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Mirrors > Home > MPE Home > Th. List > obsne0 | Structured version Visualization version GIF version |
Description: A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Ref | Expression |
---|---|
obsocv.z | β’ 0 = (0gβπ) |
Ref | Expression |
---|---|
obsne0 | β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β π΄ β 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | obsrcl 21145 | . . . . 5 β’ (π΅ β (OBasisβπ) β π β PreHil) | |
2 | phllvec 21049 | . . . . 5 β’ (π β PreHil β π β LVec) | |
3 | eqid 2733 | . . . . . 6 β’ (Scalarβπ) = (Scalarβπ) | |
4 | 3 | lvecdrng 20581 | . . . . 5 β’ (π β LVec β (Scalarβπ) β DivRing) |
5 | 1, 2, 4 | 3syl 18 | . . . 4 β’ (π΅ β (OBasisβπ) β (Scalarβπ) β DivRing) |
6 | 5 | adantr 482 | . . 3 β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β (Scalarβπ) β DivRing) |
7 | eqid 2733 | . . . 4 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
8 | eqid 2733 | . . . 4 β’ (1rβ(Scalarβπ)) = (1rβ(Scalarβπ)) | |
9 | 7, 8 | drngunz 20215 | . . 3 β’ ((Scalarβπ) β DivRing β (1rβ(Scalarβπ)) β (0gβ(Scalarβπ))) |
10 | 6, 9 | syl 17 | . 2 β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β (1rβ(Scalarβπ)) β (0gβ(Scalarβπ))) |
11 | eqid 2733 | . . . . . 6 β’ (Β·πβπ) = (Β·πβπ) | |
12 | 11, 3, 8 | obsipid 21144 | . . . . 5 β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β (π΄(Β·πβπ)π΄) = (1rβ(Scalarβπ))) |
13 | 12 | eqeq1d 2735 | . . . 4 β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β ((π΄(Β·πβπ)π΄) = (0gβ(Scalarβπ)) β (1rβ(Scalarβπ)) = (0gβ(Scalarβπ)))) |
14 | eqid 2733 | . . . . . . 7 β’ (Baseβπ) = (Baseβπ) | |
15 | 14 | obsss 21146 | . . . . . 6 β’ (π΅ β (OBasisβπ) β π΅ β (Baseβπ)) |
16 | 15 | sselda 3945 | . . . . 5 β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β π΄ β (Baseβπ)) |
17 | obsocv.z | . . . . . 6 β’ 0 = (0gβπ) | |
18 | 3, 11, 14, 7, 17 | ipeq0 21058 | . . . . 5 β’ ((π β PreHil β§ π΄ β (Baseβπ)) β ((π΄(Β·πβπ)π΄) = (0gβ(Scalarβπ)) β π΄ = 0 )) |
19 | 1, 16, 18 | syl2an2r 684 | . . . 4 β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β ((π΄(Β·πβπ)π΄) = (0gβ(Scalarβπ)) β π΄ = 0 )) |
20 | 13, 19 | bitr3d 281 | . . 3 β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β ((1rβ(Scalarβπ)) = (0gβ(Scalarβπ)) β π΄ = 0 )) |
21 | 20 | necon3bid 2985 | . 2 β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β ((1rβ(Scalarβπ)) β (0gβ(Scalarβπ)) β π΄ β 0 )) |
22 | 10, 21 | mpbid 231 | 1 β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β π΄ β 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2940 βcfv 6497 (class class class)co 7358 Basecbs 17088 Scalarcsca 17141 Β·πcip 17143 0gc0g 17326 1rcur 19918 DivRingcdr 20197 LVecclvec 20578 PreHilcphl 21044 OBasiscobs 21124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-ghm 19011 df-mgp 19902 df-ur 19919 df-ring 19971 df-oppr 20054 df-dvdsr 20075 df-unit 20076 df-drng 20199 df-lmod 20338 df-lmhm 20498 df-lvec 20579 df-sra 20649 df-rgmod 20650 df-phl 21046 df-obs 21127 |
This theorem is referenced by: obselocv 21150 obs2ss 21151 obslbs 21152 |
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