dihjatc - Mathbox for Norm Megill

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Theorem dihjatc 39439

Description: Isomorphism H of lattice join of an element under the fiducial hyperplane with atom not under it. (Contributed by NM, 26-Aug-2014.)

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

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

**Proof of Theorem dihjatc**
Step	Hyp	Ref	Expression
1		dihjatc.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2	1	simpld 495	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL)
3		hlop 37384	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
4	2, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ OP)
5		dihjatc.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
6		eqid 2738	. . . . 5 ⊢ (1.‘𝐾) = (1.‘𝐾)
7	5, 6	op1cl 37207	. . . 4 ⊢ (𝐾 ∈ OP → (1.‘𝐾) ∈ 𝐵)
8	4, 7	syl 17	. . 3 ⊢ (𝜑 → (1.‘𝐾) ∈ 𝐵)
9		dihjatc.x	. . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
10	9	simpld 495	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11		dihjatc.p	. . 3 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
12	11	simpld 495	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
13		dihjatc.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
14	5, 13	atbase 37311	. . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
15	12, 14	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
16		dihjatc.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
17	5, 16, 6	ople1 37213	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑃 ∈ 𝐵) → 𝑃 ≤ (1.‘𝐾))
18	4, 15, 17	syl2anc 584	. . 3 ⊢ (𝜑 → 𝑃 ≤ (1.‘𝐾))
19		hlol 37383	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
20	2, 19	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ OL)
21		eqid 2738	. . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾)
22	5, 21, 6	olm12 37250	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ((1.‘𝐾)(meet‘𝐾)𝑋) = 𝑋)
23	20, 10, 22	syl2anc 584	. . . 4 ⊢ (𝜑 → ((1.‘𝐾)(meet‘𝐾)𝑋) = 𝑋)
24	9	simprd 496	. . . 4 ⊢ (𝜑 → 𝑋 ≤ 𝑊)
25	23, 24	eqbrtrd 5095	. . 3 ⊢ (𝜑 → ((1.‘𝐾)(meet‘𝐾)𝑋) ≤ 𝑊)
26		dihjatc.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
27		dihjatc.j	. . . 4 ⊢ ∨ = (join‘𝐾)
28		dihjatc.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
29		dihjatc.s	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
30		dihjatc.i	. . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
31	5, 16, 26, 27, 21, 13, 28, 29, 30	dihjatc3 39335	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1.‘𝐾) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑃 ≤ (1.‘𝐾) ∧ ((1.‘𝐾)(meet‘𝐾)𝑋) ≤ 𝑊)) → (𝐼‘(((1.‘𝐾)(meet‘𝐾)𝑋) ∨ 𝑃)) = ((𝐼‘((1.‘𝐾)(meet‘𝐾)𝑋)) ⊕ (𝐼‘𝑃)))
32	1, 8, 10, 11, 18, 25, 31	syl312anc 1390	. 2 ⊢ (𝜑 → (𝐼‘(((1.‘𝐾)(meet‘𝐾)𝑋) ∨ 𝑃)) = ((𝐼‘((1.‘𝐾)(meet‘𝐾)𝑋)) ⊕ (𝐼‘𝑃)))
33	23	fvoveq1d 7289	. 2 ⊢ (𝜑 → (𝐼‘(((1.‘𝐾)(meet‘𝐾)𝑋) ∨ 𝑃)) = (𝐼‘(𝑋 ∨ 𝑃)))
34	23	fveq2d 6770	. . 3 ⊢ (𝜑 → (𝐼‘((1.‘𝐾)(meet‘𝐾)𝑋)) = (𝐼‘𝑋))
35	34	oveq1d 7282	. 2 ⊢ (𝜑 → ((𝐼‘((1.‘𝐾)(meet‘𝐾)𝑋)) ⊕ (𝐼‘𝑃)) = ((𝐼‘𝑋) ⊕ (𝐼‘𝑃)))
36	32, 33, 35	3eqtr3d 2786	1 ⊢ (𝜑 → (𝐼‘(𝑋 ∨ 𝑃)) = ((𝐼‘𝑋) ⊕ (𝐼‘𝑃)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 class class class wbr 5073 ‘cfv 6426 (class class class)co 7267 Basecbs 16922 lecple 16979 joincjn 18039 meetcmee 18040 1.cp1 18152 LSSumclsm 19249 OPcops 37194 OLcol 37196 Atomscatm 37285 HLchlt 37372 LHypclh 38006 DVecHcdvh 39100 DIsoHcdih 39250

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-riotaBAD 36975

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-tpos 8029 df-undef 8076 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-map 8604 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-n0 12244 df-z 12330 df-uz 12593 df-fz 13250 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-sca 16988 df-vsca 16989 df-0g 17162 df-proset 18023 df-poset 18041 df-plt 18058 df-lub 18074 df-glb 18075 df-join 18076 df-meet 18077 df-p0 18153 df-p1 18154 df-lat 18160 df-clat 18227 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-submnd 18441 df-grp 18590 df-minusg 18591 df-sbg 18592 df-subg 18762 df-cntz 18933 df-lsm 19251 df-cmn 19398 df-abl 19399 df-mgp 19731 df-ur 19748 df-ring 19795 df-oppr 19872 df-dvdsr 19893 df-unit 19894 df-invr 19924 df-dvr 19935 df-drng 20003 df-lmod 20135 df-lss 20204 df-lsp 20244 df-lvec 20375 df-oposet 37198 df-ol 37200 df-oml 37201 df-covers 37288 df-ats 37289 df-atl 37320 df-cvlat 37344 df-hlat 37373 df-llines 37520 df-lplanes 37521 df-lvols 37522 df-lines 37523 df-psubsp 37525 df-pmap 37526 df-padd 37818 df-lhyp 38010 df-laut 38011 df-ldil 38126 df-ltrn 38127 df-trl 38181 df-tendo 38777 df-edring 38779 df-disoa 39051 df-dvech 39101 df-dib 39161 df-dic 39195 df-dih 39251

This theorem is referenced by: dihjat 39445 dihprrnlem1N 39446

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