![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > hgmapeq0 | Structured version Visualization version Unicode version |
Description: The scalar sigma map is zero iff its argument is zero. (Contributed by NM, 12-Jun-2015.) |
Ref | Expression |
---|---|
hgmapeq0.h |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
hgmapeq0.u |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
hgmapeq0.r |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
hgmapeq0.b |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
hgmapeq0.o |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
hgmapeq0.g |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
hgmapeq0.k |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
hgmapeq0.x |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
hgmapeq0 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hgmapeq0.h |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | hgmapeq0.u |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | hgmapeq0.r |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | hgmapeq0.o |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | hgmapeq0.g |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | hgmapeq0.k |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 1, 2, 3, 4, 5, 6 | hgmapval0 35475 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | 7 | eqeq2d 2463 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | hgmapeq0.b |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | hgmapeq0.x |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 1, 2, 6 | dvhlmod 34690 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 3, 9, 4 | lmod0cl 18129 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | 11, 12 | syl 17 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 1, 2, 3, 9, 5, 6, 10, 13 | hgmap11 35485 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 8, 14 | bitr3d 259 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 ax-cnex 9600 ax-resscn 9601 ax-1cn 9602 ax-icn 9603 ax-addcl 9604 ax-addrcl 9605 ax-mulcl 9606 ax-mulrcl 9607 ax-mulcom 9608 ax-addass 9609 ax-mulass 9610 ax-distr 9611 ax-i2m1 9612 ax-1ne0 9613 ax-1rid 9614 ax-rnegex 9615 ax-rrecex 9616 ax-cnre 9617 ax-pre-lttri 9618 ax-pre-lttrn 9619 ax-pre-ltadd 9620 ax-pre-mulgt0 9621 ax-riotaBAD 32537 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-fal 1452 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-nel 2627 df-ral 2744 df-rex 2745 df-reu 2746 df-rmo 2747 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-pss 3422 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-tp 3975 df-op 3977 df-ot 3979 df-uni 4202 df-int 4238 df-iun 4283 df-iin 4284 df-br 4406 df-opab 4465 df-mpt 4466 df-tr 4501 df-eprel 4748 df-id 4752 df-po 4758 df-so 4759 df-fr 4796 df-we 4798 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-pred 5383 df-ord 5429 df-on 5430 df-lim 5431 df-suc 5432 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-riota 6257 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-of 6536 df-om 6698 df-1st 6798 df-2nd 6799 df-tpos 6978 df-undef 7025 df-wrecs 7033 df-recs 7095 df-rdg 7133 df-1o 7187 df-oadd 7191 df-er 7368 df-map 7479 df-en 7575 df-dom 7576 df-sdom 7577 df-fin 7578 df-pnf 9682 df-mnf 9683 df-xr 9684 df-ltxr 9685 df-le 9686 df-sub 9867 df-neg 9868 df-nn 10617 df-2 10675 df-3 10676 df-4 10677 df-5 10678 df-6 10679 df-n0 10877 df-z 10945 df-uz 11167 df-fz 11792 df-struct 15135 df-ndx 15136 df-slot 15137 df-base 15138 df-sets 15139 df-ress 15140 df-plusg 15215 df-mulr 15216 df-sca 15218 df-vsca 15219 df-0g 15352 df-mre 15504 df-mrc 15505 df-acs 15507 df-preset 16185 df-poset 16203 df-plt 16216 df-lub 16232 df-glb 16233 df-join 16234 df-meet 16235 df-p0 16297 df-p1 16298 df-lat 16304 df-clat 16366 df-mgm 16500 df-sgrp 16539 df-mnd 16549 df-submnd 16595 df-grp 16685 df-minusg 16686 df-sbg 16687 df-subg 16826 df-cntz 16983 df-oppg 17009 df-lsm 17300 df-cmn 17444 df-abl 17445 df-mgp 17736 df-ur 17748 df-ring 17794 df-oppr 17863 df-dvdsr 17881 df-unit 17882 df-invr 17912 df-dvr 17923 df-drng 17989 df-lmod 18105 df-lss 18168 df-lsp 18207 df-lvec 18338 df-lsatoms 32554 df-lshyp 32555 df-lcv 32597 df-lfl 32636 df-lkr 32664 df-ldual 32702 df-oposet 32754 df-ol 32756 df-oml 32757 df-covers 32844 df-ats 32845 df-atl 32876 df-cvlat 32900 df-hlat 32929 df-llines 33075 df-lplanes 33076 df-lvols 33077 df-lines 33078 df-psubsp 33080 df-pmap 33081 df-padd 33373 df-lhyp 33565 df-laut 33566 df-ldil 33681 df-ltrn 33682 df-trl 33737 df-tgrp 34322 df-tendo 34334 df-edring 34336 df-dveca 34582 df-disoa 34609 df-dvech 34659 df-dib 34719 df-dic 34753 df-dih 34809 df-doch 34928 df-djh 34975 df-lcdual 35167 df-mapd 35205 df-hvmap 35337 df-hdmap1 35374 df-hdmap 35375 df-hgmap 35467 |
This theorem is referenced by: hgmapvvlem1 35506 hgmapvvlem2 35507 |
Copyright terms: Public domain | W3C validator |