lclkrlem2c - Mathbox for Norm Megill

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Theorem lclkrlem2c 40874

Description: Lemma for lclkr 40898. (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 2724	. . . 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 2724	. . . 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 3953	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
12		lclkrlem2a.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
13	1, 3, 4, 12, 2	dihlsprn 40696	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
14	7, 11, 13	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
15		lclkrlem2a.z	. . . . . 6 ⊢ 0 = (0_g‘𝑈)
16	1, 3, 7	dvhlmod 40475	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
17		lclkrlem2a.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
18	4, 12, 15, 9, 16, 17	lsatlspsn 38357	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ 𝐴)
19	1, 2, 3, 8, 9, 7, 14, 18	dihsmatrn 40801	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
20		lclkrlem2a.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
21	20	eldifad 3953	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
22	21	snssd 4805	. . . . 5 ⊢ (𝜑 → {𝐵} ⊆ 𝑉)
23	1, 2, 3, 4, 5	dochcl 40718	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐵} ⊆ 𝑉) → ( ⊥ ‘{𝐵}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
24	7, 22, 23	syl2anc 583	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝐵}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
25	1, 2, 3, 4, 5, 6, 7, 19, 24	dochdmm1 40775	. . 3 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵}))) = (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘{𝐵}))))
26	17	eldifad 3953	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
27	4, 12, 8, 16, 11, 26	lsmpr 20929	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
28		df-pr 4624	. . . . . . . 8 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
29	28	fveq2i 6885	. . . . . . 7 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌}))
30	27, 29	eqtr3di 2779	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = (𝑁‘({𝑋} ∪ {𝑌})))
31	30	fveq2d 6886	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) = ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌}))))
32	11	snssd 4805	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
33	26	snssd 4805	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
34	32, 33	unssd 4179	. . . . . 6 ⊢ (𝜑 → ({𝑋} ∪ {𝑌}) ⊆ 𝑉)
35	1, 3, 5, 4, 12, 7, 34	dochocsp 40744	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌}))) = ( ⊥ ‘({𝑋} ∪ {𝑌})))
36	1, 3, 4, 5	dochdmj1 40755	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
37	7, 32, 33, 36	syl3anc 1368	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
38	31, 35, 37	3eqtrd 2768	. . . 4 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
39	1, 3, 5, 4, 12, 7, 21	dochocsn 40746	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{𝐵})) = (𝑁‘{𝐵}))
40	38, 39	oveq12d 7420	. . 3 ⊢ (𝜑 → (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘{𝐵}))) = ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌}))((joinH‘𝐾)‘𝑊)(𝑁‘{𝐵})))
41	1, 2, 3, 4, 5	dochcl 40718	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
42	7, 32, 41	syl2anc 583	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
43	1, 2, 3, 4, 5	dochcl 40718	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
44	7, 33, 43	syl2anc 583	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
45	1, 2	dihmeetcl 40710	. . . . 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 834	. . . 4 ⊢ (𝜑 → (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
47	1, 3, 4, 8, 12, 2, 6, 7, 46, 21	dihjat1 40794	. . 3 ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌}))((joinH‘𝐾)‘𝑊)(𝑁‘{𝐵})) = ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})))
48	25, 40, 47	3eqtrrd 2769	. 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 40873	. . 3 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)
53	1, 3, 5, 9, 49, 7, 52	dochsatshp 40816	. 2 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵}))) ∈ 𝐽)
54	48, 53	eqeltrd 2825	1 ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})) ∈ 𝐽)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∖ cdif 3938 ∪ cun 3939 ∩ cin 3940 ⊆ wss 3941 {csn 4621 {cpr 4623 ran crn 5668 ‘cfv 6534 (class class class)co 7402 Basecbs 17145 0_gc0g 17386 LSSumclsm 19546 LSpanclspn 20810 LSAtomsclsa 38338 LSHypclsh 38339 HLchlt 38714 LHypclh 39349 DVecHcdvh 40443 DIsoHcdih 40593 ocHcoch 40712 joinHcdjh 40759

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-riotaBAD 38317

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-n0 12471 df-z 12557 df-uz 12821 df-fz 13483 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-sca 17214 df-vsca 17215 df-0g 17388 df-mre 17531 df-mrc 17532 df-acs 17534 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19042 df-cntz 19225 df-oppg 19254 df-lsm 19548 df-cmn 19694 df-abl 19695 df-mgp 20032 df-rng 20050 df-ur 20079 df-ring 20132 df-oppr 20228 df-dvdsr 20251 df-unit 20252 df-invr 20282 df-dvr 20295 df-drng 20581 df-lmod 20700 df-lss 20771 df-lsp 20811 df-lvec 20943 df-lsatoms 38340 df-lshyp 38341 df-lcv 38383 df-oposet 38540 df-ol 38542 df-oml 38543 df-covers 38630 df-ats 38631 df-atl 38662 df-cvlat 38686 df-hlat 38715 df-llines 38863 df-lplanes 38864 df-lvols 38865 df-lines 38866 df-psubsp 38868 df-pmap 38869 df-padd 39161 df-lhyp 39353 df-laut 39354 df-ldil 39469 df-ltrn 39470 df-trl 39524 df-tgrp 40108 df-tendo 40120 df-edring 40122 df-dveca 40368 df-disoa 40394 df-dvech 40444 df-dib 40504 df-dic 40538 df-dih 40594 df-doch 40713 df-djh 40760

This theorem is referenced by: lclkrlem2g 40878

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