dihmeetlem10N - Mathbox for Norm Megill

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Theorem dihmeetlem10N 38486

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
dihmeetlem10N	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝐼‘((𝑋 ∧ 𝑌) ∨ 𝑝)) = ((𝐼‘𝑋) ∩ (𝐼‘(𝑌 ∨ 𝑝))))

**Proof of Theorem dihmeetlem10N**
Step	Hyp	Ref	Expression
1		simpl1l 1219	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝐾 ∈ HL)
2		simpl2 1187	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝑋 ∈ 𝐵)
3		simpl3 1188	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝑌 ∈ 𝐵)
4		simprll 777	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝑝 ∈ 𝐴)
5		simprr 771	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝑝 ≤ 𝑋)
6		dihmeetlem9.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
7		dihmeetlem9.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
8		dihmeetlem9.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
9		dihmeetlem9.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
10		dihmeetlem9.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
11	6, 7, 8, 9, 10	dihmeetlem5 38478	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑋)) → (𝑋 ∧ (𝑌 ∨ 𝑝)) = ((𝑋 ∧ 𝑌) ∨ 𝑝))
12	1, 2, 3, 4, 5, 11	syl32anc 1373	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝑋 ∧ (𝑌 ∨ 𝑝)) = ((𝑋 ∧ 𝑌) ∨ 𝑝))
13	12	fveq2d 6667	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝐼‘(𝑋 ∧ (𝑌 ∨ 𝑝))) = (𝐼‘((𝑋 ∧ 𝑌) ∨ 𝑝)))
14		simpl1 1186	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15	1	hllatd 36534	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝐾 ∈ Lat)
16	6, 10	atbase 36459	. . . . 5 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵)
17	4, 16	syl 17	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝑝 ∈ 𝐵)
18	6, 8	latjcl 17654	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) → (𝑌 ∨ 𝑝) ∈ 𝐵)
19	15, 3, 17, 18	syl3anc 1366	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝑌 ∨ 𝑝) ∈ 𝐵)
20		dihmeetlem9.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
21	6, 7, 20, 8, 9, 10	dihmeetlem6 38479	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → ¬ (𝑋 ∧ (𝑌 ∨ 𝑝)) ≤ 𝑊)
22		dihmeetlem9.i	. . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
23	6, 7, 9, 20, 22	dihmeetcN 38472	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ (𝑌 ∨ 𝑝) ∈ 𝐵) ∧ ¬ (𝑋 ∧ (𝑌 ∨ 𝑝)) ≤ 𝑊) → (𝐼‘(𝑋 ∧ (𝑌 ∨ 𝑝))) = ((𝐼‘𝑋) ∩ (𝐼‘(𝑌 ∨ 𝑝))))
24	14, 2, 19, 21, 23	syl121anc 1370	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝐼‘(𝑋 ∧ (𝑌 ∨ 𝑝))) = ((𝐼‘𝑋) ∩ (𝐼‘(𝑌 ∨ 𝑝))))
25	13, 24	eqtr3d 2857	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝐼‘((𝑋 ∧ 𝑌) ∨ 𝑝)) = ((𝐼‘𝑋) ∩ (𝐼‘(𝑌 ∨ 𝑝))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ∩ cin 3928 class class class wbr 5059 ‘cfv 6348 (class class class)co 7149 Basecbs 16476 lecple 16565 joincjn 17547 meetcmee 17548 Latclat 17648 LSSumclsm 18752 Atomscatm 36433 HLchlt 36520 LHypclh 37154 DVecHcdvh 38248 DIsoHcdih 38398

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-riotaBAD 36123

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-tpos 7885 df-undef 7932 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 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 12890 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-mulr 16572 df-sca 16574 df-vsca 16575 df-0g 16708 df-proset 17531 df-poset 17549 df-plt 17561 df-lub 17577 df-glb 17578 df-join 17579 df-meet 17580 df-p0 17642 df-p1 17643 df-lat 17649 df-clat 17711 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-submnd 17950 df-grp 18099 df-minusg 18100 df-sbg 18101 df-subg 18269 df-cntz 18440 df-lsm 18754 df-cmn 18901 df-abl 18902 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19366 df-dvdsr 19384 df-unit 19385 df-invr 19415 df-dvr 19426 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 df-oposet 36346 df-ol 36348 df-oml 36349 df-covers 36436 df-ats 36437 df-atl 36468 df-cvlat 36492 df-hlat 36521 df-llines 36668 df-lplanes 36669 df-lvols 36670 df-lines 36671 df-psubsp 36673 df-pmap 36674 df-padd 36966 df-lhyp 37158 df-laut 37159 df-ldil 37274 df-ltrn 37275 df-trl 37329 df-tendo 37925 df-edring 37927 df-disoa 38199 df-dvech 38249 df-dib 38309 df-dic 38343 df-dih 38399

This theorem is referenced by: dihmeetlem11N 38487

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