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| Mirrors > Home > MPE Home > Th. List > lspsncmp | Structured version Visualization version GIF version | ||
| Description: Comparable spans of nonzero singletons are equal. (Contributed by NM, 27-Apr-2015.) |
| Ref | Expression |
|---|---|
| lspsncmp.v | β’ π = (Baseβπ) |
| lspsncmp.o | β’ 0 = (0gβπ) |
| lspsncmp.n | β’ π = (LSpanβπ) |
| lspsncmp.w | β’ (π β π β LVec) |
| lspsncmp.x | β’ (π β π β (π β { 0 })) |
| lspsncmp.y | β’ (π β π β π) |
| Ref | Expression |
|---|---|
| lspsncmp | β’ (π β ((πβ{π}) β (πβ{π}) β (πβ{π}) = (πβ{π}))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lspsncmp.v | . . . 4 β’ π = (Baseβπ) | |
| 2 | lspsncmp.o | . . . 4 β’ 0 = (0gβπ) | |
| 3 | lspsncmp.n | . . . 4 β’ π = (LSpanβπ) | |
| 4 | lspsncmp.w | . . . . 5 β’ (π β π β LVec) | |
| 5 | 4 | adantr 479 | . . . 4 β’ ((π β§ (πβ{π}) β (πβ{π})) β π β LVec) |
| 6 | lspsncmp.y | . . . . 5 β’ (π β π β π) | |
| 7 | 6 | adantr 479 | . . . 4 β’ ((π β§ (πβ{π}) β (πβ{π})) β π β π) |
| 8 | eqid 2725 | . . . . . 6 β’ (LSubSpβπ) = (LSubSpβπ) | |
| 9 | lveclmod 20993 | . . . . . . 7 β’ (π β LVec β π β LMod) | |
| 10 | 4, 9 | syl 17 | . . . . . 6 β’ (π β π β LMod) |
| 11 | 1, 8, 3 | lspsncl 20863 | . . . . . . 7 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
| 12 | 10, 6, 11 | syl2anc 582 | . . . . . 6 β’ (π β (πβ{π}) β (LSubSpβπ)) |
| 13 | lspsncmp.x | . . . . . . 7 β’ (π β π β (π β { 0 })) | |
| 14 | 13 | eldifad 3951 | . . . . . 6 β’ (π β π β π) |
| 15 | 1, 8, 3, 10, 12, 14 | lspsnel5 20881 | . . . . 5 β’ (π β (π β (πβ{π}) β (πβ{π}) β (πβ{π}))) |
| 16 | 15 | biimpar 476 | . . . 4 β’ ((π β§ (πβ{π}) β (πβ{π})) β π β (πβ{π})) |
| 17 | eldifsni 4787 | . . . . . 6 β’ (π β (π β { 0 }) β π β 0 ) | |
| 18 | 13, 17 | syl 17 | . . . . 5 β’ (π β π β 0 ) |
| 19 | 18 | adantr 479 | . . . 4 β’ ((π β§ (πβ{π}) β (πβ{π})) β π β 0 ) |
| 20 | 1, 2, 3, 5, 7, 16, 19 | lspsneleq 21005 | . . 3 β’ ((π β§ (πβ{π}) β (πβ{π})) β (πβ{π}) = (πβ{π})) |
| 21 | 20 | ex 411 | . 2 β’ (π β ((πβ{π}) β (πβ{π}) β (πβ{π}) = (πβ{π}))) |
| 22 | eqimss 4030 | . 2 β’ ((πβ{π}) = (πβ{π}) β (πβ{π}) β (πβ{π})) | |
| 23 | 21, 22 | impbid1 224 | 1 β’ (π β ((πβ{π}) β (πβ{π}) β (πβ{π}) = (πβ{π}))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2930 β cdif 3936 β wss 3939 {csn 4622 βcfv 6541 Basecbs 17177 0gc0g 17418 LModclmod 20745 LSubSpclss 20817 LSpanclspn 20857 LVecclvec 20989 |
| 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 2166 ax-ext 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7989 df-2nd 7990 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-0g 17420 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-grp 18895 df-minusg 18896 df-sbg 18897 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-drng 20628 df-lmod 20747 df-lss 20818 df-lsp 20858 df-lvec 20990 |
| This theorem is referenced by: lspsnne1 21007 lspabs2 21010 lspabs3 21011 lsatfixedN 38509 mapdindp0 41220 hdmaprnlem4N 41354 |
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