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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem14 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 40259. (Contributed by NM, 10-Mar-2015.) |
Ref | Expression |
---|---|
lcf1o.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcf1o.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcf1o.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcf1o.v | ⊢ 𝑉 = (Base‘𝑈) |
lcf1o.a | ⊢ + = (+g‘𝑈) |
lcf1o.t | ⊢ · = ( ·𝑠 ‘𝑈) |
lcf1o.s | ⊢ 𝑆 = (Scalar‘𝑈) |
lcf1o.r | ⊢ 𝑅 = (Base‘𝑆) |
lcf1o.z | ⊢ 0 = (0g‘𝑈) |
lcf1o.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lcf1o.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcf1o.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcf1o.q | ⊢ 𝑄 = (0g‘𝐷) |
lcf1o.c | ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} |
lcf1o.j | ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) |
lcflo.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcfrlem10.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
lcfrlem14.n | ⊢ 𝑁 = (LSpan‘𝑈) |
Ref | Expression |
---|---|
lcfrlem14 | ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) = (𝑁‘{𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcf1o.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lcf1o.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
3 | lcf1o.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | lcf1o.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
5 | lcf1o.a | . . . . 5 ⊢ + = (+g‘𝑈) | |
6 | lcf1o.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑈) | |
7 | lcf1o.s | . . . . 5 ⊢ 𝑆 = (Scalar‘𝑈) | |
8 | lcf1o.r | . . . . 5 ⊢ 𝑅 = (Base‘𝑆) | |
9 | lcf1o.z | . . . . 5 ⊢ 0 = (0g‘𝑈) | |
10 | lcf1o.f | . . . . 5 ⊢ 𝐹 = (LFnl‘𝑈) | |
11 | lcf1o.l | . . . . 5 ⊢ 𝐿 = (LKer‘𝑈) | |
12 | lcf1o.d | . . . . 5 ⊢ 𝐷 = (LDual‘𝑈) | |
13 | lcf1o.q | . . . . 5 ⊢ 𝑄 = (0g‘𝐷) | |
14 | lcf1o.c | . . . . 5 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
15 | lcf1o.j | . . . . 5 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) | |
16 | lcflo.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | lcfrlem10.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 | lcfrlem11 40227 | . . . 4 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = ( ⊥ ‘{𝑋})) |
19 | lcfrlem14.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑈) | |
20 | 17 | eldifad 3956 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
21 | 20 | snssd 4805 | . . . . 5 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
22 | 1, 3, 2, 4, 19, 16, 21 | dochocsp 40053 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘{𝑋})) |
23 | 18, 22 | eqtr4d 2774 | . . 3 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = ( ⊥ ‘(𝑁‘{𝑋}))) |
24 | 23 | fveq2d 6882 | . 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) = ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋})))) |
25 | eqid 2731 | . . . . 5 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
26 | 1, 3, 4, 19, 25 | dihlsprn 40005 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
27 | 16, 20, 26 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
28 | 1, 25, 2 | dochoc 40041 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋})) |
29 | 16, 27, 28 | syl2anc 584 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋})) |
30 | 24, 29 | eqtrd 2771 | 1 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) = (𝑁‘{𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3069 {crab 3431 ∖ cdif 3941 {csn 4622 ↦ cmpt 5224 ran crn 5670 ‘cfv 6532 ℩crio 7348 (class class class)co 7393 Basecbs 17126 +gcplusg 17179 Scalarcsca 17182 ·𝑠 cvsca 17183 0gc0g 17367 LSpanclspn 20531 LFnlclfn 37730 LKerclk 37758 LDualcld 37796 HLchlt 38023 LHypclh 38658 DVecHcdvh 39752 DIsoHcdih 39902 ocHcoch 40021 |
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 37626 |
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-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-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-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 37649 df-lshyp 37650 df-lfl 37731 df-lkr 37759 df-oposet 37849 df-ol 37851 df-oml 37852 df-covers 37939 df-ats 37940 df-atl 37971 df-cvlat 37995 df-hlat 38024 df-llines 38172 df-lplanes 38173 df-lvols 38174 df-lines 38175 df-psubsp 38177 df-pmap 38178 df-padd 38470 df-lhyp 38662 df-laut 38663 df-ldil 38778 df-ltrn 38779 df-trl 38833 df-tgrp 39417 df-tendo 39429 df-edring 39431 df-dveca 39677 df-disoa 39703 df-dvech 39753 df-dib 39813 df-dic 39847 df-dih 39903 df-doch 40022 df-djh 40069 |
This theorem is referenced by: lcfrlem15 40231 lcfrlem31 40247 |
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