lcfrlem27 - Mathbox for Norm Megill

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Theorem lcfrlem27 41526

Description: Lemma for lcfr 41542. Special case of lcfrlem37 41536 when ((𝐽‘𝑌)‘𝐼) is zero. (Contributed by NM, 11-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)
lcfrlem25.jz	⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) = 𝑄)
lcfrlem25.in	⊢ (𝜑 → 𝐼 ≠ 0 )
lcfrlem27.g	⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝐷))
lcfrlem27.gs	⊢ (𝜑 → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
lcfrlem27.e	⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
lcfrlem27.xe	⊢ (𝜑 → 𝑋 ∈ 𝐸)
lcfrlem27.ye	⊢ (𝜑 → 𝑌 ∈ 𝐸)

**Assertion**
Ref	Expression
lcfrlem27	⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

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 𝑓,𝐿 ⊥ ,𝑓 + ,𝑓 𝑅,𝑓 · ,𝑓 𝑈,𝑓 𝑓,𝑉 𝑓,𝑘,𝑣,𝑤,𝑥,𝑔 𝑔,𝐺,𝑘 𝑓,𝑔,𝐽,𝑘 𝑔,𝐿,𝑘 ⊥ ,𝑔 + ,𝑔 𝑈,𝑘 𝑔,𝑉 𝑔,𝑋 𝑓,𝑌,𝑔 𝜑,𝑔,𝑘 𝑣,𝑔,𝑤,𝑥 Allowed substitution hints: 𝜑(𝑥,𝑤,𝑣,𝑓) 𝐴(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐵(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐷(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝑄(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝑅(𝑤,𝑔) 𝑆(𝑥,𝑤,𝑣,𝑓,𝑔) · (𝑔) 𝑈(𝑥,𝑤,𝑣,𝑔) 𝐸(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐺(𝑥,𝑤,𝑣,𝑓) 𝐻(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐼(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐽(𝑥,𝑤,𝑣) 𝐾(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐿(𝑥,𝑤,𝑣) 𝑁(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝑉(𝑤,𝑘) 𝑊(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝑋(𝑓) 0 (𝑤,𝑣,𝑓,𝑔,𝑘)

**Proof of Theorem lcfrlem27**
Step	Hyp	Ref	Expression
1		lcfrlem17.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcfrlem17.o	. . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lcfrlem17.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lcfrlem17.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
5		lcfrlem17.p	. . . . 5 ⊢ + = (+_g‘𝑈)
6		lcfrlem24.t	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝑈)
7		lcfrlem24.s	. . . . 5 ⊢ 𝑆 = (Scalar‘𝑈)
8		lcfrlem24.r	. . . . 5 ⊢ 𝑅 = (Base‘𝑆)
9		lcfrlem17.z	. . . . 5 ⊢ 0 = (0_g‘𝑈)
10		eqid 2740	. . . . 5 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
11		lcfrlem24.l	. . . . 5 ⊢ 𝐿 = (LKer‘𝑈)
12		lcfrlem25.d	. . . . 5 ⊢ 𝐷 = (LDual‘𝑈)
13		eqid 2740	. . . . 5 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
14		eqid 2740	. . . . 5 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
15		lcfrlem24.j	. . . . 5 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
16		lcfrlem17.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		eqid 2740	. . . . 5 ⊢ (LSubSp‘𝐷) = (LSubSp‘𝐷)
18		lcfrlem27.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝐷))
19		lcfrlem27.gs	. . . . 5 ⊢ (𝜑 → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
20		lcfrlem27.e	. . . . 5 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
21		lcfrlem27.ye	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐸)
22		lcfrlem17.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
23		eldifsni 4815	. . . . . . 7 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 )
24	22, 23	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑌 ≠ 0 )
25		eldifsn 4811	. . . . . 6 ⊢ (𝑌 ∈ (𝐸 ∖ { 0 }) ↔ (𝑌 ∈ 𝐸 ∧ 𝑌 ≠ 0 ))
26	21, 24, 25	sylanbrc 582	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝐸 ∖ { 0 }))
27	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 26	lcfrlem16 41515	. . . 4 ⊢ (𝜑 → (𝐽‘𝑌) ∈ 𝐺)
28		lcfrlem17.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
29		lcfrlem17.a	. . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑈)
30		lcfrlem17.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
31		lcfrlem17.ne	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
32		lcfrlem22.b	. . . . 5 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
33		lcfrlem24.q	. . . . 5 ⊢ 𝑄 = (0_g‘𝑆)
34		lcfrlem24.ib	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
35		lcfrlem25.jz	. . . . 5 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) = 𝑄)
36		lcfrlem25.in	. . . . 5 ⊢ (𝜑 → 𝐼 ≠ 0 )
37	1, 2, 3, 4, 5, 9, 28, 29, 16, 30, 22, 31, 32, 6, 7, 33, 8, 15, 34, 11, 12, 35, 36	lcfrlem26 41525	. . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑌))))
38		2fveq3 6925	. . . . . 6 ⊢ (𝑔 = (𝐽‘𝑌) → ( ⊥ ‘(𝐿‘𝑔)) = ( ⊥ ‘(𝐿‘(𝐽‘𝑌))))
39	38	eleq2d 2830	. . . . 5 ⊢ (𝑔 = (𝐽‘𝑌) → ((𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)) ↔ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))))
40	39	rspcev 3635	. . . 4 ⊢ (((𝐽‘𝑌) ∈ 𝐺 ∧ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))) → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
41	27, 37, 40	syl2anc 583	. . 3 ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
42		eliun 5019	. . 3 ⊢ ((𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
43	41, 42	sylibr 234	. 2 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)))
44	43, 20	eleqtrrdi 2855	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∃wrex 3076 {crab 3443 ∖ cdif 3973 ∩ cin 3975 ⊆ wss 3976 {csn 4648 {cpr 4650 ∪ ciun 5015 ↦ cmpt 5249 ‘cfv 6573 ℩crio 7403 (class class class)co 7448 Basecbs 17258 +_gcplusg 17311 Scalarcsca 17314 ·_𝑠 cvsca 17315 0_gc0g 17499 LSubSpclss 20952 LSpanclspn 20992 LSAtomsclsa 38930 LFnlclfn 39013 LKerclk 39041 LDualcld 39079 HLchlt 39306 LHypclh 39941 DVecHcdvh 41035 ocHcoch 41304

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-riotaBAD 38909

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-tpos 8267 df-undef 8314 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-0g 17501 df-mre 17644 df-mrc 17645 df-acs 17647 df-proset 18365 df-poset 18383 df-plt 18400 df-lub 18416 df-glb 18417 df-join 18418 df-meet 18419 df-p0 18495 df-p1 18496 df-lat 18502 df-clat 18569 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-subg 19163 df-cntz 19357 df-oppg 19386 df-lsm 19678 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-drng 20753 df-lmod 20882 df-lss 20953 df-lsp 20993 df-lvec 21125 df-lsatoms 38932 df-lshyp 38933 df-lcv 38975 df-lfl 39014 df-lkr 39042 df-ldual 39080 df-oposet 39132 df-ol 39134 df-oml 39135 df-covers 39222 df-ats 39223 df-atl 39254 df-cvlat 39278 df-hlat 39307 df-llines 39455 df-lplanes 39456 df-lvols 39457 df-lines 39458 df-psubsp 39460 df-pmap 39461 df-padd 39753 df-lhyp 39945 df-laut 39946 df-ldil 40061 df-ltrn 40062 df-trl 40116 df-tgrp 40700 df-tendo 40712 df-edring 40714 df-dveca 40960 df-disoa 40986 df-dvech 41036 df-dib 41096 df-dic 41130 df-dih 41186 df-doch 41305 df-djh 41352

This theorem is referenced by: lcfrlem38 41537

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