dihjatcclem2 - Mathbox for Norm Megill

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Theorem dihjatcclem2 38549

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))
dihjatcclem2.c	⊢ (𝜑 → (𝐼‘𝑉) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))

**Assertion**
Ref	Expression
dihjatcclem2	⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))

**Proof of Theorem dihjatcclem2**
Step	Hyp	Ref	Expression
1		dihjatcclem.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		dihjatcclem.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		dihjatcclem.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
4		dihjatcclem.j	. . 3 ⊢ ∨ = (join‘𝐾)
5		dihjatcclem.m	. . 3 ⊢ ∧ = (meet‘𝐾)
6		dihjatcclem.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
7		dihjatcclem.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
8		dihjatcclem.s	. . 3 ⊢ ⊕ = (LSSum‘𝑈)
9		dihjatcclem.i	. . 3 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
10		dihjatcclem.v	. . 3 ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
11		dihjatcclem.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		dihjatcclem.p	. . 3 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
13		dihjatcclem.q	. . 3 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	dihjatcclem1 38548	. 2 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)))
15	3, 7, 11	dvhlmod 38240	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
16		eqid 2821	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
17	16	lsssssubg 19724	. . . . 5 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
18	15, 17	syl 17	. . . 4 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
19	12	simpld 497	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
20	1, 6	atbase 36419	. . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
21	19, 20	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
22	1, 3, 9, 7, 16	dihlss 38380	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐵) → (𝐼‘𝑃) ∈ (LSubSp‘𝑈))
23	11, 21, 22	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (LSubSp‘𝑈))
24	13	simpld 497	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
25	1, 6	atbase 36419	. . . . . . 7 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
26	24, 25	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
27	1, 3, 9, 7, 16	dihlss 38380	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐵) → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
28	11, 26, 27	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
29	16, 8	lsmcl 19849	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝐼‘𝑃) ∈ (LSubSp‘𝑈) ∧ (𝐼‘𝑄) ∈ (LSubSp‘𝑈)) → ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ∈ (LSubSp‘𝑈))
30	15, 23, 28, 29	syl3anc 1367	. . . 4 ⊢ (𝜑 → ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ∈ (LSubSp‘𝑈))
31	18, 30	sseldd 3967	. . 3 ⊢ (𝜑 → ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ∈ (SubGrp‘𝑈))
32	10	fveq2i 6667	. . . . 5 ⊢ (𝐼‘𝑉) = (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))
33	11	simpld 497	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ HL)
34	33	hllatd 36494	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Lat)
35	1, 4, 6	hlatjcl 36497	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵)
36	33, 19, 24, 35	syl3anc 1367	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ 𝐵)
37	11	simprd 498	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
38	1, 3	lhpbase 37128	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
39	37, 38	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
40	1, 5	latmcl 17656	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
41	34, 36, 39, 40	syl3anc 1367	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
42	1, 3, 9, 7, 16	dihlss 38380	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵) → (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (LSubSp‘𝑈))
43	11, 41, 42	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (LSubSp‘𝑈))
44	32, 43	eqeltrid 2917	. . . 4 ⊢ (𝜑 → (𝐼‘𝑉) ∈ (LSubSp‘𝑈))
45	18, 44	sseldd 3967	. . 3 ⊢ (𝜑 → (𝐼‘𝑉) ∈ (SubGrp‘𝑈))
46		dihjatcclem2.c	. . 3 ⊢ (𝜑 → (𝐼‘𝑉) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))
47	8	lsmss2 18787	. . 3 ⊢ ((((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ∈ (SubGrp‘𝑈) ∧ (𝐼‘𝑉) ∈ (SubGrp‘𝑈) ∧ (𝐼‘𝑉) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄))) → (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))
48	31, 45, 46, 47	syl3anc 1367	. 2 ⊢ (𝜑 → (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))
49	14, 48	eqtrd 2856	1 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 lecple 16566 joincjn 17548 meetcmee 17549 Latclat 17649 SubGrpcsubg 18267 LSSumclsm 18753 LModclmod 19628 LSubSpclss 19697 Atomscatm 36393 HLchlt 36480 LHypclh 37114 DVecHcdvh 38208 DIsoHcdih 38358

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-riotaBAD 36083

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-undef 7933 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-0g 16709 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cntz 18441 df-lsm 18755 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-llines 36628 df-lplanes 36629 df-lvols 36630 df-lines 36631 df-psubsp 36633 df-pmap 36634 df-padd 36926 df-lhyp 37118 df-laut 37119 df-ldil 37234 df-ltrn 37235 df-trl 37289 df-tendo 37885 df-edring 37887 df-disoa 38159 df-dvech 38209 df-dib 38269 df-dic 38303 df-dih 38359

This theorem is referenced by: dihjatcc 38552

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