dihmeetlem11N - Mathbox for Norm Megill

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Theorem dihmeetlem11N 37124

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
dihmeetlem11N	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → ((𝐼‘((𝑋 ∧ 𝑌) ∨ 𝑝)) ∩ (𝐼‘𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))

**Proof of Theorem dihmeetlem11N**
Step	Hyp	Ref	Expression
1		dihmeetlem9.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		dihmeetlem9.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		dihmeetlem9.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
4		dihmeetlem9.j	. . . 4 ⊢ ∨ = (join‘𝐾)
5		dihmeetlem9.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
6		dihmeetlem9.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
7		dihmeetlem9.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
8		dihmeetlem9.s	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
9		dihmeetlem9.i	. . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	dihmeetlem10N 37123	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝐼‘((𝑋 ∧ 𝑌) ∨ 𝑝)) = ((𝐼‘𝑋) ∩ (𝐼‘(𝑌 ∨ 𝑝))))
11	10	ineq1d 3964	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → ((𝐼‘((𝑋 ∧ 𝑌) ∨ 𝑝)) ∩ (𝐼‘𝑌)) = (((𝐼‘𝑋) ∩ (𝐼‘(𝑌 ∨ 𝑝))) ∩ (𝐼‘𝑌)))
12		inass 3972	. . 3 ⊢ (((𝐼‘𝑋) ∩ (𝐼‘(𝑌 ∨ 𝑝))) ∩ (𝐼‘𝑌)) = ((𝐼‘𝑋) ∩ ((𝐼‘(𝑌 ∨ 𝑝)) ∩ (𝐼‘𝑌)))
13		simpl1l 1278	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝐾 ∈ HL)
14	13	hllatd 35169	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝐾 ∈ Lat)
15		simpl3 1231	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝑌 ∈ 𝐵)
16		simprll 764	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝑝 ∈ 𝐴)
17	1, 6	atbase 35094	. . . . . . . 8 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵)
18	16, 17	syl 17	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝑝 ∈ 𝐵)
19	1, 2, 4	latlej1 17264	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) → 𝑌 ≤ (𝑌 ∨ 𝑝))
20	14, 15, 18, 19	syl3anc 1476	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝑌 ≤ (𝑌 ∨ 𝑝))
21		simpl1 1227	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
22	1, 4	latjcl 17255	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) → (𝑌 ∨ 𝑝) ∈ 𝐵)
23	14, 15, 18, 22	syl3anc 1476	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝑌 ∨ 𝑝) ∈ 𝐵)
24	1, 2, 3, 9	dihord 37071	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑌 ∨ 𝑝) ∈ 𝐵) → ((𝐼‘𝑌) ⊆ (𝐼‘(𝑌 ∨ 𝑝)) ↔ 𝑌 ≤ (𝑌 ∨ 𝑝)))
25	21, 15, 23, 24	syl3anc 1476	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → ((𝐼‘𝑌) ⊆ (𝐼‘(𝑌 ∨ 𝑝)) ↔ 𝑌 ≤ (𝑌 ∨ 𝑝)))
26	20, 25	mpbird 247	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝐼‘𝑌) ⊆ (𝐼‘(𝑌 ∨ 𝑝)))
27		sseqin2 3968	. . . . 5 ⊢ ((𝐼‘𝑌) ⊆ (𝐼‘(𝑌 ∨ 𝑝)) ↔ ((𝐼‘(𝑌 ∨ 𝑝)) ∩ (𝐼‘𝑌)) = (𝐼‘𝑌))
28	26, 27	sylib 208	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → ((𝐼‘(𝑌 ∨ 𝑝)) ∩ (𝐼‘𝑌)) = (𝐼‘𝑌))
29	28	ineq2d 3965	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → ((𝐼‘𝑋) ∩ ((𝐼‘(𝑌 ∨ 𝑝)) ∩ (𝐼‘𝑌))) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
30	12, 29	syl5eq 2817	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (((𝐼‘𝑋) ∩ (𝐼‘(𝑌 ∨ 𝑝))) ∩ (𝐼‘𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
31	11, 30	eqtrd 2805	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → ((𝐼‘((𝑋 ∧ 𝑌) ∨ 𝑝)) ∩ (𝐼‘𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ∩ cin 3722 ⊆ wss 3723 class class class wbr 4786 ‘cfv 6029 (class class class)co 6792 Basecbs 16060 lecple 16152 joincjn 17148 meetcmee 17149 Latclat 17249 LSSumclsm 18252 Atomscatm 35068 HLchlt 35155 LHypclh 35789 DVecHcdvh 36885 DIsoHcdih 37035

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-riotaBAD 34757

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7213 df-1st 7315 df-2nd 7316 df-tpos 7504 df-undef 7551 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-n0 11496 df-z 11581 df-uz 11890 df-fz 12530 df-struct 16062 df-ndx 16063 df-slot 16064 df-base 16066 df-sets 16067 df-ress 16068 df-plusg 16158 df-mulr 16159 df-sca 16161 df-vsca 16162 df-0g 16306 df-preset 17132 df-poset 17150 df-plt 17162 df-lub 17178 df-glb 17179 df-join 17180 df-meet 17181 df-p0 17243 df-p1 17244 df-lat 17250 df-clat 17312 df-mgm 17446 df-sgrp 17488 df-mnd 17499 df-submnd 17540 df-grp 17629 df-minusg 17630 df-sbg 17631 df-subg 17795 df-cntz 17953 df-lsm 18254 df-cmn 18398 df-abl 18399 df-mgp 18694 df-ur 18706 df-ring 18753 df-oppr 18827 df-dvdsr 18845 df-unit 18846 df-invr 18876 df-dvr 18887 df-drng 18955 df-lmod 19071 df-lss 19139 df-lsp 19181 df-lvec 19312 df-oposet 34981 df-ol 34983 df-oml 34984 df-covers 35071 df-ats 35072 df-atl 35103 df-cvlat 35127 df-hlat 35156 df-llines 35303 df-lplanes 35304 df-lvols 35305 df-lines 35306 df-psubsp 35308 df-pmap 35309 df-padd 35601 df-lhyp 35793 df-laut 35794 df-ldil 35909 df-ltrn 35910 df-trl 35965 df-tendo 36561 df-edring 36563 df-disoa 36836 df-dvech 36886 df-dib 36946 df-dic 36980 df-dih 37036

This theorem is referenced by: dihmeetlem12N 37125

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