djhljjN - Mathbox for Norm Megill

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Theorem djhljjN 41420

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 41419	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐼‘(𝑋 ∨ 𝑌)) = ((𝐼‘𝑋)𝐽(𝐼‘𝑌)))
10	1, 2, 3, 9	syl12anc 836	. . 3 ⊢ (𝜑 → (𝐼‘(𝑋 ∨ 𝑌)) = ((𝐼‘𝑋)𝐽(𝐼‘𝑌)))
11	4, 6, 7	dihcl 41288	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ ran 𝐼)
12	1, 2, 11	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑋) ∈ ran 𝐼)
13		eqid 2730	. . . . . . 7 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
14		eqid 2730	. . . . . . 7 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊))
15	6, 13, 7, 14	dihrnss 41296	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐼‘𝑋) ∈ ran 𝐼) → (𝐼‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
16	1, 12, 15	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
17	4, 6, 7	dihcl 41288	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵) → (𝐼‘𝑌) ∈ ran 𝐼)
18	1, 3, 17	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑌) ∈ ran 𝐼)
19	6, 13, 7, 14	dihrnss 41296	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐼‘𝑌) ∈ ran 𝐼) → (𝐼‘𝑌) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
20	1, 18, 19	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑌) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
21	6, 7, 13, 14, 8	djhcl 41418	. . . . 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 836	. . . 4 ⊢ (𝜑 → ((𝐼‘𝑋)𝐽(𝐼‘𝑌)) ∈ ran 𝐼)
23	6, 7	dihcnvid2 41291	. . . 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 584	. . 3 ⊢ (𝜑 → (𝐼‘(^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))) = ((𝐼‘𝑋)𝐽(𝐼‘𝑌)))
25	10, 24	eqtr4d 2768	. 2 ⊢ (𝜑 → (𝐼‘(𝑋 ∨ 𝑌)) = (𝐼‘(^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))))
26	1	simpld 494	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL)
27	26	hllatd 39382	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat)
28	4, 5	latjcl 18337	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵)
29	27, 2, 3, 28	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝑋 ∨ 𝑌) ∈ 𝐵)
30	4, 6, 7	dihcnvcl 41289	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐼‘𝑋)𝐽(𝐼‘𝑌)) ∈ ran 𝐼) → (^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌))) ∈ 𝐵)
31	1, 22, 30	syl2anc 584	. . 3 ⊢ (𝜑 → (^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌))) ∈ 𝐵)
32	4, 6, 7	dih11 41283	. . 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 1373	. 2 ⊢ (𝜑 → ((𝐼‘(𝑋 ∨ 𝑌)) = (𝐼‘(^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))) ↔ (𝑋 ∨ 𝑌) = (^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))))
34	25, 33	mpbid 232	1 ⊢ (𝜑 → (𝑋 ∨ 𝑌) = (^◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2110 ⊆ wss 3900 ^◡ccnv 5613 ran crn 5615 ‘cfv 6477 (class class class)co 7341 Basecbs 17112 joincjn 18209 Latclat 18329 HLchlt 39368 LHypclh 40002 DVecHcdvh 41096 DIsoHcdih 41246 joinHcdjh 41412

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-riotaBAD 38971

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-tpos 8151 df-undef 8198 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-er 8617 df-map 8747 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-n0 12374 df-z 12461 df-uz 12725 df-fz 13400 df-struct 17050 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-mulr 17167 df-sca 17169 df-vsca 17170 df-0g 17337 df-proset 18192 df-poset 18211 df-plt 18226 df-lub 18242 df-glb 18243 df-join 18244 df-meet 18245 df-p0 18321 df-p1 18322 df-lat 18330 df-clat 18397 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-submnd 18684 df-grp 18841 df-minusg 18842 df-sbg 18843 df-subg 19028 df-cntz 19222 df-lsm 19541 df-cmn 19687 df-abl 19688 df-mgp 20052 df-rng 20064 df-ur 20093 df-ring 20146 df-oppr 20248 df-dvdsr 20268 df-unit 20269 df-invr 20299 df-dvr 20312 df-drng 20639 df-lmod 20788 df-lss 20858 df-lsp 20898 df-lvec 21030 df-lsatoms 38994 df-oposet 39194 df-ol 39196 df-oml 39197 df-covers 39284 df-ats 39285 df-atl 39316 df-cvlat 39340 df-hlat 39369 df-llines 39516 df-lplanes 39517 df-lvols 39518 df-lines 39519 df-psubsp 39521 df-pmap 39522 df-padd 39814 df-lhyp 40006 df-laut 40007 df-ldil 40122 df-ltrn 40123 df-trl 40177 df-tendo 40773 df-edring 40775 df-disoa 41047 df-dvech 41097 df-dib 41157 df-dic 41191 df-dih 41247 df-doch 41366 df-djh 41413

This theorem is referenced by: (None)

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