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Mirrors > Home > MPE Home > Th. List > lsppratlem2 | Structured version Visualization version GIF version |
Description: Lemma for lspprat 20994. Show that if π and π are both in (πβ{π₯, π¦}) (which will be our goal for each of the two cases above), then (πβ{π, π}) β π, contradicting the hypothesis for π. (Contributed by NM, 29-Aug-2014.) (Revised by Mario Carneiro, 5-Sep-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₯}))) |
lsppratlem2.x1 | β’ (π β π β (πβ{π₯, π¦})) |
lsppratlem2.y1 | β’ (π β π β (πβ{π₯, π¦})) |
Ref | Expression |
---|---|
lsppratlem2 | β’ (π β (πβ{π, π}) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspprat.s | . . 3 β’ π = (LSubSpβπ) | |
2 | lspprat.n | . . 3 β’ π = (LSpanβπ) | |
3 | lspprat.w | . . . 4 β’ (π β π β LVec) | |
4 | lveclmod 20944 | . . . 4 β’ (π β LVec β π β LMod) | |
5 | 3, 4 | syl 17 | . . 3 β’ (π β π β LMod) |
6 | lsppratlem1.x2 | . . . . . . 7 β’ (π β π₯ β (π β { 0 })) | |
7 | 6 | eldifad 3952 | . . . . . 6 β’ (π β π₯ β π) |
8 | lsppratlem1.y2 | . . . . . . 7 β’ (π β π¦ β (π β (πβ{π₯}))) | |
9 | 8 | eldifad 3952 | . . . . . 6 β’ (π β π¦ β π) |
10 | 7, 9 | prssd 4817 | . . . . 5 β’ (π β {π₯, π¦} β π) |
11 | lspprat.u | . . . . . 6 β’ (π β π β π) | |
12 | lspprat.v | . . . . . . 7 β’ π = (Baseβπ) | |
13 | 12, 1 | lssss 20773 | . . . . . 6 β’ (π β π β π β π) |
14 | 11, 13 | syl 17 | . . . . 5 β’ (π β π β π) |
15 | 10, 14 | sstrd 3984 | . . . 4 β’ (π β {π₯, π¦} β π) |
16 | 12, 1, 2 | lspcl 20813 | . . . 4 β’ ((π β LMod β§ {π₯, π¦} β π) β (πβ{π₯, π¦}) β π) |
17 | 5, 15, 16 | syl2anc 583 | . . 3 β’ (π β (πβ{π₯, π¦}) β π) |
18 | lsppratlem2.x1 | . . 3 β’ (π β π β (πβ{π₯, π¦})) | |
19 | lsppratlem2.y1 | . . 3 β’ (π β π β (πβ{π₯, π¦})) | |
20 | 1, 2, 5, 17, 18, 19 | lspprss 20829 | . 2 β’ (π β (πβ{π, π}) β (πβ{π₯, π¦})) |
21 | 1, 2, 5, 11, 7, 9 | lspprss 20829 | . 2 β’ (π β (πβ{π₯, π¦}) β π) |
22 | 20, 21 | sstrd 3984 | 1 β’ (π β (πβ{π, π}) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β cdif 3937 β wss 3940 β wpss 3941 {csn 4620 {cpr 4622 βcfv 6533 Basecbs 17143 0gc0g 17384 LModclmod 20696 LSubSpclss 20768 LSpanclspn 20808 LVecclvec 20940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-0g 17386 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-minusg 18857 df-sbg 18858 df-mgp 20030 df-ur 20077 df-ring 20130 df-lmod 20698 df-lss 20769 df-lsp 20809 df-lvec 20941 |
This theorem is referenced by: lsppratlem5 20992 |
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