dihmeetlem9N - Mathbox for Norm Megill

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Theorem dihmeetlem9N 40938

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
dihmeetlem9.b	⊢ 𝐵 = (Base‘𝐾)
dihmeetlem9.l	⊢ ≤ = (le‘𝐾)
dihmeetlem9.h	⊢ 𝐻 = (LHyp‘𝐾)
dihmeetlem9.j	⊢ ∨ = (join‘𝐾)
dihmeetlem9.m	⊢ ∧ = (meet‘𝐾)
dihmeetlem9.a	⊢ 𝐴 = (Atoms‘𝐾)
dihmeetlem9.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihmeetlem9.s	⊢ ⊕ = (LSSum‘𝑈)
dihmeetlem9.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
dihmeetlem9N	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (((𝐼‘𝑝) ⊕ (𝐼‘(𝑋 ∧ 𝑌))) ∩ (𝐼‘𝑌)) = ((𝐼‘(𝑋 ∧ 𝑌)) ⊕ ((𝐼‘𝑝) ∩ (𝐼‘𝑌))))

**Proof of Theorem dihmeetlem9N**
Step	Hyp	Ref	Expression
1		dihmeetlem9.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
2		dihmeetlem9.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		simp1 1133	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	dvhlmod 40733	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝑈 ∈ LMod)
5		eqid 2725	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
6	5	lsssssubg 20871	. . . . 5 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
7	4, 6	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
8		simp1l 1194	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝐾 ∈ HL)
9	8	hllatd 38986	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝐾 ∈ Lat)
10		simp2l 1196	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝑋 ∈ 𝐵)
11		simp2r 1197	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝑌 ∈ 𝐵)
12		dihmeetlem9.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
13		dihmeetlem9.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
14	12, 13	latmcl 18451	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵)
15	9, 10, 11, 14	syl3anc 1368	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (𝑋 ∧ 𝑌) ∈ 𝐵)
16		dihmeetlem9.i	. . . . . 6 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
17	12, 1, 16, 2, 5	dihlss 40873	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∧ 𝑌) ∈ 𝐵) → (𝐼‘(𝑋 ∧ 𝑌)) ∈ (LSubSp‘𝑈))
18	3, 15, 17	syl2anc 582	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (𝐼‘(𝑋 ∧ 𝑌)) ∈ (LSubSp‘𝑈))
19	7, 18	sseldd 3977	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (𝐼‘(𝑋 ∧ 𝑌)) ∈ (SubGrp‘𝑈))
20		dihmeetlem9.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
21	12, 20	atbase 38911	. . . . . 6 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵)
22	21	3ad2ant3 1132	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐵)
23	12, 1, 16, 2, 5	dihlss 40873	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ 𝐵) → (𝐼‘𝑝) ∈ (LSubSp‘𝑈))
24	3, 22, 23	syl2anc 582	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (𝐼‘𝑝) ∈ (LSubSp‘𝑈))
25	7, 24	sseldd 3977	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (𝐼‘𝑝) ∈ (SubGrp‘𝑈))
26	12, 1, 16, 2, 5	dihlss 40873	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵) → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
27	3, 11, 26	syl2anc 582	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
28	7, 27	sseldd 3977	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (𝐼‘𝑌) ∈ (SubGrp‘𝑈))
29		dihmeetlem9.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
30	12, 29, 13	latmle2 18476	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌)
31	9, 10, 11, 30	syl3anc 1368	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (𝑋 ∧ 𝑌) ≤ 𝑌)
32	12, 29, 1, 16	dihord 40887	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∧ 𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝐼‘(𝑋 ∧ 𝑌)) ⊆ (𝐼‘𝑌) ↔ (𝑋 ∧ 𝑌) ≤ 𝑌))
33	3, 15, 11, 32	syl3anc 1368	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → ((𝐼‘(𝑋 ∧ 𝑌)) ⊆ (𝐼‘𝑌) ↔ (𝑋 ∧ 𝑌) ≤ 𝑌))
34	31, 33	mpbird 256	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (𝐼‘(𝑋 ∧ 𝑌)) ⊆ (𝐼‘𝑌))
35		dihmeetlem9.s	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
36	35	lsmmod 19659	. . 3 ⊢ ((((𝐼‘(𝑋 ∧ 𝑌)) ∈ (SubGrp‘𝑈) ∧ (𝐼‘𝑝) ∈ (SubGrp‘𝑈) ∧ (𝐼‘𝑌) ∈ (SubGrp‘𝑈)) ∧ (𝐼‘(𝑋 ∧ 𝑌)) ⊆ (𝐼‘𝑌)) → ((𝐼‘(𝑋 ∧ 𝑌)) ⊕ ((𝐼‘𝑝) ∩ (𝐼‘𝑌))) = (((𝐼‘(𝑋 ∧ 𝑌)) ⊕ (𝐼‘𝑝)) ∩ (𝐼‘𝑌)))
37	19, 25, 28, 34, 36	syl31anc 1370	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → ((𝐼‘(𝑋 ∧ 𝑌)) ⊕ ((𝐼‘𝑝) ∩ (𝐼‘𝑌))) = (((𝐼‘(𝑋 ∧ 𝑌)) ⊕ (𝐼‘𝑝)) ∩ (𝐼‘𝑌)))
38		lmodabl 20821	. . . . 5 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Abel)
39	4, 38	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝑈 ∈ Abel)
40	35	lsmcom 19842	. . . 4 ⊢ ((𝑈 ∈ Abel ∧ (𝐼‘(𝑋 ∧ 𝑌)) ∈ (SubGrp‘𝑈) ∧ (𝐼‘𝑝) ∈ (SubGrp‘𝑈)) → ((𝐼‘(𝑋 ∧ 𝑌)) ⊕ (𝐼‘𝑝)) = ((𝐼‘𝑝) ⊕ (𝐼‘(𝑋 ∧ 𝑌))))
41	39, 19, 25, 40	syl3anc 1368	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → ((𝐼‘(𝑋 ∧ 𝑌)) ⊕ (𝐼‘𝑝)) = ((𝐼‘𝑝) ⊕ (𝐼‘(𝑋 ∧ 𝑌))))
42	41	ineq1d 4209	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (((𝐼‘(𝑋 ∧ 𝑌)) ⊕ (𝐼‘𝑝)) ∩ (𝐼‘𝑌)) = (((𝐼‘𝑝) ⊕ (𝐼‘(𝑋 ∧ 𝑌))) ∩ (𝐼‘𝑌)))
43	37, 42	eqtr2d 2766	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (((𝐼‘𝑝) ⊕ (𝐼‘(𝑋 ∧ 𝑌))) ∩ (𝐼‘𝑌)) = ((𝐼‘(𝑋 ∧ 𝑌)) ⊕ ((𝐼‘𝑝) ∩ (𝐼‘𝑌))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∩ cin 3943 ⊆ wss 3944 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 Basecbs 17199 lecple 17259 joincjn 18322 meetcmee 18323 Latclat 18442 SubGrpcsubg 19100 LSSumclsm 19618 Abelcabl 19765 LModclmod 20772 LSubSpclss 20844 Atomscatm 38885 HLchlt 38972 LHypclh 39607 DVecHcdvh 40701 DIsoHcdih 40851

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-riotaBAD 38575

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17135 df-sets 17152 df-slot 17170 df-ndx 17182 df-base 17200 df-ress 17229 df-plusg 17265 df-mulr 17266 df-sca 17268 df-vsca 17269 df-0g 17442 df-mre 17585 df-mrc 17586 df-acs 17588 df-proset 18306 df-poset 18324 df-plt 18341 df-lub 18357 df-glb 18358 df-join 18359 df-meet 18360 df-p0 18436 df-p1 18437 df-lat 18443 df-clat 18510 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-submnd 18760 df-grp 18917 df-minusg 18918 df-sbg 18919 df-subg 19103 df-cntz 19297 df-lsm 19620 df-cmn 19766 df-abl 19767 df-mgp 20104 df-rng 20122 df-ur 20151 df-ring 20204 df-oppr 20302 df-dvdsr 20325 df-unit 20326 df-invr 20356 df-dvr 20369 df-drng 20655 df-lmod 20774 df-lss 20845 df-lsp 20885 df-lvec 21017 df-oposet 38798 df-ol 38800 df-oml 38801 df-covers 38888 df-ats 38889 df-atl 38920 df-cvlat 38944 df-hlat 38973 df-llines 39121 df-lplanes 39122 df-lvols 39123 df-lines 39124 df-psubsp 39126 df-pmap 39127 df-padd 39419 df-lhyp 39611 df-laut 39612 df-ldil 39727 df-ltrn 39728 df-trl 39782 df-tendo 40378 df-edring 40380 df-disoa 40652 df-dvech 40702 df-dib 40762 df-dic 40796 df-dih 40852

This theorem is referenced by: dihmeetlem12N 40941

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