	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑃 ∨ 𝑄))
60	21, 11, 24, 59	syl3anc 1372	. . . . . 6 ⊢ (𝜑 → 𝑄 ≤ (𝑃 ∨ 𝑄))
61	12, 48, 25, 31, 13, 1, 16, 2, 43	dihvalcq2 38873	. . . . . 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 1380	. . . . 5 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑄) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))))
63	56	oveq2i 7175	. . . . 5 ⊢ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉)) = ((𝐼‘𝑄) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊)))
64	62, 63	eqtr4di 2791	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑄) ⊕ (𝐼‘𝑉)))
65	58, 64	oveq12d 7182	. . 3 ⊢ (𝜑 → ((𝐼‘(𝑃 ∨ 𝑄)) ⊕ (𝐼‘(𝑃 ∨ 𝑄))) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) ⊕ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉))))
66	12, 1, 16, 2, 7	dihlss 38876	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝐼‘(𝑃 ∨ 𝑄)) ∈ (LSubSp‘𝑈))
67	3, 27, 66	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) ∈ (LSubSp‘𝑈))
68	9, 67	sseldd 3876	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) ∈ (SubGrp‘𝑈))
69	43	lsmidm 18899	. . . 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 2775	. 2 ⊢ (𝜑 → (((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) ⊕ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉))) = (𝐼‘(𝑃 ∨ 𝑄)))
72	43	lsmidm 18899	. . . 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 7180	. 2 ⊢ (𝜑 → (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ ((𝐼‘𝑉) ⊕ (𝐼‘𝑉))) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)))
75	45, 71, 74	3eqtr3d 2781	1 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2113   ⊆ wss 3841   class class class wbr 5027  ‘cfv 6333  (class class class)co 7164  Basecbs 16579  lecple 16668  joincjn 17663  meetcmee 17664  Latclat 17764  SubGrpcsubg 18384  LSSumclsm 18870  Abelcabl 19018  LModclmod 19746  LSubSpclss 19815  Atomscatm 36889  HLchlt 36976  LHypclh 37610  DVecHcdvh 38704  DIsoHcdih 38854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-cnex 10664  ax-resscn 10665  ax-1cn 10666  ax-icn 10667  ax-addcl 10668  ax-addrcl 10669  ax-mulcl 10670  ax-mulrcl 10671  ax-mulcom 10672  ax-addass 10673  ax-mulass 10674  ax-distr 10675  ax-i2m1 10676  ax-1ne0 10677  ax-1rid 10678  ax-rnegex 10679  ax-rrecex 10680  ax-cnre 10681  ax-pre-lttri 10682  ax-pre-lttrn 10683  ax-pre-ltadd 10684  ax-pre-mulgt0 10685  ax-riotaBAD 36579

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-int 4834  df-iun 4880  df-iin 4881  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-tpos 7914  df-undef 7961  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-1o 8124  df-er 8313  df-map 8432  df-en 8549  df-dom 8550  df-sdom 8551  df-fin 8552  df-pnf 10748  df-mnf 10749  df-xr 10750  df-ltxr 10751  df-le 10752  df-sub 10943  df-neg 10944  df-nn 11710  df-2 11772  df-3 11773  df-4 11774  df-5 11775  df-6 11776  df-n0 11970  df-z 12056  df-uz 12318  df-fz 12975  df-struct 16581  df-ndx 16582  df-slot 16583  df-base 16585  df-sets 16586  df-ress 16587  df-plusg 16674  df-mulr 16675  df-sca 16677  df-vsca 16678  df-0g 16811  df-proset 17647  df-poset 17665  df-plt 17677  df-lub 17693  df-glb 17694  df-join 17695  df-meet 17696  df-p0 17758  df-p1 17759  df-lat 17765  df-clat 17827  df-mgm 17961  df-sgrp 18010  df-mnd 18021  df-submnd 18066  df-grp 18215  df-minusg 18216  df-sbg 18217  df-subg 18387  df-cntz 18558  df-lsm 18872  df-cmn 19019  df-abl 19020  df-mgp 19352  df-ur 19364  df-ring 19411  df-oppr 19488  df-dvdsr 19506  df-unit 19507  df-invr 19537  df-dvr 19548  df-drng 19616  df-lmod 19748  df-lss 19816  df-lsp 19856  df-lvec 19987  df-oposet 36802  df-ol 36804  df-oml 36805  df-covers 36892  df-ats 36893  df-atl 36924  df-cvlat 36948  df-hlat 36977  df-llines 37124  df-lplanes 37125  df-lvols 37126  df-lines 37127  df-psubsp 37129  df-pmap 37130  df-padd 37422  df-lhyp 37614  df-laut 37615  df-ldil 37730  df-ltrn 37731  df-trl 37785  df-tendo 38381  df-edring 38383  df-disoa 38655  df-dvech 38705  df-dib 38765  df-dic 38799  df-dih 38855

This theorem is referenced by:  dihjatcclem2  39045