Theorem dihjatcclem1 38569

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 38261	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		lmodabl 19681	. . . 4 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Abel)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝑈 ∈ Abel)
7		eqid 2821	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8	7	lsssssubg 19730	. . . . 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 497	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
12		dihjatcclem.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
13		dihjatcclem.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
14	12, 13	atbase 36440	. . . . . 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 38401	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐵) → (𝐼‘𝑃) ∈ (LSubSp‘𝑈))
18	3, 15, 17	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (LSubSp‘𝑈))
19	9, 18	sseldd 3968	. . 3 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (SubGrp‘𝑈))
20		dihjatcclem.v	. . . . . 6 ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
21	3	simpld 497	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ HL)
22	21	hllatd 36515	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Lat)
23		dihjatcclem.q	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
24	23	simpld 497	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
25		dihjatcclem.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
26	12, 25, 13	hlatjcl 36518	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵)
27	21, 11, 24, 26	syl3anc 1367	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ 𝐵)
28	3	simprd 498	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
29	12, 1	lhpbase 37149	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
30	28, 29	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
31		dihjatcclem.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
32	12, 31	latmcl 17662	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
33	22, 27, 30, 32	syl3anc 1367	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
34	20, 33	eqeltrid 2917	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
35	12, 1, 16, 2, 7	dihlss 38401	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ 𝐵) → (𝐼‘𝑉) ∈ (LSubSp‘𝑈))
36	3, 34, 35	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝐼‘𝑉) ∈ (LSubSp‘𝑈))
37	9, 36	sseldd 3968	. . 3 ⊢ (𝜑 → (𝐼‘𝑉) ∈ (SubGrp‘𝑈))
38	12, 13	atbase 36440	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
39	24, 38	syl 17	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
40	12, 1, 16, 2, 7	dihlss 38401	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐵) → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
41	3, 39, 40	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
42	9, 41	sseldd 3968	. . 3 ⊢ (𝜑 → (𝐼‘𝑄) ∈ (SubGrp‘𝑈))
43		dihjatcclem.s	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
44	43	lsm4 18980	. . 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 1375	. 2 ⊢ (𝜑 → (((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) ⊕ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉))) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ ((𝐼‘𝑉) ⊕ (𝐼‘𝑉))))
46	23	simprd 498	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑄 ≤ 𝑊)
47	46	intnand 491	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊))
48		dihjatcclem.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
49	12, 48, 25	latjle12 17672	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊) ↔ (𝑃 ∨ 𝑄) ≤ 𝑊))
50	22, 15, 39, 30, 49	syl13anc 1368	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊) ↔ (𝑃 ∨ 𝑄) ≤ 𝑊))
51	47, 50	mtbid 326	. . . . . 6 ⊢ (𝜑 → ¬ (𝑃 ∨ 𝑄) ≤ 𝑊)
52	48, 25, 13	hlatlej1 36526	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑄))
53	21, 11, 24, 52	syl3anc 1367	. . . . . 6 ⊢ (𝜑 → 𝑃 ≤ (𝑃 ∨ 𝑄))
54	12, 48, 25, 31, 13, 1, 16, 2, 43	dihvalcq2 38398	. . . . . 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 1375	. . . . 5 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))))
56	20	fveq2i 6673	. . . . . 6 ⊢ (𝐼‘𝑉) = (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))
57	56	oveq2i 7167	. . . . 5 ⊢ ((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) = ((𝐼‘𝑃) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊)))
58	55, 57	syl6eqr 2874	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑉)))
59	48, 25, 13	hlatlej2 36527	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑃 ∨ 𝑄))
60	21, 11, 24, 59	syl3anc 1367	. . . . . 6 ⊢ (𝜑 → 𝑄 ≤ (𝑃 ∨ 𝑄))
61	12, 48, 25, 31, 13, 1, 16, 2, 43	dihvalcq2 38398	. . . . . 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 1375	. . . . 5 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑄) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))))
63	56	oveq2i 7167	. . . . 5 ⊢ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉)) = ((𝐼‘𝑄) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊)))
64	62, 63	syl6eqr 2874	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑄) ⊕ (𝐼‘𝑉)))
65	58, 64	oveq12d 7174	. . 3 ⊢ (𝜑 → ((𝐼‘(𝑃 ∨ 𝑄)) ⊕ (𝐼‘(𝑃 ∨ 𝑄))) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) ⊕ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉))))
66	12, 1, 16, 2, 7	dihlss 38401	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝐼‘(𝑃 ∨ 𝑄)) ∈ (LSubSp‘𝑈))
67	3, 27, 66	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) ∈ (LSubSp‘𝑈))
68	9, 67	sseldd 3968	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) ∈ (SubGrp‘𝑈))
69	43	lsmidm 18788	. . . 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 2858	. 2 ⊢ (𝜑 → (((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) ⊕ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉))) = (𝐼‘(𝑃 ∨ 𝑄)))
72	43	lsmidm 18788	. . . 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 7172	. 2 ⊢ (𝜑 → (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ ((𝐼‘𝑉) ⊕ (𝐼‘𝑉))) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)))
75	45, 71, 74	3eqtr3d 2864	1 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ⊆ wss 3936   class class class wbr 5066  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  lecple 16572  joincjn 17554  meetcmee 17555  Latclat 17655  SubGrpcsubg 18273  LSSumclsm 18759  Abelcabl 18907  LModclmod 19634  LSubSpclss 19703  Atomscatm 36414  HLchlt 36501  LHypclh 37135  DVecHcdvh 38229  DIsoHcdih 38379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-riotaBAD 36104

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-tpos 7892  df-undef 7939  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-sca 16581  df-vsca 16582  df-0g 16715  df-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-p1 17650  df-lat 17656  df-clat 17718  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-submnd 17957  df-grp 18106  df-minusg 18107  df-sbg 18108  df-subg 18276  df-cntz 18447  df-lsm 18761  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-ring 19299  df-oppr 19373  df-dvdsr 19391  df-unit 19392  df-invr 19422  df-dvr 19433  df-drng 19504  df-lmod 19636  df-lss 19704  df-lsp 19744  df-lvec 19875  df-oposet 36327  df-ol 36329  df-oml 36330  df-covers 36417  df-ats 36418  df-atl 36449  df-cvlat 36473  df-hlat 36502  df-llines 36649  df-lplanes 36650  df-lvols 36651  df-lines 36652  df-psubsp 36654  df-pmap 36655  df-padd 36947  df-lhyp 37139  df-laut 37140  df-ldil 37255  df-ltrn 37256  df-trl 37310  df-tendo 37906  df-edring 37908  df-disoa 38180  df-dvech 38230  df-dib 38290  df-dic 38324  df-dih 38380

This theorem is referenced by:  dihjatcclem2  38570