dihjat5N - Mathbox for Norm Megill

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Theorem dihjat5N 39188

Description: Transfer lattice join with atom to subspace sum. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
dihjat5.b	⊢ 𝐵 = (Base‘𝐾)
dihjat5.h	⊢ 𝐻 = (LHyp‘𝐾)
dihjat5.j	⊢ ∨ = (join‘𝐾)
dihjat5.a	⊢ 𝐴 = (Atoms‘𝐾)
dihjat5.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihjat5.s	⊢ ⊕ = (LSSum‘𝑈)
dihjat5.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihjat5.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dihjat5.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
dihjat5.p	⊢ (𝜑 → 𝑃 ∈ 𝐴)

**Assertion**
Ref	Expression
dihjat5N	⊢ (𝜑 → (𝑋 ∨ 𝑃) = (^◡𝐼‘((𝐼‘𝑋) ⊕ (𝐼‘𝑃))))

**Proof of Theorem dihjat5N**
Step	Hyp	Ref	Expression
1		dihjat5.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		dihjat5.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
3		dihjat5.j	. . . 4 ⊢ ∨ = (join‘𝐾)
4		dihjat5.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
5		dihjat5.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6		dihjat5.s	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
7		dihjat5.i	. . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
8		dihjat5.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9		dihjat5.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10		dihjat5.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	dihjat3 39183	. . 3 ⊢ (𝜑 → (𝐼‘(𝑋 ∨ 𝑃)) = ((𝐼‘𝑋) ⊕ (𝐼‘𝑃)))
12		eqid 2737	. . . . 5 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
13	1, 2, 7	dihcl 39021	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ ran 𝐼)
14	8, 9, 13	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑋) ∈ ran 𝐼)
15	4, 2, 5, 7, 12	dihatlat 39085	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴) → (𝐼‘𝑃) ∈ (LSAtoms‘𝑈))
16	8, 10, 15	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (LSAtoms‘𝑈))
17	2, 7, 5, 6, 12, 8, 14, 16	dihsmatrn 39187	. . . 4 ⊢ (𝜑 → ((𝐼‘𝑋) ⊕ (𝐼‘𝑃)) ∈ ran 𝐼)
18	2, 7	dihcnvid2 39024	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐼‘𝑋) ⊕ (𝐼‘𝑃)) ∈ ran 𝐼) → (𝐼‘(^◡𝐼‘((𝐼‘𝑋) ⊕ (𝐼‘𝑃)))) = ((𝐼‘𝑋) ⊕ (𝐼‘𝑃)))
19	8, 17, 18	syl2anc 587	. . 3 ⊢ (𝜑 → (𝐼‘(^◡𝐼‘((𝐼‘𝑋) ⊕ (𝐼‘𝑃)))) = ((𝐼‘𝑋) ⊕ (𝐼‘𝑃)))
20	11, 19	eqtr4d 2780	. 2 ⊢ (𝜑 → (𝐼‘(𝑋 ∨ 𝑃)) = (𝐼‘(^◡𝐼‘((𝐼‘𝑋) ⊕ (𝐼‘𝑃)))))
21	8	simpld 498	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL)
22	21	hllatd 37115	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat)
23	1, 4	atbase 37040	. . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
24	10, 23	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
25	1, 3	latjcl 17945	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → (𝑋 ∨ 𝑃) ∈ 𝐵)
26	22, 9, 24, 25	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝑋 ∨ 𝑃) ∈ 𝐵)
27	1, 2, 7	dihcnvcl 39022	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐼‘𝑋) ⊕ (𝐼‘𝑃)) ∈ ran 𝐼) → (^◡𝐼‘((𝐼‘𝑋) ⊕ (𝐼‘𝑃))) ∈ 𝐵)
28	8, 17, 27	syl2anc 587	. . 3 ⊢ (𝜑 → (^◡𝐼‘((𝐼‘𝑋) ⊕ (𝐼‘𝑃))) ∈ 𝐵)
29	1, 2, 7	dih11 39016	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∨ 𝑃) ∈ 𝐵 ∧ (^◡𝐼‘((𝐼‘𝑋) ⊕ (𝐼‘𝑃))) ∈ 𝐵) → ((𝐼‘(𝑋 ∨ 𝑃)) = (𝐼‘(^◡𝐼‘((𝐼‘𝑋) ⊕ (𝐼‘𝑃)))) ↔ (𝑋 ∨ 𝑃) = (^◡𝐼‘((𝐼‘𝑋) ⊕ (𝐼‘𝑃)))))
30	8, 26, 28, 29	syl3anc 1373	. 2 ⊢ (𝜑 → ((𝐼‘(𝑋 ∨ 𝑃)) = (𝐼‘(^◡𝐼‘((𝐼‘𝑋) ⊕ (𝐼‘𝑃)))) ↔ (𝑋 ∨ 𝑃) = (^◡𝐼‘((𝐼‘𝑋) ⊕ (𝐼‘𝑃)))))
31	20, 30	mpbid 235	1 ⊢ (𝜑 → (𝑋 ∨ 𝑃) = (^◡𝐼‘((𝐼‘𝑋) ⊕ (𝐼‘𝑃))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ^◡ccnv 5550 ran crn 5552 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 joincjn 17818 Latclat 17937 LSSumclsm 19023 LSAtomsclsa 36725 Atomscatm 37014 HLchlt 37101 LHypclh 37735 DVecHcdvh 38829 DIsoHcdih 38979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-riotaBAD 36704

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-tpos 7968 df-undef 8015 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-0g 16946 df-proset 17802 df-poset 17820 df-plt 17836 df-lub 17852 df-glb 17853 df-join 17854 df-meet 17855 df-p0 17931 df-p1 17932 df-lat 17938 df-clat 18005 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-subg 18540 df-cntz 18711 df-lsm 19025 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-dvr 19701 df-drng 19769 df-lmod 19901 df-lss 19969 df-lsp 20009 df-lvec 20140 df-lsatoms 36727 df-oposet 36927 df-ol 36929 df-oml 36930 df-covers 37017 df-ats 37018 df-atl 37049 df-cvlat 37073 df-hlat 37102 df-llines 37249 df-lplanes 37250 df-lvols 37251 df-lines 37252 df-psubsp 37254 df-pmap 37255 df-padd 37547 df-lhyp 37739 df-laut 37740 df-ldil 37855 df-ltrn 37856 df-trl 37910 df-tgrp 38494 df-tendo 38506 df-edring 38508 df-dveca 38754 df-disoa 38780 df-dvech 38830 df-dib 38890 df-dic 38924 df-dih 38980 df-doch 39099 df-djh 39146

This theorem is referenced by: (None)

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