lcfrlem36 - Mathbox for Norm Megill

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Theorem lcfrlem36 41616

Description: Lemma for lcfr 41623. (Contributed by NM, 6-Mar-2015.)

**Hypotheses**
Ref	Expression
lcfrlem17.h	⊢ 𝐻 = (LHyp‘𝐾)
lcfrlem17.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcfrlem17.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcfrlem17.v	⊢ 𝑉 = (Base‘𝑈)
lcfrlem17.p	⊢ + = (+_g‘𝑈)
lcfrlem17.z	⊢ 0 = (0_g‘𝑈)
lcfrlem17.n	⊢ 𝑁 = (LSpan‘𝑈)
lcfrlem17.a	⊢ 𝐴 = (LSAtoms‘𝑈)
lcfrlem17.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfrlem17.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lcfrlem17.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
lcfrlem17.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
lcfrlem22.b	⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
lcfrlem24.t	⊢ · = ( ·_𝑠 ‘𝑈)
lcfrlem24.s	⊢ 𝑆 = (Scalar‘𝑈)
lcfrlem24.q	⊢ 𝑄 = (0_g‘𝑆)
lcfrlem24.r	⊢ 𝑅 = (Base‘𝑆)
lcfrlem24.j	⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
lcfrlem24.ib	⊢ (𝜑 → 𝐼 ∈ 𝐵)
lcfrlem24.l	⊢ 𝐿 = (LKer‘𝑈)
lcfrlem25.d	⊢ 𝐷 = (LDual‘𝑈)
lcfrlem28.jn	⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)
lcfrlem29.i	⊢ 𝐹 = (inv_r‘𝑆)
lcfrlem30.m	⊢ − = (-_g‘𝐷)
lcfrlem30.c	⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌)))

**Assertion**
Ref	Expression
lcfrlem36	⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶)))

Distinct variable groups: 𝑣,𝑘,𝑤,𝑥, ⊥ + ,𝑘,𝑣,𝑤,𝑥 𝑅,𝑘,𝑣,𝑥 𝑆,𝑘 · ,𝑘,𝑣,𝑤,𝑥 𝑣,𝑉,𝑥 𝑘,𝑋,𝑣,𝑤,𝑥 𝑘,𝑌,𝑣,𝑤,𝑥 𝑥, 0 Allowed substitution hints: 𝜑(𝑥,𝑤,𝑣,𝑘) 𝐴(𝑥,𝑤,𝑣,𝑘) 𝐵(𝑥,𝑤,𝑣,𝑘) 𝐶(𝑥,𝑤,𝑣,𝑘) 𝐷(𝑥,𝑤,𝑣,𝑘) 𝑄(𝑥,𝑤,𝑣,𝑘) 𝑅(𝑤) 𝑆(𝑥,𝑤,𝑣) 𝑈(𝑥,𝑤,𝑣,𝑘) 𝐹(𝑥,𝑤,𝑣,𝑘) 𝐻(𝑥,𝑤,𝑣,𝑘) 𝐼(𝑥,𝑤,𝑣,𝑘) 𝐽(𝑥,𝑤,𝑣,𝑘) 𝐾(𝑥,𝑤,𝑣,𝑘) 𝐿(𝑥,𝑤,𝑣,𝑘) − (𝑥,𝑤,𝑣,𝑘) 𝑁(𝑥,𝑤,𝑣,𝑘) 𝑉(𝑤,𝑘) 𝑊(𝑥,𝑤,𝑣,𝑘) 0 (𝑤,𝑣,𝑘)

**Proof of Theorem lcfrlem36**
Step	Hyp	Ref	Expression
1		lcfrlem17.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcfrlem17.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		lcfrlem17.o	. . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
4		lcfrlem17.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
5		lcfrlem17.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
6		lcfrlem17.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7		lcfrlem17.p	. . . . . . 7 ⊢ + = (+_g‘𝑈)
8		lcfrlem17.z	. . . . . . 7 ⊢ 0 = (0_g‘𝑈)
9		lcfrlem17.a	. . . . . . 7 ⊢ 𝐴 = (LSAtoms‘𝑈)
10		lcfrlem17.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
11		lcfrlem17.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
12		lcfrlem17.ne	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
13	1, 3, 2, 4, 7, 8, 5, 9, 6, 10, 11, 12	lcfrlem17 41597	. . . . . 6 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3914	. . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
15	1, 2, 3, 4, 5, 6, 14	dochocsn 41419	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)})) = (𝑁‘{(𝑋 + 𝑌)}))
16		lcfrlem22.b	. . . . . 6 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
17		lcfrlem24.t	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑈)
18		lcfrlem24.s	. . . . . 6 ⊢ 𝑆 = (Scalar‘𝑈)
19		lcfrlem24.q	. . . . . 6 ⊢ 𝑄 = (0_g‘𝑆)
20		lcfrlem24.r	. . . . . 6 ⊢ 𝑅 = (Base‘𝑆)
21		lcfrlem24.j	. . . . . 6 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
22		lcfrlem24.ib	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
23		lcfrlem24.l	. . . . . 6 ⊢ 𝐿 = (LKer‘𝑈)
24		lcfrlem25.d	. . . . . 6 ⊢ 𝐷 = (LDual‘𝑈)
25		lcfrlem28.jn	. . . . . 6 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)
26		lcfrlem29.i	. . . . . 6 ⊢ 𝐹 = (inv_r‘𝑆)
27		lcfrlem30.m	. . . . . 6 ⊢ − = (-_g‘𝐷)
28		lcfrlem30.c	. . . . . 6 ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌)))
29	1, 3, 2, 4, 7, 8, 5, 9, 6, 10, 11, 12, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28	lcfrlem35 41615	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) = (𝐿‘𝐶))
30	29	fveq2d 6826	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)})) = ( ⊥ ‘(𝐿‘𝐶)))
31	15, 30	eqtr3d 2768	. . 3 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) = ( ⊥ ‘(𝐿‘𝐶)))
32		eqimss 3993	. . 3 ⊢ ((𝑁‘{(𝑋 + 𝑌)}) = ( ⊥ ‘(𝐿‘𝐶)) → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ( ⊥ ‘(𝐿‘𝐶)))
33	31, 32	syl 17	. 2 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ( ⊥ ‘(𝐿‘𝐶)))
34		eqid 2731	. . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
35	1, 2, 6	dvhlmod 41148	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
36		eqid 2731	. . . . 5 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
37	1, 3, 2, 4, 7, 8, 5, 9, 6, 10, 11, 12, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28	lcfrlem30 41610	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (LFnl‘𝑈))
38	4, 36, 23, 35, 37	lkrssv 39134	. . . 4 ⊢ (𝜑 → (𝐿‘𝐶) ⊆ 𝑉)
39	1, 2, 4, 34, 3	dochlss 41392	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐶) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘𝐶)) ∈ (LSubSp‘𝑈))
40	6, 38, 39	syl2anc 584	. . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐶)) ∈ (LSubSp‘𝑈))
41	4, 34, 5, 35, 40, 14	ellspsn5b 20926	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶)) ↔ (𝑁‘{(𝑋 + 𝑌)}) ⊆ ( ⊥ ‘(𝐿‘𝐶))))
42	33, 41	mpbird 257	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∃wrex 3056 ∖ cdif 3899 ∩ cin 3901 ⊆ wss 3902 {csn 4576 {cpr 4578 ↦ cmpt 5172 ‘cfv 6481 ℩crio 7302 (class class class)co 7346 Basecbs 17117 +_gcplusg 17158 ._rcmulr 17159 Scalarcsca 17161 ·_𝑠 cvsca 17162 0_gc0g 17340 -_gcsg 18845 inv_rcinvr 20303 LSubSpclss 20862 LSpanclspn 20902 LSAtomsclsa 39012 LFnlclfn 39095 LKerclk 39123 LDualcld 39161 HLchlt 39388 LHypclh 40022 DVecHcdvh 41116 ocHcoch 41385

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-riotaBAD 38991

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-undef 8203 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-n0 12379 df-z 12466 df-uz 12730 df-fz 13405 df-struct 17055 df-sets 17072 df-slot 17090 df-ndx 17102 df-base 17118 df-ress 17139 df-plusg 17171 df-mulr 17172 df-sca 17174 df-vsca 17175 df-0g 17342 df-mre 17485 df-mrc 17486 df-acs 17488 df-proset 18197 df-poset 18216 df-plt 18231 df-lub 18247 df-glb 18248 df-join 18249 df-meet 18250 df-p0 18326 df-p1 18327 df-lat 18335 df-clat 18402 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-submnd 18689 df-grp 18846 df-minusg 18847 df-sbg 18848 df-subg 19033 df-cntz 19227 df-oppg 19256 df-lsm 19546 df-cmn 19692 df-abl 19693 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-nzr 20426 df-rlreg 20607 df-domn 20608 df-drng 20644 df-lmod 20793 df-lss 20863 df-lsp 20903 df-lvec 21035 df-lsatoms 39014 df-lshyp 39015 df-lcv 39057 df-lfl 39096 df-lkr 39124 df-ldual 39162 df-oposet 39214 df-ol 39216 df-oml 39217 df-covers 39304 df-ats 39305 df-atl 39336 df-cvlat 39360 df-hlat 39389 df-llines 39536 df-lplanes 39537 df-lvols 39538 df-lines 39539 df-psubsp 39541 df-pmap 39542 df-padd 39834 df-lhyp 40026 df-laut 40027 df-ldil 40142 df-ltrn 40143 df-trl 40197 df-tgrp 40781 df-tendo 40793 df-edring 40795 df-dveca 41041 df-disoa 41067 df-dvech 41117 df-dib 41177 df-dic 41211 df-dih 41267 df-doch 41386 df-djh 41433

This theorem is referenced by: lcfrlem37 41617

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