lcfrlem27 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem27		Structured version Visualization version GIF version

Theorem lcfrlem27 41551

Description: Lemma for lcfr 41567. Special case of lcfrlem37 41561 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 2734	. . . . 5 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
11		lcfrlem24.l	. . . . 5 ⊢ 𝐿 = (LKer‘𝑈)
12		lcfrlem25.d	. . . . 5 ⊢ 𝐷 = (LDual‘𝑈)
13		eqid 2734	. . . . 5 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
14		eqid 2734	. . . . 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 2734	. . . . 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 4794	. . . . . . 7 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 )
24	22, 23	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑌 ≠ 0 )
25		eldifsn 4790	. . . . . 6 ⊢ (𝑌 ∈ (𝐸 ∖ { 0 }) ↔ (𝑌 ∈ 𝐸 ∧ 𝑌 ≠ 0 ))
26	21, 24, 25	sylanbrc 583	. . . . 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 41540	. . . 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 41550	. . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑌))))
38		2fveq3 6911	. . . . . 6 ⊢ (𝑔 = (𝐽‘𝑌) → ( ⊥ ‘(𝐿‘𝑔)) = ( ⊥ ‘(𝐿‘(𝐽‘𝑌))))
39	38	eleq2d 2824	. . . . 5 ⊢ (𝑔 = (𝐽‘𝑌) → ((𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)) ↔ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))))
40	39	rspcev 3621	. . . 4 ⊢ (((𝐽‘𝑌) ∈ 𝐺 ∧ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))) → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
41	27, 37, 40	syl2anc 584	. . 3 ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
42		eliun 4999	. . 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 2849	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∃wrex 3067 {crab 3432 ∖ cdif 3959 ∩ cin 3961 ⊆ wss 3962 {csn 4630 {cpr 4632 ∪ ciun 4995 ↦ cmpt 5230 ‘cfv 6562 ℩crio 7386 (class class class)co 7430 Basecbs 17244 +_gcplusg 17297 Scalarcsca 17300 ·_𝑠 cvsca 17301 0_gc0g 17485 LSubSpclss 20946 LSpanclspn 20986 LSAtomsclsa 38955 LFnlclfn 39038 LKerclk 39066 LDualcld 39104 HLchlt 39331 LHypclh 39966 DVecHcdvh 41060 ocHcoch 41329

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-riotaBAD 38934

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-undef 8296 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17487 df-mre 17630 df-mrc 17631 df-acs 17633 df-proset 18351 df-poset 18370 df-plt 18387 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-p0 18482 df-p1 18483 df-lat 18489 df-clat 18556 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-cntz 19347 df-oppg 19376 df-lsm 19668 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-drng 20747 df-lmod 20876 df-lss 20947 df-lsp 20987 df-lvec 21119 df-lsatoms 38957 df-lshyp 38958 df-lcv 39000 df-lfl 39039 df-lkr 39067 df-ldual 39105 df-oposet 39157 df-ol 39159 df-oml 39160 df-covers 39247 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-llines 39480 df-lplanes 39481 df-lvols 39482 df-lines 39483 df-psubsp 39485 df-pmap 39486 df-padd 39778 df-lhyp 39970 df-laut 39971 df-ldil 40086 df-ltrn 40087 df-trl 40141 df-tgrp 40725 df-tendo 40737 df-edring 40739 df-dveca 40985 df-disoa 41011 df-dvech 41061 df-dib 41121 df-dic 41155 df-dih 41211 df-doch 41330 df-djh 41377

This theorem is referenced by: lcfrlem38 41562

W3C validator