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| Mirrors > Home > MPE Home > Th. List > lsppratlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for lspprat 20912. 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 20902 (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 480 | . . . . 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 4812 | . . . . . 6 β’ (π β {π} β π) |
| 5 | 4 | adantr 480 | . . . . 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 480 | . . . . 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 4097 | . . . . . . . . 9 β’ (π β π β (πβ{π, π})) |
| 10 | lsppratlem1.x2 | . . . . . . . . . 10 β’ (π β π₯ β (π β { 0 })) | |
| 11 | 10 | eldifad 3960 | . . . . . . . . 9 β’ (π β π₯ β π) |
| 12 | 9, 11 | sseldd 3983 | . . . . . . . 8 β’ (π β π₯ β (πβ{π, π})) |
| 13 | prcom 4736 | . . . . . . . . . 10 β’ {π, π} = {π, π} | |
| 14 | df-pr 4631 | . . . . . . . . . 10 β’ {π, π} = ({π} βͺ {π}) | |
| 15 | 13, 14 | eqtri 2759 | . . . . . . . . 9 β’ {π, π} = ({π} βͺ {π}) |
| 16 | 15 | fveq2i 6894 | . . . . . . . 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 614 | . . . . . 6 β’ ((π β§ Β¬ π₯ β (πβ{π})) β (π₯ β (πβ({π} βͺ {π})) β§ Β¬ π₯ β (πβ{π}))) |
| 19 | eldif 3958 | . . . . . 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 20902 | . . . . 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 1371 | . . . 4 β’ ((π β§ Β¬ π₯ β (πβ{π})) β π β (πβ({π} βͺ {π₯}))) |
| 26 | df-pr 4631 | . . . . . 6 β’ {π, π₯} = ({π} βͺ {π₯}) | |
| 27 | prcom 4736 | . . . . . 6 β’ {π, π₯} = {π₯, π} | |
| 28 | 26, 27 | eqtr3i 2761 | . . . . 5 β’ ({π} βͺ {π₯}) = {π₯, π} |
| 29 | 28 | fveq2i 6894 | . . . 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 412 | . 2 β’ (π β (Β¬ π₯ β (πβ{π}) β π β (πβ{π₯, π}))) |
| 32 | 31 | orrd 860 | 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 395 β¨ wo 844 = wceq 1540 β wcel 2105 β cdif 3945 βͺ cun 3946 β wss 3948 β wpss 3949 {csn 4628 {cpr 4630 βcfv 6543 Basecbs 17149 0gc0g 17390 LSubSpclss 20687 LSpanclspn 20727 LVecclvec 20858 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8215 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-0g 17392 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-minusg 18860 df-sbg 18861 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-drng 20503 df-lmod 20617 df-lss 20688 df-lsp 20728 df-lvec 20859 |
| This theorem is referenced by: lsppratlem5 20910 |
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