lcfrlem34 - Mathbox for Norm Megill

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Theorem lcfrlem34 40252

Description: Lemma for lcfr 40261. (Contributed by NM, 10-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
lcfrlem34	⊢ (𝜑 → 𝐶 ≠ (0_g‘𝐷))

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 lcfrlem34**
Step	Hyp	Ref	Expression
1		lcfrlem17.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcfrlem17.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lcfrlem17.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lcfrlem17.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
5		lcfrlem17.p	. . 3 ⊢ + = (+_g‘𝑈)
6		lcfrlem17.z	. . 3 ⊢ 0 = (0_g‘𝑈)
7		lcfrlem17.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
8		lcfrlem17.a	. . 3 ⊢ 𝐴 = (LSAtoms‘𝑈)
9		lcfrlem17.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10	9	adantr 481	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑋)‘𝐼) = 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		lcfrlem17.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
12	11	adantr 481	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑋)‘𝐼) = 𝑄) → 𝑋 ∈ (𝑉 ∖ { 0 }))
13		lcfrlem17.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	13	adantr 481	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑋)‘𝐼) = 𝑄) → 𝑌 ∈ (𝑉 ∖ { 0 }))
15		lcfrlem17.ne	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
16	15	adantr 481	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑋)‘𝐼) = 𝑄) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
17		lcfrlem22.b	. . 3 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
18		lcfrlem24.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝑈)
19		lcfrlem24.s	. . 3 ⊢ 𝑆 = (Scalar‘𝑈)
20		lcfrlem24.q	. . 3 ⊢ 𝑄 = (0_g‘𝑆)
21		lcfrlem24.r	. . 3 ⊢ 𝑅 = (Base‘𝑆)
22		lcfrlem24.j	. . 3 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
23		lcfrlem24.ib	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
24	23	adantr 481	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑋)‘𝐼) = 𝑄) → 𝐼 ∈ 𝐵)
25		lcfrlem24.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
26		lcfrlem25.d	. . 3 ⊢ 𝐷 = (LDual‘𝑈)
27		lcfrlem28.jn	. . . 4 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)
28	27	adantr 481	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑋)‘𝐼) = 𝑄) → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)
29		lcfrlem29.i	. . 3 ⊢ 𝐹 = (inv_r‘𝑆)
30		lcfrlem30.m	. . 3 ⊢ − = (-_g‘𝐷)
31		lcfrlem30.c	. . 3 ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌)))
32		simpr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑋)‘𝐼) = 𝑄) → ((𝐽‘𝑋)‘𝐼) = 𝑄)
33	1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 29, 30, 31, 32	lcfrlem33 40251	. 2 ⊢ ((𝜑 ∧ ((𝐽‘𝑋)‘𝐼) = 𝑄) → 𝐶 ≠ (0_g‘𝐷))
34	9	adantr 481	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑋)‘𝐼) ≠ 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
35	11	adantr 481	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑋)‘𝐼) ≠ 𝑄) → 𝑋 ∈ (𝑉 ∖ { 0 }))
36	13	adantr 481	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑋)‘𝐼) ≠ 𝑄) → 𝑌 ∈ (𝑉 ∖ { 0 }))
37	15	adantr 481	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑋)‘𝐼) ≠ 𝑄) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
38	23	adantr 481	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑋)‘𝐼) ≠ 𝑄) → 𝐼 ∈ 𝐵)
39	27	adantr 481	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑋)‘𝐼) ≠ 𝑄) → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)
40		simpr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑋)‘𝐼) ≠ 𝑄) → ((𝐽‘𝑋)‘𝐼) ≠ 𝑄)
41	1, 2, 3, 4, 5, 6, 7, 8, 34, 35, 36, 37, 17, 18, 19, 20, 21, 22, 38, 25, 26, 39, 29, 30, 31, 40	lcfrlem32 40250	. 2 ⊢ ((𝜑 ∧ ((𝐽‘𝑋)‘𝐼) ≠ 𝑄) → 𝐶 ≠ (0_g‘𝐷))
42	33, 41	pm2.61dane 3028	1 ⊢ (𝜑 → 𝐶 ≠ (0_g‘𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∃wrex 3069 ∖ cdif 3941 ∩ cin 3943 {csn 4622 {cpr 4624 ↦ cmpt 5224 ‘cfv 6532 ℩crio 7348 (class class class)co 7393 Basecbs 17126 +_gcplusg 17179 ._rcmulr 17180 Scalarcsca 17182 ·_𝑠 cvsca 17183 0_gc0g 17367 -_gcsg 18796 inv_rcinvr 20153 LSpanclspn 20531 LSAtomsclsa 37649 LKerclk 37760 LDualcld 37798 HLchlt 38025 LHypclh 38660 DVecHcdvh 39754 ocHcoch 40023

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-riotaBAD 37628

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-om 7839 df-1st 7957 df-2nd 7958 df-tpos 8193 df-undef 8240 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-n0 12455 df-z 12541 df-uz 12805 df-fz 13467 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-0g 17369 df-mre 17512 df-mrc 17513 df-acs 17515 df-proset 18230 df-poset 18248 df-plt 18265 df-lub 18281 df-glb 18282 df-join 18283 df-meet 18284 df-p0 18360 df-p1 18361 df-lat 18367 df-clat 18434 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-submnd 18648 df-grp 18797 df-minusg 18798 df-sbg 18799 df-subg 18975 df-cntz 19147 df-oppg 19174 df-lsm 19468 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-oppr 20102 df-dvdsr 20123 df-unit 20124 df-invr 20154 df-dvr 20165 df-drng 20267 df-lmod 20422 df-lss 20492 df-lsp 20532 df-lvec 20663 df-lsatoms 37651 df-lshyp 37652 df-lcv 37694 df-lfl 37733 df-lkr 37761 df-ldual 37799 df-oposet 37851 df-ol 37853 df-oml 37854 df-covers 37941 df-ats 37942 df-atl 37973 df-cvlat 37997 df-hlat 38026 df-llines 38174 df-lplanes 38175 df-lvols 38176 df-lines 38177 df-psubsp 38179 df-pmap 38180 df-padd 38472 df-lhyp 38664 df-laut 38665 df-ldil 38780 df-ltrn 38781 df-trl 38835 df-tgrp 39419 df-tendo 39431 df-edring 39433 df-dveca 39679 df-disoa 39705 df-dvech 39755 df-dib 39815 df-dic 39849 df-dih 39905 df-doch 40024 df-djh 40071

This theorem is referenced by: lcfrlem35 40253

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