dihmeetlem11N - Mathbox for Norm Megill

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Theorem dihmeetlem11N 37997

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 37996	. . 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 4110	. 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 4118	. . 3 ⊢ (((𝐼‘𝑋) ∩ (𝐼‘(𝑌 ∨ 𝑝))) ∩ (𝐼‘𝑌)) = ((𝐼‘𝑋) ∩ ((𝐼‘(𝑌 ∨ 𝑝)) ∩ (𝐼‘𝑌)))
13		simpl1l 1217	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝐾 ∈ HL)
14	13	hllatd 36044	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝐾 ∈ Lat)
15		simpl3 1186	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝑌 ∈ 𝐵)
16		simprll 775	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝑝 ∈ 𝐴)
17	1, 6	atbase 35969	. . . . . . . 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 17499	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) → 𝑌 ≤ (𝑌 ∨ 𝑝))
20	14, 15, 18, 19	syl3anc 1364	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝑌 ≤ (𝑌 ∨ 𝑝))
21		simpl1 1184	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
22	1, 4	latjcl 17490	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) → (𝑌 ∨ 𝑝) ∈ 𝐵)
23	14, 15, 18, 22	syl3anc 1364	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝑌 ∨ 𝑝) ∈ 𝐵)
24	1, 2, 3, 9	dihord 37944	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑌 ∨ 𝑝) ∈ 𝐵) → ((𝐼‘𝑌) ⊆ (𝐼‘(𝑌 ∨ 𝑝)) ↔ 𝑌 ≤ (𝑌 ∨ 𝑝)))
25	21, 15, 23, 24	syl3anc 1364	. . . . . 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 258	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝐼‘𝑌) ⊆ (𝐼‘(𝑌 ∨ 𝑝)))
27		sseqin2 4114	. . . . 5 ⊢ ((𝐼‘𝑌) ⊆ (𝐼‘(𝑌 ∨ 𝑝)) ↔ ((𝐼‘(𝑌 ∨ 𝑝)) ∩ (𝐼‘𝑌)) = (𝐼‘𝑌))
28	26, 27	sylib 219	. . . 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 4111	. . 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 2842	. 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 2830	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 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2080 ∩ cin 3860 ⊆ wss 3861 class class class wbr 4964 ‘cfv 6228 (class class class)co 7019 Basecbs 16312 lecple 16401 joincjn 17383 meetcmee 17384 Latclat 17484 LSSumclsm 18489 Atomscatm 35943 HLchlt 36030 LHypclh 36664 DVecHcdvh 37758 DIsoHcdih 37908

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 ax-riotaBAD 35633

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-int 4785 df-iun 4829 df-iin 4830 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-1st 7548 df-2nd 7549 df-tpos 7746 df-undef 7793 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-1o 7956 df-oadd 7960 df-er 8142 df-map 8261 df-en 8361 df-dom 8362 df-sdom 8363 df-fin 8364 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-nn 11489 df-2 11550 df-3 11551 df-4 11552 df-5 11553 df-6 11554 df-n0 11748 df-z 11832 df-uz 12094 df-fz 12743 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-sca 16410 df-vsca 16411 df-0g 16544 df-proset 17367 df-poset 17385 df-plt 17397 df-lub 17413 df-glb 17414 df-join 17415 df-meet 17416 df-p0 17478 df-p1 17479 df-lat 17485 df-clat 17547 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-grp 17864 df-minusg 17865 df-sbg 17866 df-subg 18030 df-cntz 18188 df-lsm 18491 df-cmn 18635 df-abl 18636 df-mgp 18930 df-ur 18942 df-ring 18989 df-oppr 19063 df-dvdsr 19081 df-unit 19082 df-invr 19112 df-dvr 19123 df-drng 19194 df-lmod 19326 df-lss 19394 df-lsp 19434 df-lvec 19565 df-oposet 35856 df-ol 35858 df-oml 35859 df-covers 35946 df-ats 35947 df-atl 35978 df-cvlat 36002 df-hlat 36031 df-llines 36178 df-lplanes 36179 df-lvols 36180 df-lines 36181 df-psubsp 36183 df-pmap 36184 df-padd 36476 df-lhyp 36668 df-laut 36669 df-ldil 36784 df-ltrn 36785 df-trl 36839 df-tendo 37435 df-edring 37437 df-disoa 37709 df-dvech 37759 df-dib 37819 df-dic 37853 df-dih 37909

This theorem is referenced by: dihmeetlem12N 37998

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