![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem26 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 40048. Special case of lcfrlem36 40041 when ((𝐽‘𝑌)‘𝐼) is zero. (Contributed by NM, 11-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) |
lcfrlem25.jz | ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) = 𝑄) |
lcfrlem25.in | ⊢ (𝜑 → 𝐼 ≠ 0 ) |
Ref | Expression |
---|---|
lcfrlem26 | ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem17.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lcfrlem17.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | lcfrlem17.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
4 | lcfrlem17.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
5 | lcfrlem17.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑈) | |
6 | lcfrlem17.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | lcfrlem17.p | . . . . . . 7 ⊢ + = (+g‘𝑈) | |
8 | lcfrlem17.z | . . . . . . 7 ⊢ 0 = (0g‘𝑈) | |
9 | lcfrlem17.a | . . . . . . 7 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
10 | lcfrlem17.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
11 | lcfrlem17.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
12 | lcfrlem17.ne | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
13 | 1, 3, 2, 4, 7, 8, 5, 9, 6, 10, 11, 12 | lcfrlem17 40022 | . . . . . 6 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 })) |
14 | 13 | eldifad 3922 | . . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉) |
15 | 1, 2, 3, 4, 5, 6, 14 | dochocsn 39844 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)})) = (𝑁‘{(𝑋 + 𝑌)})) |
16 | lcfrlem22.b | . . . . . 6 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) | |
17 | lcfrlem24.t | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑈) | |
18 | lcfrlem24.s | . . . . . 6 ⊢ 𝑆 = (Scalar‘𝑈) | |
19 | lcfrlem24.q | . . . . . 6 ⊢ 𝑄 = (0g‘𝑆) | |
20 | lcfrlem24.r | . . . . . 6 ⊢ 𝑅 = (Base‘𝑆) | |
21 | lcfrlem24.j | . . . . . 6 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) | |
22 | lcfrlem24.ib | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐵) | |
23 | lcfrlem24.l | . . . . . 6 ⊢ 𝐿 = (LKer‘𝑈) | |
24 | lcfrlem25.d | . . . . . 6 ⊢ 𝐷 = (LDual‘𝑈) | |
25 | lcfrlem25.jz | . . . . . 6 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) = 𝑄) | |
26 | lcfrlem25.in | . . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 0 ) | |
27 | 1, 3, 2, 4, 7, 8, 5, 9, 6, 10, 11, 12, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26 | lcfrlem25 40030 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) = (𝐿‘(𝐽‘𝑌))) |
28 | 27 | fveq2d 6846 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)})) = ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))) |
29 | 15, 28 | eqtr3d 2778 | . . 3 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) = ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))) |
30 | eqimss 4000 | . . 3 ⊢ ((𝑁‘{(𝑋 + 𝑌)}) = ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))) | |
31 | 29, 30 | syl 17 | . 2 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))) |
32 | eqid 2736 | . . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
33 | 1, 2, 6 | dvhlmod 39573 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
34 | eqid 2736 | . . . . 5 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
35 | eqid 2736 | . . . . . 6 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
36 | eqid 2736 | . . . . . 6 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
37 | 1, 3, 2, 4, 7, 17, 18, 20, 8, 34, 23, 24, 35, 36, 21, 6, 11 | lcfrlem10 40015 | . . . . 5 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈)) |
38 | 4, 34, 23, 33, 37 | lkrssv 37558 | . . . 4 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑌)) ⊆ 𝑉) |
39 | 1, 2, 4, 32, 3 | dochlss 39817 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘(𝐽‘𝑌)) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) ∈ (LSubSp‘𝑈)) |
40 | 6, 38, 39 | syl2anc 584 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) ∈ (LSubSp‘𝑈)) |
41 | 4, 32, 5, 33, 40, 14 | lspsnel5 20456 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) ↔ (𝑁‘{(𝑋 + 𝑌)}) ⊆ ( ⊥ ‘(𝐿‘(𝐽‘𝑌))))) |
42 | 31, 41 | mpbird 256 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 {crab 3407 ∖ cdif 3907 ∩ cin 3909 ⊆ wss 3910 {csn 4586 {cpr 4588 ↦ cmpt 5188 ‘cfv 6496 ℩crio 7312 (class class class)co 7357 Basecbs 17083 +gcplusg 17133 Scalarcsca 17136 ·𝑠 cvsca 17137 0gc0g 17321 LSubSpclss 20392 LSpanclspn 20432 LSAtomsclsa 37436 LFnlclfn 37519 LKerclk 37547 LDualcld 37585 HLchlt 37812 LHypclh 38447 DVecHcdvh 39541 ocHcoch 39810 |
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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-riotaBAD 37415 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-undef 8204 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-0g 17323 df-mre 17466 df-mrc 17467 df-acs 17469 df-proset 18184 df-poset 18202 df-plt 18219 df-lub 18235 df-glb 18236 df-join 18237 df-meet 18238 df-p0 18314 df-p1 18315 df-lat 18321 df-clat 18388 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-cntz 19097 df-oppg 19124 df-lsm 19418 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-drng 20187 df-lmod 20324 df-lss 20393 df-lsp 20433 df-lvec 20564 df-lsatoms 37438 df-lshyp 37439 df-lcv 37481 df-lfl 37520 df-lkr 37548 df-ldual 37586 df-oposet 37638 df-ol 37640 df-oml 37641 df-covers 37728 df-ats 37729 df-atl 37760 df-cvlat 37784 df-hlat 37813 df-llines 37961 df-lplanes 37962 df-lvols 37963 df-lines 37964 df-psubsp 37966 df-pmap 37967 df-padd 38259 df-lhyp 38451 df-laut 38452 df-ldil 38567 df-ltrn 38568 df-trl 38622 df-tgrp 39206 df-tendo 39218 df-edring 39220 df-dveca 39466 df-disoa 39492 df-dvech 39542 df-dib 39602 df-dic 39636 df-dih 39692 df-doch 39811 df-djh 39858 |
This theorem is referenced by: lcfrlem27 40032 |
Copyright terms: Public domain | W3C validator |