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| 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 41276 | . . 3 β’ (π β (πΊβ 0 ) = 0 ) |
| 8 | 7 | eqeq2d 2737 | . 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 40494 | . . . 4 β’ (π β π β LMod) |
| 12 | 3, 9, 4 | lmod0cl 20734 | . . . 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 41286 | . 2 β’ (π β ((πΊβπ) = (πΊβ 0 ) β π = 0 )) |
| 15 | 8, 14 | bitr3d 281 | 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 395 = wceq 1533 β wcel 2098 βcfv 6537 Basecbs 17153 Scalarcsca 17209 0gc0g 17394 LModclmod 20706 HLchlt 38733 LHypclh 39368 DVecHcdvh 40462 HGMapchg 41267 |
| 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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-riotaBAD 38336 |
| 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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-ot 4632 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-undef 8259 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-0g 17396 df-mre 17539 df-mrc 17540 df-acs 17542 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-cntz 19233 df-oppg 19262 df-lsm 19556 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-drng 20589 df-lmod 20708 df-lss 20779 df-lsp 20819 df-lvec 20951 df-lsatoms 38359 df-lshyp 38360 df-lcv 38402 df-lfl 38441 df-lkr 38469 df-ldual 38507 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-llines 38882 df-lplanes 38883 df-lvols 38884 df-lines 38885 df-psubsp 38887 df-pmap 38888 df-padd 39180 df-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 df-trl 39543 df-tgrp 40127 df-tendo 40139 df-edring 40141 df-dveca 40387 df-disoa 40413 df-dvech 40463 df-dib 40523 df-dic 40557 df-dih 40613 df-doch 40732 df-djh 40779 df-lcdual 40971 df-mapd 41009 df-hvmap 41141 df-hdmap1 41177 df-hdmap 41178 df-hgmap 41268 |
| This theorem is referenced by: hgmapvvlem1 41307 hgmapvvlem2 41308 |
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