lclkrlem2c - Mathbox for Norm Megill

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Theorem lclkrlem2c 42316

Description: Lemma for lclkr 42340. (Contributed by NM, 16-Jan-2015.)

**Hypotheses**
Ref	Expression
lclkrlem2a.h	⊢ 𝐻 = (LHyp‘𝐾)
lclkrlem2a.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lclkrlem2a.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lclkrlem2a.v	⊢ 𝑉 = (Base‘𝑈)
lclkrlem2a.z	⊢ 0 = (0_g‘𝑈)
lclkrlem2a.p	⊢ ⊕ = (LSSum‘𝑈)
lclkrlem2a.n	⊢ 𝑁 = (LSpan‘𝑈)
lclkrlem2a.a	⊢ 𝐴 = (LSAtoms‘𝑈)
lclkrlem2a.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lclkrlem2a.b	⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
lclkrlem2a.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lclkrlem2a.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
lclkrlem2a.e	⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))
lclkrlem2b.da	⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
lclkrlem2c.j	⊢ 𝐽 = (LSHyp‘𝑈)

**Assertion**
Ref	Expression
lclkrlem2c	⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})) ∈ 𝐽)

**Proof of Theorem lclkrlem2c**
Step	Hyp	Ref	Expression
1		lclkrlem2a.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		eqid 2765	. . . 4 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
3		lclkrlem2a.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lclkrlem2a.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
5		lclkrlem2a.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
6		eqid 2765	. . . 4 ⊢ ((joinH‘𝐾)‘𝑊) = ((joinH‘𝐾)‘𝑊)
7		lclkrlem2a.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8		lclkrlem2a.p	. . . . 5 ⊢ ⊕ = (LSSum‘𝑈)
9		lclkrlem2a.a	. . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑈)
10		lclkrlem2a.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
11	10	eldifad 3918	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
12		lclkrlem2a.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
13	1, 3, 4, 12, 2	dihlsprn 42138	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
14	7, 11, 13	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
15		lclkrlem2a.z	. . . . . 6 ⊢ 0 = (0_g‘𝑈)
16	1, 3, 7	dvhlmod 41917	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
17		lclkrlem2a.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
18	4, 12, 15, 9, 16, 17	lsatlspsn 39800	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ 𝐴)
19	1, 2, 3, 8, 9, 7, 14, 18	dihsmatrn 42243	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
20		lclkrlem2a.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
21	20	eldifad 3918	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
22	21	snssd 4754	. . . . 5 ⊢ (𝜑 → {𝐵} ⊆ 𝑉)
23	1, 2, 3, 4, 5	dochcl 42160	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐵} ⊆ 𝑉) → ( ⊥ ‘{𝐵}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
24	7, 22, 23	syl2anc 596	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝐵}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
25	1, 2, 3, 4, 5, 6, 7, 19, 24	dochdmm1 42217	. . 3 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵}))) = (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘{𝐵}))))
26	17	eldifad 3918	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
27	4, 12, 8, 16, 11, 26	lsmpr 21239	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
28		df-pr 4594	. . . . . . . 8 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
29	28	fveq2i 6888	. . . . . . 7 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌}))
30	27, 29	eqtr3di 2815	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = (𝑁‘({𝑋} ∪ {𝑌})))
31	30	fveq2d 6889	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) = ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌}))))
32	11	snssd 4754	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
33	26	snssd 4754	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
34	32, 33	unssd 4145	. . . . . 6 ⊢ (𝜑 → ({𝑋} ∪ {𝑌}) ⊆ 𝑉)
35	1, 3, 5, 4, 12, 7, 34	dochocsp 42186	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌}))) = ( ⊥ ‘({𝑋} ∪ {𝑌})))
36	1, 3, 4, 5	dochdmj1 42197	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
37	7, 32, 33, 36	syl3anc 1398	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
38	31, 35, 37	3eqtrd 2804	. . . 4 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
39	1, 3, 5, 4, 12, 7, 21	dochocsn 42188	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{𝐵})) = (𝑁‘{𝐵}))
40	38, 39	oveq12d 7434	. . 3 ⊢ (𝜑 → (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘{𝐵}))) = ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌}))((joinH‘𝐾)‘𝑊)(𝑁‘{𝐵})))
41	1, 2, 3, 4, 5	dochcl 42160	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
42	7, 32, 41	syl2anc 596	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
43	1, 2, 3, 4, 5	dochcl 42160	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
44	7, 33, 43	syl2anc 596	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
45	1, 2	dihmeetcl 42152	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊) ∧ ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))) → (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
46	7, 42, 44, 45	syl12anc 850	. . . 4 ⊢ (𝜑 → (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
47	1, 3, 4, 8, 12, 2, 6, 7, 46, 21	dihjat1 42236	. . 3 ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌}))((joinH‘𝐾)‘𝑊)(𝑁‘{𝐵})) = ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})))
48	25, 40, 47	3eqtrrd 2805	. 2 ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})) = ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵}))))
49		lclkrlem2c.j	. . 3 ⊢ 𝐽 = (LSHyp‘𝑈)
50		lclkrlem2a.e	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))
51		lclkrlem2b.da	. . . 4 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
52	1, 5, 3, 4, 15, 8, 12, 9, 7, 20, 10, 17, 50, 51	lclkrlem2b 42315	. . 3 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)
53	1, 3, 5, 9, 49, 7, 52	dochsatshp 42258	. 2 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵}))) ∈ 𝐽)
54	48, 53	eqeltrd 2865	1 ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})) ∈ 𝐽)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∖ cdif 3903 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 {csn 4591 {cpr 4593 ran crn 5664 ‘cfv 6540 (class class class)co 7416 Basecbs 17286 0_gc0g 17509 LSSumclsm 19727 LSpanclspn 21121 LSAtomsclsa 39781 LSHypclsh 39782 HLchlt 40157 LHypclh 40791 DVecHcdvh 41885 DIsoHcdih 42035 ocHcoch 42154 joinHcdjh 42201

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-riotaBAD 39760

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7865 df-1st 7988 df-2nd 7989 df-tpos 8224 df-undef 8271 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-n0 12516 df-z 12603 df-uz 12874 df-fz 13547 df-struct 17224 df-sets 17241 df-slot 17259 df-ndx 17271 df-base 17287 df-ress 17308 df-plusg 17340 df-mulr 17341 df-sca 17343 df-vsca 17344 df-0g 17511 df-mre 17655 df-mrc 17656 df-acs 17658 df-proset 18367 df-poset 18386 df-plt 18401 df-lub 18417 df-glb 18418 df-join 18419 df-meet 18420 df-p0 18496 df-p1 18497 df-lat 18505 df-clat 18572 df-mgm 18715 df-sgrp 18798 df-mnd 18814 df-submnd 18865 df-grp 19026 df-minusg 19027 df-sbg 19028 df-subg 19212 df-cntz 19410 df-oppg 19439 df-lsm 19729 df-cmn 19875 df-abl 19876 df-mgp 20240 df-rng 20254 df-ur 20287 df-ring 20340 df-oppr 20444 df-dvdsr 20464 df-unit 20465 df-invr 20495 df-dvr 20508 df-drng 20858 df-lmod 21012 df-lss 21082 df-lsp 21122 df-lvec 21253 df-lsatoms 39783 df-lshyp 39784 df-lcv 39826 df-oposet 39983 df-ol 39985 df-oml 39986 df-covers 40073 df-ats 40074 df-atl 40105 df-cvlat 40129 df-hlat 40158 df-llines 40305 df-lplanes 40306 df-lvols 40307 df-lines 40308 df-psubsp 40310 df-pmap 40311 df-padd 40603 df-lhyp 40795 df-laut 40796 df-ldil 40911 df-ltrn 40912 df-trl 40966 df-tgrp 41550 df-tendo 41562 df-edring 41564 df-dveca 41810 df-disoa 41836 df-dvech 41886 df-dib 41946 df-dic 41980 df-dih 42036 df-doch 42155 df-djh 42202

This theorem is used by: lclkrlem2g 42320

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