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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem28 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 39986. TODO: This can be a hypothesis since the zero version of (𝐽‘𝑌)‘𝐼 needs it. (Contributed by NM, 9-Mar-2015.) |
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 = (0g‘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 = (0g‘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) |
Ref | Expression |
---|---|
lcfrlem28 | ⊢ (𝜑 → 𝐼 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem28.jn | . 2 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) | |
2 | lcfrlem17.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | lcfrlem17.u | . . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | lcfrlem17.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
5 | 2, 3, 4 | dvhlmod 39511 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
6 | lcfrlem17.o | . . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
7 | lcfrlem17.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
8 | lcfrlem17.p | . . . . . 6 ⊢ + = (+g‘𝑈) | |
9 | lcfrlem24.t | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑈) | |
10 | lcfrlem24.s | . . . . . 6 ⊢ 𝑆 = (Scalar‘𝑈) | |
11 | lcfrlem24.r | . . . . . 6 ⊢ 𝑅 = (Base‘𝑆) | |
12 | lcfrlem17.z | . . . . . 6 ⊢ 0 = (0g‘𝑈) | |
13 | eqid 2737 | . . . . . 6 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
14 | lcfrlem24.l | . . . . . 6 ⊢ 𝐿 = (LKer‘𝑈) | |
15 | lcfrlem25.d | . . . . . 6 ⊢ 𝐷 = (LDual‘𝑈) | |
16 | eqid 2737 | . . . . . 6 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
17 | eqid 2737 | . . . . . 6 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
18 | lcfrlem24.j | . . . . . 6 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) | |
19 | lcfrlem17.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
20 | 2, 6, 3, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 4, 19 | lcfrlem10 39953 | . . . . 5 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈)) |
21 | lcfrlem24.q | . . . . . 6 ⊢ 𝑄 = (0g‘𝑆) | |
22 | 10, 21, 12, 13 | lfl0 37465 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑌) ∈ (LFnl‘𝑈)) → ((𝐽‘𝑌)‘ 0 ) = 𝑄) |
23 | 5, 20, 22 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((𝐽‘𝑌)‘ 0 ) = 𝑄) |
24 | fveqeq2 6848 | . . . 4 ⊢ (𝐼 = 0 → (((𝐽‘𝑌)‘𝐼) = 𝑄 ↔ ((𝐽‘𝑌)‘ 0 ) = 𝑄)) | |
25 | 23, 24 | syl5ibrcom 246 | . . 3 ⊢ (𝜑 → (𝐼 = 0 → ((𝐽‘𝑌)‘𝐼) = 𝑄)) |
26 | 25 | necon3d 2962 | . 2 ⊢ (𝜑 → (((𝐽‘𝑌)‘𝐼) ≠ 𝑄 → 𝐼 ≠ 0 )) |
27 | 1, 26 | mpd 15 | 1 ⊢ (𝜑 → 𝐼 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∃wrex 3071 {crab 3405 ∖ cdif 3905 ∩ cin 3907 {csn 4584 {cpr 4586 ↦ cmpt 5186 ‘cfv 6493 ℩crio 7306 (class class class)co 7351 Basecbs 17043 +gcplusg 17093 Scalarcsca 17096 ·𝑠 cvsca 17097 0gc0g 17281 LModclmod 20275 LSpanclspn 20385 LSAtomsclsa 37374 LFnlclfn 37457 LKerclk 37485 LDualcld 37523 HLchlt 37750 LHypclh 38385 DVecHcdvh 39479 ocHcoch 39748 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-riotaBAD 37353 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-tpos 8149 df-undef 8196 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-sca 17109 df-vsca 17110 df-0g 17283 df-proset 18144 df-poset 18162 df-plt 18179 df-lub 18195 df-glb 18196 df-join 18197 df-meet 18198 df-p0 18274 df-p1 18275 df-lat 18281 df-clat 18348 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-submnd 18562 df-grp 18711 df-minusg 18712 df-sbg 18713 df-subg 18884 df-cntz 19056 df-lsm 19377 df-cmn 19523 df-abl 19524 df-mgp 19856 df-ur 19873 df-ring 19920 df-oppr 20002 df-dvdsr 20023 df-unit 20024 df-invr 20054 df-dvr 20065 df-drng 20140 df-lmod 20277 df-lss 20346 df-lsp 20386 df-lvec 20517 df-lsatoms 37376 df-lshyp 37377 df-lfl 37458 df-oposet 37576 df-ol 37578 df-oml 37579 df-covers 37666 df-ats 37667 df-atl 37698 df-cvlat 37722 df-hlat 37751 df-llines 37899 df-lplanes 37900 df-lvols 37901 df-lines 37902 df-psubsp 37904 df-pmap 37905 df-padd 38197 df-lhyp 38389 df-laut 38390 df-ldil 38505 df-ltrn 38506 df-trl 38560 df-tgrp 39144 df-tendo 39156 df-edring 39158 df-dveca 39404 df-disoa 39430 df-dvech 39480 df-dib 39540 df-dic 39574 df-dih 39630 df-doch 39749 df-djh 39796 |
This theorem is referenced by: lcfrlem35 39978 |
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