Theorem dihjatcclem1 39881

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 39573	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		lmodabl 20369	. . . 4 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Abel)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝑈 ∈ Abel)
7		eqid 2736	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8	7	lsssssubg 20419	. . . . 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 495	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
12		dihjatcclem.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
13		dihjatcclem.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
14	12, 13	atbase 37751	. . . . . 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 39713	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐵) → (𝐼‘𝑃) ∈ (LSubSp‘𝑈))
18	3, 15, 17	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (LSubSp‘𝑈))
19	9, 18	sseldd 3945	. . 3 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (SubGrp‘𝑈))
20		dihjatcclem.v	. . . . . 6 ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
21	3	simpld 495	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ HL)
22	21	hllatd 37826	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Lat)
23		dihjatcclem.q	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
24	23	simpld 495	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
25		dihjatcclem.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
26	12, 25, 13	hlatjcl 37829	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵)
27	21, 11, 24, 26	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ 𝐵)
28	3	simprd 496	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
29	12, 1	lhpbase 38461	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
30	28, 29	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
31		dihjatcclem.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
32	12, 31	latmcl 18329	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
33	22, 27, 30, 32	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
34	20, 33	eqeltrid 2842	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
35	12, 1, 16, 2, 7	dihlss 39713	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ 𝐵) → (𝐼‘𝑉) ∈ (LSubSp‘𝑈))
36	3, 34, 35	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐼‘𝑉) ∈ (LSubSp‘𝑈))
37	9, 36	sseldd 3945	. . 3 ⊢ (𝜑 → (𝐼‘𝑉) ∈ (SubGrp‘𝑈))
38	12, 13	atbase 37751	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
39	24, 38	syl 17	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
40	12, 1, 16, 2, 7	dihlss 39713	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐵) → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
41	3, 39, 40	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
42	9, 41	sseldd 3945	. . 3 ⊢ (𝜑 → (𝐼‘𝑄) ∈ (SubGrp‘𝑈))
43		dihjatcclem.s	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
44	43	lsm4 19638	. . 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 1379	. 2 ⊢ (𝜑 → (((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) ⊕ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉))) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ ((𝐼‘𝑉) ⊕ (𝐼‘𝑉))))
46	23	simprd 496	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑄 ≤ 𝑊)
47	46	intnand 489	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊))
48		dihjatcclem.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
49	12, 48, 25	latjle12 18339	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊) ↔ (𝑃 ∨ 𝑄) ≤ 𝑊))
50	22, 15, 39, 30, 49	syl13anc 1372	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊) ↔ (𝑃 ∨ 𝑄) ≤ 𝑊))
51	47, 50	mtbid 323	. . . . . 6 ⊢ (𝜑 → ¬ (𝑃 ∨ 𝑄) ≤ 𝑊)
52	48, 25, 13	hlatlej1 37837	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑄))
53	21, 11, 24, 52	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → 𝑃 ≤ (𝑃 ∨ 𝑄))
54	12, 48, 25, 31, 13, 1, 16, 2, 43	dihvalcq2 39710	. . . . . 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 1379	. . . . 5 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))))
56	20	fveq2i 6845	. . . . . 6 ⊢ (𝐼‘𝑉) = (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))
57	56	oveq2i 7368	. . . . 5 ⊢ ((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) = ((𝐼‘𝑃) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊)))
58	55, 57	eqtr4di 2794	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑉)))
59	48, 25, 13	hlatlej2 37838	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑃 ∨ 𝑄))
60	21, 11, 24, 59	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → 𝑄 ≤ (𝑃 ∨ 𝑄))
61	12, 48, 25, 31, 13, 1, 16, 2, 43	dihvalcq2 39710	. . . . . 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 1379	. . . . 5 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑄) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))))
63	56	oveq2i 7368	. . . . 5 ⊢ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉)) = ((𝐼‘𝑄) ⊕ (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊)))
64	62, 63	eqtr4di 2794	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑄) ⊕ (𝐼‘𝑉)))
65	58, 64	oveq12d 7375	. . 3 ⊢ (𝜑 → ((𝐼‘(𝑃 ∨ 𝑄)) ⊕ (𝐼‘(𝑃 ∨ 𝑄))) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) ⊕ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉))))
66	12, 1, 16, 2, 7	dihlss 39713	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝐼‘(𝑃 ∨ 𝑄)) ∈ (LSubSp‘𝑈))
67	3, 27, 66	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) ∈ (LSubSp‘𝑈))
68	9, 67	sseldd 3945	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) ∈ (SubGrp‘𝑈))
69	43	lsmidm 19445	. . . 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 2778	. 2 ⊢ (𝜑 → (((𝐼‘𝑃) ⊕ (𝐼‘𝑉)) ⊕ ((𝐼‘𝑄) ⊕ (𝐼‘𝑉))) = (𝐼‘(𝑃 ∨ 𝑄)))
72	43	lsmidm 19445	. . . 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 7373	. 2 ⊢ (𝜑 → (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ ((𝐼‘𝑉) ⊕ (𝐼‘𝑉))) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)))
75	45, 71, 74	3eqtr3d 2784	1 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ⊆ wss 3910   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  meetcmee 18201  Latclat 18320  SubGrpcsubg 18922  LSSumclsm 19416  Abelcabl 19563  LModclmod 20322  LSubSpclss 20392  Atomscatm 37725  HLchlt 37812  LHypclh 38447  DVecHcdvh 39541  DIsoHcdih 39691

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-riotaBAD 37415

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-undef 8204  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-0g 17323  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-p1 18315  df-lat 18321  df-clat 18388  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-subg 18925  df-cntz 19097  df-lsm 19418  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-dvr 20112  df-drng 20187  df-lmod 20324  df-lss 20393  df-lsp 20433  df-lvec 20564  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962  df-lvols 37963  df-lines 37964  df-psubsp 37966  df-pmap 37967  df-padd 38259  df-lhyp 38451  df-laut 38452  df-ldil 38567  df-ltrn 38568  df-trl 38622  df-tendo 39218  df-edring 39220  df-disoa 39492  df-dvech 39542  df-dib 39602  df-dic 39636  df-dih 39692

This theorem is referenced by:  dihjatcclem2  39882