Theorem dihjatcclem1 41713

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 41405	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		lmodabl 20862	. . . 4 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Abel)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝑈 ∈ Abel)
7		eqid 2735	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8	7	lsssssubg 20911	. . . . 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 39584	. . . . . 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 41545	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐵) → (𝐼‘𝑃) ∈ (LSubSp‘𝑈))
18	3, 15, 17	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (LSubSp‘𝑈))
19	9, 18	sseldd 3933	. . 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 39659	. . . . . . 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 39662	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵)
27	21, 11, 24, 26	syl3anc 1374	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ 𝐵)
28	3	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
29	12, 1	lhpbase 40293	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
30	28, 29	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
31		dihjatcclem.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
32	12, 31	latmcl 18365	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
33	22, 27, 30, 32	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
34	20, 33	eqeltrid 2839	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
35	12, 1, 16, 2, 7	dihlss 41545	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ 𝐵) → (𝐼‘𝑉) ∈ (LSubSp‘𝑈))
36	3, 34, 35	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝐼‘𝑉) ∈ (LSubSp‘𝑈))
37	9, 36	sseldd 3933	. . 3 ⊢ (𝜑 → (𝐼‘𝑉) ∈ (SubGrp‘𝑈))
38	12, 13	atbase 39584	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
39	24, 38	syl 17	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
40	12, 1, 16, 2, 7	dihlss 41545	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐵) → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
41	3, 39, 40	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
42	9, 41	sseldd 3933	. . 3 ⊢ (𝜑 → (𝐼‘𝑄) ∈ (SubGrp‘𝑈))
43		dihjatcclem.s	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
44	43	lsm4 19791	. . 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 1382	. 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 18375	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊) ↔ (𝑃 ∨ 𝑄) ≤ 𝑊))
50	22, 15, 39, 30, 49	syl13anc 1375	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊) ↔ (𝑃 ∨ 𝑄) ≤ 𝑊))
51	47, 50	mtbid 324	. . . . . 6 ⊢ (𝜑 → ¬ (𝑃 ∨ 𝑄) ≤ 𝑊)
52	48, 25, 13	hlatlej1 39670	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑄))
53	21, 11, 24, 52	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → 𝑃 ≤ (𝑃 ∨ 𝑄))
54	12, 48, 25, 31, 13, 1, 16, 2, 43	dihvalcq2 41542	. . . . . 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 1382	. . . . 5 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))))
56	20	fveq2i 6836	. . . . . 6 ⊢ (𝐼‘𝑉) = (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))
57	56	oveq2i 7369	. . . . 5 ⊢ ((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) = ((𝐼‘𝑃) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊)))
58	55, 57	eqtr4di 2788	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑉)))
59	48, 25, 13	hlatlej2 39671	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑃 ∨ 𝑄))
60	21, 11, 24, 59	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → 𝑄 ≤ (𝑃 ∨ 𝑄))
61	12, 48, 25, 31, 13, 1, 16, 2, 43	dihvalcq2 41542	. . . . . 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 1382	. . . . 5 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑄) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))))
63	56	oveq2i 7369	. . . . 5 ⊢ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉)) = ((𝐼‘𝑄) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊)))
64	62, 63	eqtr4di 2788	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑄) ⊕ (𝐼‘𝑉)))
65	58, 64	oveq12d 7376	. . 3 ⊢ (𝜑 → ((𝐼‘(𝑃 ∨ 𝑄)) ⊕ (𝐼‘(𝑃 ∨ 𝑄))) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) ⊕ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉))))
66	12, 1, 16, 2, 7	dihlss 41545	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝐼‘(𝑃 ∨ 𝑄)) ∈ (LSubSp‘𝑈))
67	3, 27, 66	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) ∈ (LSubSp‘𝑈))
68	9, 67	sseldd 3933	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) ∈ (SubGrp‘𝑈))
69	43	lsmidm 19594	. . . 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 2772	. 2 ⊢ (𝜑 → (((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) ⊕ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉))) = (𝐼‘(𝑃 ∨ 𝑄)))
72	43	lsmidm 19594	. . . 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 7374	. 2 ⊢ (𝜑 → (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ ((𝐼‘𝑉) ⊕ (𝐼‘𝑉))) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)))
75	45, 71, 74	3eqtr3d 2778	1 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3900   class class class wbr 5097  ‘cfv 6491  (class class class)co 7358  Basecbs 17138  lecple 17186  joincjn 18236  meetcmee 18237  Latclat 18356  SubGrpcsubg 19052  LSSumclsm 19565  Abelcabl 19712  LModclmod 20813  LSubSpclss 20884  Atomscatm 39558  HLchlt 39645  LHypclh 40279  DVecHcdvh 41373  DIsoHcdih 41523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-riotaBAD 39248

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-iin 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-tpos 8168  df-undef 8215  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8767  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-n0 12404  df-z 12491  df-uz 12754  df-fz 13426  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-mulr 17193  df-sca 17195  df-vsca 17196  df-0g 17363  df-proset 18219  df-poset 18238  df-plt 18253  df-lub 18269  df-glb 18270  df-join 18271  df-meet 18272  df-p0 18348  df-p1 18349  df-lat 18357  df-clat 18424  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-subg 19055  df-cntz 19248  df-lsm 19567  df-cmn 19713  df-abl 19714  df-mgp 20078  df-rng 20090  df-ur 20119  df-ring 20172  df-oppr 20275  df-dvdsr 20295  df-unit 20296  df-invr 20326  df-dvr 20339  df-drng 20666  df-lmod 20815  df-lss 20885  df-lsp 20925  df-lvec 21057  df-oposet 39471  df-ol 39473  df-oml 39474  df-covers 39561  df-ats 39562  df-atl 39593  df-cvlat 39617  df-hlat 39646  df-llines 39793  df-lplanes 39794  df-lvols 39795  df-lines 39796  df-psubsp 39798  df-pmap 39799  df-padd 40091  df-lhyp 40283  df-laut 40284  df-ldil 40399  df-ltrn 40400  df-trl 40454  df-tendo 41050  df-edring 41052  df-disoa 41324  df-dvech 41374  df-dib 41434  df-dic 41468  df-dih 41524

This theorem is referenced by:  dihjatcclem2  41714