![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > islshpkrN | Structured version Visualization version GIF version |
Description: The predicate "is a hyperplane" (of a left module or left vector space). TODO: should it be π = (πΎβπ) or (πΎβπ) = π as in lshpkrex 38646? Both standards seem to be used randomly throughout set.mm; we should decide on a preferred one. (Contributed by NM, 7-Oct-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lshpset2.v | β’ π = (Baseβπ) |
lshpset2.d | β’ π· = (Scalarβπ) |
lshpset2.z | β’ 0 = (0gβπ·) |
lshpset2.h | β’ π» = (LSHypβπ) |
lshpset2.f | β’ πΉ = (LFnlβπ) |
lshpset2.k | β’ πΎ = (LKerβπ) |
Ref | Expression |
---|---|
islshpkrN | β’ (π β LVec β (π β π» β βπ β πΉ (π β (π Γ { 0 }) β§ π = (πΎβπ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lshpset2.v | . . . 4 β’ π = (Baseβπ) | |
2 | lshpset2.d | . . . 4 β’ π· = (Scalarβπ) | |
3 | lshpset2.z | . . . 4 β’ 0 = (0gβπ·) | |
4 | lshpset2.h | . . . 4 β’ π» = (LSHypβπ) | |
5 | lshpset2.f | . . . 4 β’ πΉ = (LFnlβπ) | |
6 | lshpset2.k | . . . 4 β’ πΎ = (LKerβπ) | |
7 | 1, 2, 3, 4, 5, 6 | lshpset2N 38647 | . . 3 β’ (π β LVec β π» = {π β£ βπ β πΉ (π β (π Γ { 0 }) β§ π = (πΎβπ))}) |
8 | 7 | eleq2d 2811 | . 2 β’ (π β LVec β (π β π» β π β {π β£ βπ β πΉ (π β (π Γ { 0 }) β§ π = (πΎβπ))})) |
9 | elex 3482 | . . . 4 β’ (π β {π β£ βπ β πΉ (π β (π Γ { 0 }) β§ π = (πΎβπ))} β π β V) | |
10 | 9 | adantl 480 | . . 3 β’ ((π β LVec β§ π β {π β£ βπ β πΉ (π β (π Γ { 0 }) β§ π = (πΎβπ))}) β π β V) |
11 | fvex 6905 | . . . . . . 7 β’ (πΎβπ) β V | |
12 | eleq1 2813 | . . . . . . 7 β’ (π = (πΎβπ) β (π β V β (πΎβπ) β V)) | |
13 | 11, 12 | mpbiri 257 | . . . . . 6 β’ (π = (πΎβπ) β π β V) |
14 | 13 | adantl 480 | . . . . 5 β’ ((π β (π Γ { 0 }) β§ π = (πΎβπ)) β π β V) |
15 | 14 | rexlimivw 3141 | . . . 4 β’ (βπ β πΉ (π β (π Γ { 0 }) β§ π = (πΎβπ)) β π β V) |
16 | 15 | adantl 480 | . . 3 β’ ((π β LVec β§ βπ β πΉ (π β (π Γ { 0 }) β§ π = (πΎβπ))) β π β V) |
17 | eqeq1 2729 | . . . . . 6 β’ (π = π β (π = (πΎβπ) β π = (πΎβπ))) | |
18 | 17 | anbi2d 628 | . . . . 5 β’ (π = π β ((π β (π Γ { 0 }) β§ π = (πΎβπ)) β (π β (π Γ { 0 }) β§ π = (πΎβπ)))) |
19 | 18 | rexbidv 3169 | . . . 4 β’ (π = π β (βπ β πΉ (π β (π Γ { 0 }) β§ π = (πΎβπ)) β βπ β πΉ (π β (π Γ { 0 }) β§ π = (πΎβπ)))) |
20 | 19 | elabg 3657 | . . 3 β’ (π β V β (π β {π β£ βπ β πΉ (π β (π Γ { 0 }) β§ π = (πΎβπ))} β βπ β πΉ (π β (π Γ { 0 }) β§ π = (πΎβπ)))) |
21 | 10, 16, 20 | pm5.21nd 800 | . 2 β’ (π β LVec β (π β {π β£ βπ β πΉ (π β (π Γ { 0 }) β§ π = (πΎβπ))} β βπ β πΉ (π β (π Γ { 0 }) β§ π = (πΎβπ)))) |
22 | 8, 21 | bitrd 278 | 1 β’ (π β LVec β (π β π» β βπ β πΉ (π β (π Γ { 0 }) β§ π = (πΎβπ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 {cab 2702 β wne 2930 βwrex 3060 Vcvv 3463 {csn 4624 Γ cxp 5670 βcfv 6543 Basecbs 17179 Scalarcsca 17235 0gc0g 17420 LVecclvec 20991 LSHypclsh 38503 LFnlclfn 38585 LKerclk 38613 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-subg 19082 df-cntz 19272 df-lsm 19595 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-drng 20630 df-lmod 20749 df-lss 20820 df-lsp 20860 df-lvec 20992 df-lshyp 38505 df-lfl 38586 df-lkr 38614 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |