lclkrlem2c - Mathbox for Norm Megill

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Theorem lclkrlem2c 42673

Description: Lemma for lclkr 42697. (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 2760	. . . 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 2760	. . . 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 4050	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
12		lclkrlem2a.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
13	1, 3, 4, 12, 2	dihlsprn 42495	. . . . . 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 42274	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
17		lclkrlem2a.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
18	4, 12, 15, 9, 16, 17	lsatlspsn 40157	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ 𝐴)
19	1, 2, 3, 8, 9, 7, 14, 18	dihsmatrn 42600	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
20		lclkrlem2a.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
21	20	eldifad 4050	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
22	21	snssd 4743	. . . . 5 ⊢ (𝜑 → {𝐵} ⊆ 𝑉)
23	1, 2, 3, 4, 5	dochcl 42517	. . . . 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 42574	. . 3 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵}))) = (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘{𝐵}))))
26	17	eldifad 4050	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
27	4, 12, 8, 16, 11, 26	lsmpr 21371	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
28		df-pr 4587	. . . . . . . 8 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
29	28	fveq2i 6891	. . . . . . 7 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌}))
30	27, 29	eqtr3di 2810	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = (𝑁‘({𝑋} ∪ {𝑌})))
31	30	fveq2d 6892	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) = ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌}))))
32	11	snssd 4743	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
33	26	snssd 4743	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
34	32, 33	unssd 4189	. . . . . 6 ⊢ (𝜑 → ({𝑋} ∪ {𝑌}) ⊆ 𝑉)
35	1, 3, 5, 4, 12, 7, 34	dochocsp 42543	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌}))) = ( ⊥ ‘({𝑋} ∪ {𝑌})))
36	1, 3, 4, 5	dochdmj1 42554	. . . . . 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 2799	. . . 4 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
39	1, 3, 5, 4, 12, 7, 21	dochocsn 42545	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{𝐵})) = (𝑁‘{𝐵}))
40	38, 39	oveq12d 7440	. . 3 ⊢ (𝜑 → (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘{𝐵}))) = ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌}))((joinH‘𝐾)‘𝑊)(𝑁‘{𝐵})))
41	1, 2, 3, 4, 5	dochcl 42517	. . . . . 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 42517	. . . . . 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 42509	. . . . 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 42593	. . 3 ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌}))((joinH‘𝐾)‘𝑊)(𝑁‘{𝐵})) = ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})))
48	25, 40, 47	3eqtrrd 2800	. 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 42672	. . 3 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)
53	1, 3, 5, 9, 49, 7, 52	dochsatshp 42615	. 2 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵}))) ∈ 𝐽)
54	48, 53	eqeltrd 2860	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 1569 ∈ wcel 2144 ≠ wne 2955 ⊆ wss 3896 ∖ cdif 4044 ∩ cin 4088 ∪ cun 4148 {csn 4584 {cpr 4586 ran crn 5651 ‘cfv 6541 (class class class)co 7422 Basecbs 17394 0_gc0g 17617 LSSumclsm 19855 LSpanclspn 21253 LSAtomsclsa 40138 LSHypclsh 40139 HLchlt 40514 LHypclh 41148 DVecHcdvh 42242 DIsoHcdih 42392 ocHcoch 42511 joinHcdjh 42558

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-cnex 11261 ax-resscn 11262 ax-1cn 11263 ax-icn 11264 ax-addcl 11265 ax-addrcl 11266 ax-mulcl 11267 ax-mulrcl 11268 ax-mulcom 11269 ax-addass 11270 ax-mulass 11271 ax-distr 11272 ax-i2m1 11273 ax-1ne0 11274 ax-1rid 11275 ax-rnegex 11276 ax-rrecex 11277 ax-cnre 11278 ax-pre-lttri 11279 ax-pre-lttrn 11280 ax-pre-ltadd 11281 ax-pre-mulgt0 11282 ax-riotaBAD 40117

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7379 df-ov 7425 df-oprab 7426 df-mpo 7427 df-om 7880 df-1st 8003 df-2nd 8004 df-tpos 8248 df-undef 8295 df-frecs 8304 df-wrecs 8335 df-recs 8384 df-rdg 8423 df-1o 8481 df-2o 8482 df-er 8722 df-map 8854 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-pnf 11350 df-mnf 11351 df-xr 11352 df-ltxr 11353 df-le 11354 df-sub 11548 df-neg 11549 df-nn 12341 df-2 12410 df-3 12411 df-4 12412 df-5 12413 df-6 12414 df-n0 12612 df-z 12699 df-uz 12971 df-fz 13645 df-struct 17332 df-sets 17349 df-slot 17367 df-ndx 17379 df-base 17395 df-ress 17416 df-plusg 17448 df-mulr 17449 df-sca 17451 df-vsca 17452 df-0g 17619 df-mre 17763 df-mrc 17764 df-acs 17766 df-proset 18475 df-poset 18494 df-plt 18509 df-lub 18525 df-glb 18526 df-join 18527 df-meet 18528 df-p0 18604 df-p1 18605 df-lat 18613 df-clat 18680 df-mgm 18823 df-sgrp 18915 df-mnd 18931 df-submnd 18986 df-grp 19154 df-minusg 19155 df-sbg 19156 df-subg 19340 df-cntz 19538 df-oppg 19567 df-lsm 19857 df-cmn 20003 df-abl 20004 df-mgp 20368 df-rng 20382 df-ur 20415 df-ring 20468 df-oppr 20574 df-dvdsr 20594 df-unit 20595 df-invr 20625 df-dvr 20638 df-drng 20989 df-lmod 21144 df-lss 21214 df-lsp 21254 df-lvec 21385 df-lsatoms 40140 df-lshyp 40141 df-lcv 40183 df-oposet 40340 df-ol 40342 df-oml 40343 df-covers 40430 df-ats 40431 df-atl 40462 df-cvlat 40486 df-hlat 40515 df-llines 40662 df-lplanes 40663 df-lvols 40664 df-lines 40665 df-psubsp 40667 df-pmap 40668 df-padd 40960 df-lhyp 41152 df-laut 41153 df-ldil 41268 df-ltrn 41269 df-trl 41323 df-tgrp 41907 df-tendo 41919 df-edring 41921 df-dveca 42167 df-disoa 42193 df-dvech 42243 df-dib 42303 df-dic 42337 df-dih 42393 df-doch 42512 df-djh 42559

This theorem is used by: lclkrlem2g 42677

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