Nearby theorems
|
|
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > hgmapeq0 | Structured version Visualization version GIF version | ||
| Description: The scalar sigma map is zero iff its argument is zero. (Contributed by NM, 12-Jun-2015.) |
| Ref | Expression |
|---|---|
| hgmapeq0.h | β’ π» = (LHypβπΎ) |
| hgmapeq0.u | β’ π = ((DVecHβπΎ)βπ) |
| hgmapeq0.r | β’ π = (Scalarβπ) |
| hgmapeq0.b | β’ π΅ = (Baseβπ ) |
| hgmapeq0.o | β’ 0 = (0gβπ ) |
| hgmapeq0.g | β’ πΊ = ((HGMapβπΎ)βπ) |
| hgmapeq0.k | β’ (π β (πΎ β HL β§ π β π»)) |
| hgmapeq0.x | β’ (π β π β π΅) |
| Ref | Expression |
|---|---|
| hgmapeq0 | β’ (π β ((πΊβπ) = 0 β π = 0 )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hgmapeq0.h | . . . 4 β’ π» = (LHypβπΎ) | |
| 2 | hgmapeq0.u | . . . 4 β’ π = ((DVecHβπΎ)βπ) | |
| 3 | hgmapeq0.r | . . . 4 β’ π = (Scalarβπ) | |
| 4 | hgmapeq0.o | . . . 4 β’ 0 = (0gβπ ) | |
| 5 | hgmapeq0.g | . . . 4 β’ πΊ = ((HGMapβπΎ)βπ) | |
| 6 | hgmapeq0.k | . . . 4 β’ (π β (πΎ β HL β§ π β π»)) | |
| 7 | 1, 2, 3, 4, 5, 6 | hgmapval0 41405 | . . 3 β’ (π β (πΊβ 0 ) = 0 ) |
| 8 | 7 | eqeq2d 2739 | . 2 β’ (π β ((πΊβπ) = (πΊβ 0 ) β (πΊβπ) = 0 )) |
| 9 | hgmapeq0.b | . . 3 β’ π΅ = (Baseβπ ) | |
| 10 | hgmapeq0.x | . . 3 β’ (π β π β π΅) | |
| 11 | 1, 2, 6 | dvhlmod 40623 | . . . 4 β’ (π β π β LMod) |
| 12 | 3, 9, 4 | lmod0cl 20785 | . . . 4 β’ (π β LMod β 0 β π΅) |
| 13 | 11, 12 | syl 17 | . . 3 β’ (π β 0 β π΅) |
| 14 | 1, 2, 3, 9, 5, 6, 10, 13 | hgmap11 41415 | . 2 β’ (π β ((πΊβπ) = (πΊβ 0 ) β π = 0 )) |
| 15 | 8, 14 | bitr3d 280 | 1 β’ (π β ((πΊβπ) = 0 β π = 0 )) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 βcfv 6553 Basecbs 17189 Scalarcsca 17245 0gc0g 17430 LModclmod 20757 HLchlt 38862 LHypclh 39497 DVecHcdvh 40591 HGMapchg 41396 |
| 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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-riotaBAD 38465 |
| 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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-ot 4641 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8001 df-2nd 8002 df-tpos 8240 df-undef 8287 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-n0 12513 df-z 12599 df-uz 12863 df-fz 13527 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-sca 17258 df-vsca 17259 df-0g 17432 df-mre 17575 df-mrc 17576 df-acs 17578 df-proset 18296 df-poset 18314 df-plt 18331 df-lub 18347 df-glb 18348 df-join 18349 df-meet 18350 df-p0 18426 df-p1 18427 df-lat 18433 df-clat 18500 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-submnd 18750 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19092 df-cntz 19282 df-oppg 19311 df-lsm 19605 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-oppr 20287 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-dvr 20354 df-drng 20640 df-lmod 20759 df-lss 20830 df-lsp 20870 df-lvec 21002 df-lsatoms 38488 df-lshyp 38489 df-lcv 38531 df-lfl 38570 df-lkr 38598 df-ldual 38636 df-oposet 38688 df-ol 38690 df-oml 38691 df-covers 38778 df-ats 38779 df-atl 38810 df-cvlat 38834 df-hlat 38863 df-llines 39011 df-lplanes 39012 df-lvols 39013 df-lines 39014 df-psubsp 39016 df-pmap 39017 df-padd 39309 df-lhyp 39501 df-laut 39502 df-ldil 39617 df-ltrn 39618 df-trl 39672 df-tgrp 40256 df-tendo 40268 df-edring 40270 df-dveca 40516 df-disoa 40542 df-dvech 40592 df-dib 40652 df-dic 40686 df-dih 40742 df-doch 40861 df-djh 40908 df-lcdual 41100 df-mapd 41138 df-hvmap 41270 df-hdmap1 41306 df-hdmap 41307 df-hgmap 41397 |
| This theorem is referenced by: hgmapvvlem1 41436 hgmapvvlem2 41437 |
| Copyright terms: Public domain | W3C validator |