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| Mirrors > Home > MPE Home > Th. List > lspindp1 | Structured version Visualization version GIF version | ||
| Description: Alternate way to say 3 vectors are mutually independent (swap 1st and 2nd). (Contributed by NM, 11-Apr-2015.) |
| Ref | Expression |
|---|---|
| lspindp1.v | β’ π = (Baseβπ) |
| lspindp1.o | β’ 0 = (0gβπ) |
| lspindp1.n | β’ π = (LSpanβπ) |
| lspindp1.w | β’ (π β π β LVec) |
| lspindp1.y | β’ (π β π β (π β { 0 })) |
| lspindp1.z | β’ (π β π β π) |
| lspindp1.x | β’ (π β π β π) |
| lspindp1.q | β’ (π β (πβ{π}) β (πβ{π})) |
| lspindp1.e | β’ (π β Β¬ π β (πβ{π, π})) |
| Ref | Expression |
|---|---|
| lspindp1 | β’ (π β ((πβ{π}) β (πβ{π}) β§ Β¬ π β (πβ{π, π}))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lspindp1.v | . . . 4 β’ π = (Baseβπ) | |
| 2 | lspindp1.n | . . . 4 β’ π = (LSpanβπ) | |
| 3 | lspindp1.w | . . . 4 β’ (π β π β LVec) | |
| 4 | lspindp1.x | . . . 4 β’ (π β π β π) | |
| 5 | lspindp1.y | . . . . 5 β’ (π β π β (π β { 0 })) | |
| 6 | 5 | eldifad 3952 | . . . 4 β’ (π β π β π) |
| 7 | lspindp1.z | . . . 4 β’ (π β π β π) | |
| 8 | lspindp1.e | . . . 4 β’ (π β Β¬ π β (πβ{π, π})) | |
| 9 | 1, 2, 3, 4, 6, 7, 8 | lspindpi 20968 | . . 3 β’ (π β ((πβ{π}) β (πβ{π}) β§ (πβ{π}) β (πβ{π}))) |
| 10 | 9 | simprd 495 | . 2 β’ (π β (πβ{π}) β (πβ{π})) |
| 11 | lspindp1.o | . . . 4 β’ 0 = (0gβπ) | |
| 12 | 3 | adantr 480 | . . . 4 β’ ((π β§ π β (πβ{π, π})) β π β LVec) |
| 13 | 5 | adantr 480 | . . . 4 β’ ((π β§ π β (πβ{π, π})) β π β (π β { 0 })) |
| 14 | 4 | adantr 480 | . . . 4 β’ ((π β§ π β (πβ{π, π})) β π β π) |
| 15 | 7 | adantr 480 | . . . 4 β’ ((π β§ π β (πβ{π, π})) β π β π) |
| 16 | lspindp1.q | . . . . 5 β’ (π β (πβ{π}) β (πβ{π})) | |
| 17 | 16 | adantr 480 | . . . 4 β’ ((π β§ π β (πβ{π, π})) β (πβ{π}) β (πβ{π})) |
| 18 | simpr 484 | . . . 4 β’ ((π β§ π β (πβ{π, π})) β π β (πβ{π, π})) | |
| 19 | 1, 11, 2, 12, 13, 14, 15, 17, 18 | lspexch 20965 | . . 3 β’ ((π β§ π β (πβ{π, π})) β π β (πβ{π, π})) |
| 20 | 8, 19 | mtand 813 | . 2 β’ (π β Β¬ π β (πβ{π, π})) |
| 21 | 10, 20 | jca 511 | 1 β’ (π β ((πβ{π}) β (πβ{π}) β§ Β¬ π β (πβ{π, π}))) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2932 β cdif 3937 {csn 4620 {cpr 4622 βcfv 6533 Basecbs 17140 0gc0g 17381 LSpanclspn 20803 LVecclvec 20935 |
| 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 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
| 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-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-0g 17383 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-submnd 18701 df-grp 18853 df-minusg 18854 df-sbg 18855 df-subg 19035 df-cntz 19218 df-lsm 19541 df-cmn 19687 df-abl 19688 df-mgp 20025 df-rng 20043 df-ur 20072 df-ring 20125 df-oppr 20221 df-dvdsr 20244 df-unit 20245 df-invr 20275 df-drng 20574 df-lmod 20693 df-lss 20764 df-lsp 20804 df-lvec 20936 |
| This theorem is referenced by: lspindp2l 20970 lspindp2 20971 mapdindp3 41049 mapdindp4 41050 mapdheq4lem 41058 mapdheq4 41059 mapdh6lem1N 41060 mapdh6lem2N 41061 mapdh6aN 41062 mapdh6dN 41066 mapdh6eN 41067 mapdh6fN 41068 mapdh7dN 41077 hdmap1l6lem1 41134 hdmap1l6lem2 41135 hdmap1l6a 41136 hdmap1l6d 41140 hdmap1l6e 41141 hdmap1l6f 41142 |
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