dihmeetlem11N - Mathbox for Norm Megill

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Theorem dihmeetlem11N 37120

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 37119	. . 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 3962	. 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 3970	. . 3 ⊢ (((𝐼‘𝑋) ∩ (𝐼‘(𝑌 ∨ 𝑝))) ∩ (𝐼‘𝑌)) = ((𝐼‘𝑋) ∩ ((𝐼‘(𝑌 ∨ 𝑝)) ∩ (𝐼‘𝑌)))
13		simpl1l 1277	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝐾 ∈ HL)
14		hllat 35165	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
15	13, 14	syl 17	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝐾 ∈ Lat)
16		simpl3 1230	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝑌 ∈ 𝐵)
17		simprll 756	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝑝 ∈ 𝐴)
18	1, 6	atbase 35091	. . . . . . . 8 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵)
19	17, 18	syl 17	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝑝 ∈ 𝐵)
20	1, 2, 4	latlej1 17267	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) → 𝑌 ≤ (𝑌 ∨ 𝑝))
21	15, 16, 19, 20	syl3anc 1475	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → 𝑌 ≤ (𝑌 ∨ 𝑝))
22		simpl1 1226	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23	1, 4	latjcl 17258	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) → (𝑌 ∨ 𝑝) ∈ 𝐵)
24	15, 16, 19, 23	syl3anc 1475	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝑌 ∨ 𝑝) ∈ 𝐵)
25	1, 2, 3, 9	dihord 37067	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑌 ∨ 𝑝) ∈ 𝐵) → ((𝐼‘𝑌) ⊆ (𝐼‘(𝑌 ∨ 𝑝)) ↔ 𝑌 ≤ (𝑌 ∨ 𝑝)))
26	22, 16, 24, 25	syl3anc 1475	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → ((𝐼‘𝑌) ⊆ (𝐼‘(𝑌 ∨ 𝑝)) ↔ 𝑌 ≤ (𝑌 ∨ 𝑝)))
27	21, 26	mpbird 247	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝐼‘𝑌) ⊆ (𝐼‘(𝑌 ∨ 𝑝)))
28		sseqin2 3966	. . . . 5 ⊢ ((𝐼‘𝑌) ⊆ (𝐼‘(𝑌 ∨ 𝑝)) ↔ ((𝐼‘(𝑌 ∨ 𝑝)) ∩ (𝐼‘𝑌)) = (𝐼‘𝑌))
29	27, 28	sylib 208	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → ((𝐼‘(𝑌 ∨ 𝑝)) ∩ (𝐼‘𝑌)) = (𝐼‘𝑌))
30	29	ineq2d 3963	. . 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)))
31	12, 30	syl5eq 2816	. 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)))
32	11, 31	eqtrd 2804	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 1070 = wceq 1630 ∈ wcel 2144 ∩ cin 3720 ⊆ wss 3721 class class class wbr 4784 ‘cfv 6031 (class class class)co 6792 Basecbs 16063 lecple 16155 joincjn 17151 meetcmee 17152 Latclat 17252 LSSumclsm 18255 Atomscatm 35065 HLchlt 35152 LHypclh 35785 DVecHcdvh 36881 DIsoHcdih 37031

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-riotaBAD 34754

This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-fal 1636 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-int 4610 df-iun 4654 df-iin 4655 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 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 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-tpos 7503 df-undef 7550 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-oadd 7716 df-er 7895 df-map 8010 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-n0 11494 df-z 11579 df-uz 11888 df-fz 12533 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-sca 16164 df-vsca 16165 df-0g 16309 df-preset 17135 df-poset 17153 df-plt 17165 df-lub 17181 df-glb 17182 df-join 17183 df-meet 17184 df-p0 17246 df-p1 17247 df-lat 17253 df-clat 17315 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-submnd 17543 df-grp 17632 df-minusg 17633 df-sbg 17634 df-subg 17798 df-cntz 17956 df-lsm 18257 df-cmn 18401 df-abl 18402 df-mgp 18697 df-ur 18709 df-ring 18756 df-oppr 18830 df-dvdsr 18848 df-unit 18849 df-invr 18879 df-dvr 18890 df-drng 18958 df-lmod 19074 df-lss 19142 df-lsp 19184 df-lvec 19315 df-oposet 34978 df-ol 34980 df-oml 34981 df-covers 35068 df-ats 35069 df-atl 35100 df-cvlat 35124 df-hlat 35153 df-llines 35299 df-lplanes 35300 df-lvols 35301 df-lines 35302 df-psubsp 35304 df-pmap 35305 df-padd 35597 df-lhyp 35789 df-laut 35790 df-ldil 35905 df-ltrn 35906 df-trl 35961 df-tendo 36557 df-edring 36559 df-disoa 36832 df-dvech 36882 df-dib 36942 df-dic 36976 df-dih 37032

This theorem is referenced by: dihmeetlem12N 37121

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