lclkrlem2c - Mathbox for Norm Megill

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Theorem lclkrlem2c 41879

Description: Lemma for lclkr 41903. (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 2737	. . . 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 2737	. . . 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 3915	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
12		lclkrlem2a.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
13	1, 3, 4, 12, 2	dihlsprn 41701	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
14	7, 11, 13	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
15		lclkrlem2a.z	. . . . . 6 ⊢ 0 = (0_g‘𝑈)
16	1, 3, 7	dvhlmod 41480	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
17		lclkrlem2a.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
18	4, 12, 15, 9, 16, 17	lsatlspsn 39363	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ 𝐴)
19	1, 2, 3, 8, 9, 7, 14, 18	dihsmatrn 41806	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
20		lclkrlem2a.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
21	20	eldifad 3915	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
22	21	snssd 4767	. . . . 5 ⊢ (𝜑 → {𝐵} ⊆ 𝑉)
23	1, 2, 3, 4, 5	dochcl 41723	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐵} ⊆ 𝑉) → ( ⊥ ‘{𝐵}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
24	7, 22, 23	syl2anc 585	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝐵}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
25	1, 2, 3, 4, 5, 6, 7, 19, 24	dochdmm1 41780	. . 3 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵}))) = (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘{𝐵}))))
26	17	eldifad 3915	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
27	4, 12, 8, 16, 11, 26	lsmpr 21053	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
28		df-pr 4585	. . . . . . . 8 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
29	28	fveq2i 6845	. . . . . . 7 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌}))
30	27, 29	eqtr3di 2787	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = (𝑁‘({𝑋} ∪ {𝑌})))
31	30	fveq2d 6846	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) = ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌}))))
32	11	snssd 4767	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
33	26	snssd 4767	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
34	32, 33	unssd 4146	. . . . . 6 ⊢ (𝜑 → ({𝑋} ∪ {𝑌}) ⊆ 𝑉)
35	1, 3, 5, 4, 12, 7, 34	dochocsp 41749	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌}))) = ( ⊥ ‘({𝑋} ∪ {𝑌})))
36	1, 3, 4, 5	dochdmj1 41760	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
37	7, 32, 33, 36	syl3anc 1374	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
38	31, 35, 37	3eqtrd 2776	. . . 4 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
39	1, 3, 5, 4, 12, 7, 21	dochocsn 41751	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{𝐵})) = (𝑁‘{𝐵}))
40	38, 39	oveq12d 7386	. . 3 ⊢ (𝜑 → (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘{𝐵}))) = ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌}))((joinH‘𝐾)‘𝑊)(𝑁‘{𝐵})))
41	1, 2, 3, 4, 5	dochcl 41723	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
42	7, 32, 41	syl2anc 585	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
43	1, 2, 3, 4, 5	dochcl 41723	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
44	7, 33, 43	syl2anc 585	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
45	1, 2	dihmeetcl 41715	. . . . 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 837	. . . 4 ⊢ (𝜑 → (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
47	1, 3, 4, 8, 12, 2, 6, 7, 46, 21	dihjat1 41799	. . 3 ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌}))((joinH‘𝐾)‘𝑊)(𝑁‘{𝐵})) = ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})))
48	25, 40, 47	3eqtrrd 2777	. 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 41878	. . 3 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)
53	1, 3, 5, 9, 49, 7, 52	dochsatshp 41821	. 2 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵}))) ∈ 𝐽)
54	48, 53	eqeltrd 2837	1 ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})) ∈ 𝐽)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3900 ∪ cun 3901 ∩ cin 3902 ⊆ wss 3903 {csn 4582 {cpr 4584 ran crn 5633 ‘cfv 6500 (class class class)co 7368 Basecbs 17148 0_gc0g 17371 LSSumclsm 19575 LSpanclspn 20934 LSAtomsclsa 39344 LSHypclsh 39345 HLchlt 39720 LHypclh 40354 DVecHcdvh 41448 DIsoHcdih 41598 ocHcoch 41717 joinHcdjh 41764

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-riotaBAD 39323

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-tpos 8178 df-undef 8225 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-0g 17373 df-mre 17517 df-mrc 17518 df-acs 17520 df-proset 18229 df-poset 18248 df-plt 18263 df-lub 18279 df-glb 18280 df-join 18281 df-meet 18282 df-p0 18358 df-p1 18359 df-lat 18367 df-clat 18434 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19065 df-cntz 19258 df-oppg 19287 df-lsm 19577 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-dvr 20349 df-drng 20676 df-lmod 20825 df-lss 20895 df-lsp 20935 df-lvec 21067 df-lsatoms 39346 df-lshyp 39347 df-lcv 39389 df-oposet 39546 df-ol 39548 df-oml 39549 df-covers 39636 df-ats 39637 df-atl 39668 df-cvlat 39692 df-hlat 39721 df-llines 39868 df-lplanes 39869 df-lvols 39870 df-lines 39871 df-psubsp 39873 df-pmap 39874 df-padd 40166 df-lhyp 40358 df-laut 40359 df-ldil 40474 df-ltrn 40475 df-trl 40529 df-tgrp 41113 df-tendo 41125 df-edring 41127 df-dveca 41373 df-disoa 41399 df-dvech 41449 df-dib 41509 df-dic 41543 df-dih 41599 df-doch 41718 df-djh 41765

This theorem is referenced by: lclkrlem2g 41883

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