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| Mirrors > Home > MPE Home > Th. List > lsppratlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for lspprat 20712. Let π₯ β (π β {0}) (if there is no such π₯ then π is the zero subspace), and let π¦ β (π β (πβ{π₯})) (assuming the conclusion is false). The goal is to write π, π in terms of π₯, π¦, which would normally be done by solving the system of linear equations. The span equivalent of this process is lspsolv 20702 (hence the name), which we use extensively below. In this lemma, we show that since π₯ β (πβ{π, π}), either π₯ β (πβ{π}) or π β (πβ{π₯, π}). (Contributed by NM, 29-Aug-2014.) |
| Ref | Expression |
|---|---|
| lspprat.v | β’ π = (Baseβπ) |
| lspprat.s | β’ π = (LSubSpβπ) |
| lspprat.n | β’ π = (LSpanβπ) |
| lspprat.w | β’ (π β π β LVec) |
| lspprat.u | β’ (π β π β π) |
| lspprat.x | β’ (π β π β π) |
| lspprat.y | β’ (π β π β π) |
| lspprat.p | β’ (π β π β (πβ{π, π})) |
| lsppratlem1.o | β’ 0 = (0gβπ) |
| lsppratlem1.x2 | β’ (π β π₯ β (π β { 0 })) |
| lsppratlem1.y2 | β’ (π β π¦ β (π β (πβ{π₯}))) |
| Ref | Expression |
|---|---|
| lsppratlem1 | β’ (π β (π₯ β (πβ{π}) β¨ π β (πβ{π₯, π}))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lspprat.w | . . . . . 6 β’ (π β π β LVec) | |
| 2 | 1 | adantr 481 | . . . . 5 β’ ((π β§ Β¬ π₯ β (πβ{π})) β π β LVec) |
| 3 | lspprat.y | . . . . . . 7 β’ (π β π β π) | |
| 4 | 3 | snssd 4804 | . . . . . 6 β’ (π β {π} β π) |
| 5 | 4 | adantr 481 | . . . . 5 β’ ((π β§ Β¬ π₯ β (πβ{π})) β {π} β π) |
| 6 | lspprat.x | . . . . . 6 β’ (π β π β π) | |
| 7 | 6 | adantr 481 | . . . . 5 β’ ((π β§ Β¬ π₯ β (πβ{π})) β π β π) |
| 8 | lspprat.p | . . . . . . . . . 10 β’ (π β π β (πβ{π, π})) | |
| 9 | 8 | pssssd 4092 | . . . . . . . . 9 β’ (π β π β (πβ{π, π})) |
| 10 | lsppratlem1.x2 | . . . . . . . . . 10 β’ (π β π₯ β (π β { 0 })) | |
| 11 | 10 | eldifad 3955 | . . . . . . . . 9 β’ (π β π₯ β π) |
| 12 | 9, 11 | sseldd 3978 | . . . . . . . 8 β’ (π β π₯ β (πβ{π, π})) |
| 13 | prcom 4728 | . . . . . . . . . 10 β’ {π, π} = {π, π} | |
| 14 | df-pr 4624 | . . . . . . . . . 10 β’ {π, π} = ({π} βͺ {π}) | |
| 15 | 13, 14 | eqtri 2759 | . . . . . . . . 9 β’ {π, π} = ({π} βͺ {π}) |
| 16 | 15 | fveq2i 6880 | . . . . . . . 8 β’ (πβ{π, π}) = (πβ({π} βͺ {π})) |
| 17 | 12, 16 | eleqtrdi 2842 | . . . . . . 7 β’ (π β π₯ β (πβ({π} βͺ {π}))) |
| 18 | 17 | anim1i 615 | . . . . . 6 β’ ((π β§ Β¬ π₯ β (πβ{π})) β (π₯ β (πβ({π} βͺ {π})) β§ Β¬ π₯ β (πβ{π}))) |
| 19 | eldif 3953 | . . . . . 6 β’ (π₯ β ((πβ({π} βͺ {π})) β (πβ{π})) β (π₯ β (πβ({π} βͺ {π})) β§ Β¬ π₯ β (πβ{π}))) | |
| 20 | 18, 19 | sylibr 233 | . . . . 5 β’ ((π β§ Β¬ π₯ β (πβ{π})) β π₯ β ((πβ({π} βͺ {π})) β (πβ{π}))) |
| 21 | lspprat.v | . . . . . 6 β’ π = (Baseβπ) | |
| 22 | lspprat.s | . . . . . 6 β’ π = (LSubSpβπ) | |
| 23 | lspprat.n | . . . . . 6 β’ π = (LSpanβπ) | |
| 24 | 21, 22, 23 | lspsolv 20702 | . . . . 5 β’ ((π β LVec β§ ({π} β π β§ π β π β§ π₯ β ((πβ({π} βͺ {π})) β (πβ{π})))) β π β (πβ({π} βͺ {π₯}))) |
| 25 | 2, 5, 7, 20, 24 | syl13anc 1372 | . . . 4 β’ ((π β§ Β¬ π₯ β (πβ{π})) β π β (πβ({π} βͺ {π₯}))) |
| 26 | df-pr 4624 | . . . . . 6 β’ {π, π₯} = ({π} βͺ {π₯}) | |
| 27 | prcom 4728 | . . . . . 6 β’ {π, π₯} = {π₯, π} | |
| 28 | 26, 27 | eqtr3i 2761 | . . . . 5 β’ ({π} βͺ {π₯}) = {π₯, π} |
| 29 | 28 | fveq2i 6880 | . . . 4 β’ (πβ({π} βͺ {π₯})) = (πβ{π₯, π}) |
| 30 | 25, 29 | eleqtrdi 2842 | . . 3 β’ ((π β§ Β¬ π₯ β (πβ{π})) β π β (πβ{π₯, π})) |
| 31 | 30 | ex 413 | . 2 β’ (π β (Β¬ π₯ β (πβ{π}) β π β (πβ{π₯, π}))) |
| 32 | 31 | orrd 861 | 1 β’ (π β (π₯ β (πβ{π}) β¨ π β (πβ{π₯, π}))) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 β¨ wo 845 = wceq 1541 β wcel 2106 β cdif 3940 βͺ cun 3941 β wss 3943 β wpss 3944 {csn 4621 {cpr 4623 βcfv 6531 Basecbs 17125 0gc0g 17366 LSubSpclss 20488 LSpanclspn 20528 LVecclvec 20659 |
| 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 5277 ax-sep 5291 ax-nul 5298 ax-pow 5355 ax-pr 5419 ax-un 7707 ax-cnex 11147 ax-resscn 11148 ax-1cn 11149 ax-icn 11150 ax-addcl 11151 ax-addrcl 11152 ax-mulcl 11153 ax-mulrcl 11154 ax-mulcom 11155 ax-addass 11156 ax-mulass 11157 ax-distr 11158 ax-i2m1 11159 ax-1ne0 11160 ax-1rid 11161 ax-rnegex 11162 ax-rrecex 11163 ax-cnre 11164 ax-pre-lttri 11165 ax-pre-lttrn 11166 ax-pre-ltadd 11167 ax-pre-mulgt0 11168 |
| 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 3474 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4943 df-iun 4991 df-br 5141 df-opab 5203 df-mpt 5224 df-tr 5258 df-id 5566 df-eprel 5572 df-po 5580 df-so 5581 df-fr 5623 df-we 5625 df-xp 5674 df-rel 5675 df-cnv 5676 df-co 5677 df-dm 5678 df-rn 5679 df-res 5680 df-ima 5681 df-pred 6288 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7348 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7838 df-1st 7956 df-2nd 7957 df-tpos 8192 df-frecs 8247 df-wrecs 8278 df-recs 8352 df-rdg 8391 df-er 8685 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11231 df-mnf 11232 df-xr 11233 df-ltxr 11234 df-le 11235 df-sub 11427 df-neg 11428 df-nn 12194 df-2 12256 df-3 12257 df-sets 17078 df-slot 17096 df-ndx 17108 df-base 17126 df-ress 17155 df-plusg 17191 df-mulr 17192 df-0g 17368 df-mgm 18542 df-sgrp 18591 df-mnd 18602 df-grp 18796 df-minusg 18797 df-sbg 18798 df-cmn 19613 df-abl 19614 df-mgp 19946 df-ur 19963 df-ring 20015 df-oppr 20101 df-dvdsr 20122 df-unit 20123 df-invr 20153 df-drng 20266 df-lmod 20419 df-lss 20489 df-lsp 20529 df-lvec 20660 |
| This theorem is referenced by: lsppratlem5 20710 |
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