djhljjN - Mathbox for Norm Megill

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Theorem djhljjN 37212

Description: Lattice join in terms of DVecH vector space closed subspace join. (Contributed by NM, 17-Aug-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
djhlj.b	⊢ 𝐵 = (Base‘𝐾)
djhlj.k	⊢ ∨ = (join‘𝐾)
djhlj.h	⊢ 𝐻 = (LHyp‘𝐾)
djhlj.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
djhlj.j	⊢ 𝐽 = ((joinH‘𝐾)‘𝑊)
djhljj.w	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
djhljj.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
djhljj.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

**Assertion**
Ref	Expression
djhljjN	⊢ (𝜑 → (𝑋 ∨ 𝑌) = (^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌))))

**Proof of Theorem djhljjN**
Step	Hyp	Ref	Expression
1		djhljj.w	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		djhljj.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		djhljj.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
4		djhlj.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
5		djhlj.k	. . . . 5 ⊢ ∨ = (join‘𝐾)
6		djhlj.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
7		djhlj.i	. . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
8		djhlj.j	. . . . 5 ⊢ 𝐽 = ((joinH‘𝐾)‘𝑊)
9	4, 5, 6, 7, 8	djhlj 37211	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐼‘(𝑋 ∨ 𝑌)) = ((𝐼‘𝑋)𝐽(𝐼‘𝑌)))
10	1, 2, 3, 9	syl12anc 1474	. . 3 ⊢ (𝜑 → (𝐼‘(𝑋 ∨ 𝑌)) = ((𝐼‘𝑋)𝐽(𝐼‘𝑌)))
11	4, 6, 7	dihcl 37080	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ ran 𝐼)
12	1, 2, 11	syl2anc 573	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑋) ∈ ran 𝐼)
13		eqid 2771	. . . . . . 7 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
14		eqid 2771	. . . . . . 7 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊))
15	6, 13, 7, 14	dihrnss 37088	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐼‘𝑋) ∈ ran 𝐼) → (𝐼‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
16	1, 12, 15	syl2anc 573	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
17	4, 6, 7	dihcl 37080	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵) → (𝐼‘𝑌) ∈ ran 𝐼)
18	1, 3, 17	syl2anc 573	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑌) ∈ ran 𝐼)
19	6, 13, 7, 14	dihrnss 37088	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐼‘𝑌) ∈ ran 𝐼) → (𝐼‘𝑌) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
20	1, 18, 19	syl2anc 573	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑌) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
21	6, 7, 13, 14, 8	djhcl 37210	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐼‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ (𝐼‘𝑌) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))) → ((𝐼‘𝑋)𝐽(𝐼‘𝑌)) ∈ ran 𝐼)
22	1, 16, 20, 21	syl12anc 1474	. . . 4 ⊢ (𝜑 → ((𝐼‘𝑋)𝐽(𝐼‘𝑌)) ∈ ran 𝐼)
23	6, 7	dihcnvid2 37083	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐼‘𝑋)𝐽(𝐼‘𝑌)) ∈ ran 𝐼) → (𝐼‘(^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))) = ((𝐼‘𝑋)𝐽(𝐼‘𝑌)))
24	1, 22, 23	syl2anc 573	. . 3 ⊢ (𝜑 → (𝐼‘(^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))) = ((𝐼‘𝑋)𝐽(𝐼‘𝑌)))
25	10, 24	eqtr4d 2808	. 2 ⊢ (𝜑 → (𝐼‘(𝑋 ∨ 𝑌)) = (𝐼‘(^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))))
26	1	simpld 482	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL)
27	26	hllatd 35173	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat)
28	4, 5	latjcl 17259	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵)
29	27, 2, 3, 28	syl3anc 1476	. . 3 ⊢ (𝜑 → (𝑋 ∨ 𝑌) ∈ 𝐵)
30	4, 6, 7	dihcnvcl 37081	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐼‘𝑋)𝐽(𝐼‘𝑌)) ∈ ran 𝐼) → (^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌))) ∈ 𝐵)
31	1, 22, 30	syl2anc 573	. . 3 ⊢ (𝜑 → (^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌))) ∈ 𝐵)
32	4, 6, 7	dih11 37075	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ (^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌))) ∈ 𝐵) → ((𝐼‘(𝑋 ∨ 𝑌)) = (𝐼‘(^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))) ↔ (𝑋 ∨ 𝑌) = (^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))))
33	1, 29, 31, 32	syl3anc 1476	. 2 ⊢ (𝜑 → ((𝐼‘(𝑋 ∨ 𝑌)) = (𝐼‘(^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))) ↔ (𝑋 ∨ 𝑌) = (^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))))
34	25, 33	mpbid 222	1 ⊢ (𝜑 → (𝑋 ∨ 𝑌) = (^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ⊆ wss 3723 ^◡ccnv 5249 ran crn 5251 ‘cfv 6030 (class class class)co 6796 Basecbs 16064 joincjn 17152 Latclat 17253 HLchlt 35159 LHypclh 35793 DVecHcdvh 36888 DIsoHcdih 37038 joinHcdjh 37204

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 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-riotaBAD 34761

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 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-iin 4658 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-tpos 7508 df-undef 7555 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-er 7900 df-map 8015 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-nn 11227 df-2 11285 df-3 11286 df-4 11287 df-5 11288 df-6 11289 df-n0 11500 df-z 11585 df-uz 11894 df-fz 12534 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-sca 16165 df-vsca 16166 df-0g 16310 df-preset 17136 df-poset 17154 df-plt 17166 df-lub 17182 df-glb 17183 df-join 17184 df-meet 17185 df-p0 17247 df-p1 17248 df-lat 17254 df-clat 17316 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-grp 17633 df-minusg 17634 df-sbg 17635 df-subg 17799 df-cntz 17957 df-lsm 18258 df-cmn 18402 df-abl 18403 df-mgp 18698 df-ur 18710 df-ring 18757 df-oppr 18831 df-dvdsr 18849 df-unit 18850 df-invr 18880 df-dvr 18891 df-drng 18959 df-lmod 19075 df-lss 19143 df-lsp 19185 df-lvec 19316 df-lsatoms 34785 df-oposet 34985 df-ol 34987 df-oml 34988 df-covers 35075 df-ats 35076 df-atl 35107 df-cvlat 35131 df-hlat 35160 df-llines 35307 df-lplanes 35308 df-lvols 35309 df-lines 35310 df-psubsp 35312 df-pmap 35313 df-padd 35605 df-lhyp 35797 df-laut 35798 df-ldil 35913 df-ltrn 35914 df-trl 35969 df-tendo 36565 df-edring 36567 df-disoa 36839 df-dvech 36889 df-dib 36949 df-dic 36983 df-dih 37039 df-doch 37158 df-djh 37205

This theorem is referenced by: (None)

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