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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdn0 | Structured version Visualization version GIF version |
Description: Transfer nonzero property from domain to range of projectivity mapd. (Contributed by NM, 12-Apr-2015.) |
Ref | Expression |
---|---|
mapdindp.h | β’ π» = (LHypβπΎ) |
mapdindp.m | β’ π = ((mapdβπΎ)βπ) |
mapdindp.u | β’ π = ((DVecHβπΎ)βπ) |
mapdindp.v | β’ π = (Baseβπ) |
mapdindp.n | β’ π = (LSpanβπ) |
mapdindp.c | β’ πΆ = ((LCDualβπΎ)βπ) |
mapdindp.d | β’ π· = (BaseβπΆ) |
mapdindp.j | β’ π½ = (LSpanβπΆ) |
mapdindp.k | β’ (π β (πΎ β HL β§ π β π»)) |
mapdindp.f | β’ (π β πΉ β π·) |
mapdindp.mx | β’ (π β (πβ(πβ{π})) = (π½β{πΉ})) |
mapdn0.o | β’ 0 = (0gβπ) |
mapdn0.z | β’ π = (0gβπΆ) |
mapdn0.x | β’ (π β π β (π β { 0 })) |
Ref | Expression |
---|---|
mapdn0 | β’ (π β πΉ β (π· β {π})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdindp.f | . 2 β’ (π β πΉ β π·) | |
2 | mapdn0.x | . . . 4 β’ (π β π β (π β { 0 })) | |
3 | eldifsni 4789 | . . . 4 β’ (π β (π β { 0 }) β π β 0 ) | |
4 | 2, 3 | syl 17 | . . 3 β’ (π β π β 0 ) |
5 | mapdindp.mx | . . . . . . . 8 β’ (π β (πβ(πβ{π})) = (π½β{πΉ})) | |
6 | sneq 4634 | . . . . . . . . 9 β’ (πΉ = π β {πΉ} = {π}) | |
7 | 6 | fveq2d 6895 | . . . . . . . 8 β’ (πΉ = π β (π½β{πΉ}) = (π½β{π})) |
8 | 5, 7 | sylan9eq 2787 | . . . . . . 7 β’ ((π β§ πΉ = π) β (πβ(πβ{π})) = (π½β{π})) |
9 | mapdindp.h | . . . . . . . . . 10 β’ π» = (LHypβπΎ) | |
10 | mapdindp.m | . . . . . . . . . 10 β’ π = ((mapdβπΎ)βπ) | |
11 | mapdindp.u | . . . . . . . . . 10 β’ π = ((DVecHβπΎ)βπ) | |
12 | mapdn0.o | . . . . . . . . . 10 β’ 0 = (0gβπ) | |
13 | mapdindp.c | . . . . . . . . . 10 β’ πΆ = ((LCDualβπΎ)βπ) | |
14 | mapdn0.z | . . . . . . . . . 10 β’ π = (0gβπΆ) | |
15 | mapdindp.k | . . . . . . . . . 10 β’ (π β (πΎ β HL β§ π β π»)) | |
16 | 9, 10, 11, 12, 13, 14, 15 | mapd0 41127 | . . . . . . . . 9 β’ (π β (πβ{ 0 }) = {π}) |
17 | 9, 13, 15 | lcdlmod 41054 | . . . . . . . . . 10 β’ (π β πΆ β LMod) |
18 | mapdindp.j | . . . . . . . . . . 11 β’ π½ = (LSpanβπΆ) | |
19 | 14, 18 | lspsn0 20885 | . . . . . . . . . 10 β’ (πΆ β LMod β (π½β{π}) = {π}) |
20 | 17, 19 | syl 17 | . . . . . . . . 9 β’ (π β (π½β{π}) = {π}) |
21 | 16, 20 | eqtr4d 2770 | . . . . . . . 8 β’ (π β (πβ{ 0 }) = (π½β{π})) |
22 | 21 | adantr 480 | . . . . . . 7 β’ ((π β§ πΉ = π) β (πβ{ 0 }) = (π½β{π})) |
23 | 8, 22 | eqtr4d 2770 | . . . . . 6 β’ ((π β§ πΉ = π) β (πβ(πβ{π})) = (πβ{ 0 })) |
24 | 23 | ex 412 | . . . . 5 β’ (π β (πΉ = π β (πβ(πβ{π})) = (πβ{ 0 }))) |
25 | eqid 2727 | . . . . . . 7 β’ (LSubSpβπ) = (LSubSpβπ) | |
26 | 9, 11, 15 | dvhlmod 40572 | . . . . . . . 8 β’ (π β π β LMod) |
27 | 2 | eldifad 3956 | . . . . . . . 8 β’ (π β π β π) |
28 | mapdindp.v | . . . . . . . . 9 β’ π = (Baseβπ) | |
29 | mapdindp.n | . . . . . . . . 9 β’ π = (LSpanβπ) | |
30 | 28, 25, 29 | lspsncl 20854 | . . . . . . . 8 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
31 | 26, 27, 30 | syl2anc 583 | . . . . . . 7 β’ (π β (πβ{π}) β (LSubSpβπ)) |
32 | 12, 25 | lsssn0 20825 | . . . . . . . 8 β’ (π β LMod β { 0 } β (LSubSpβπ)) |
33 | 26, 32 | syl 17 | . . . . . . 7 β’ (π β { 0 } β (LSubSpβπ)) |
34 | 9, 11, 25, 10, 15, 31, 33 | mapd11 41101 | . . . . . 6 β’ (π β ((πβ(πβ{π})) = (πβ{ 0 }) β (πβ{π}) = { 0 })) |
35 | 28, 12, 29 | lspsneq0 20889 | . . . . . . 7 β’ ((π β LMod β§ π β π) β ((πβ{π}) = { 0 } β π = 0 )) |
36 | 26, 27, 35 | syl2anc 583 | . . . . . 6 β’ (π β ((πβ{π}) = { 0 } β π = 0 )) |
37 | 34, 36 | bitrd 279 | . . . . 5 β’ (π β ((πβ(πβ{π})) = (πβ{ 0 }) β π = 0 )) |
38 | 24, 37 | sylibd 238 | . . . 4 β’ (π β (πΉ = π β π = 0 )) |
39 | 38 | necon3d 2956 | . . 3 β’ (π β (π β 0 β πΉ β π)) |
40 | 4, 39 | mpd 15 | . 2 β’ (π β πΉ β π) |
41 | eldifsn 4786 | . 2 β’ (πΉ β (π· β {π}) β (πΉ β π· β§ πΉ β π)) | |
42 | 1, 40, 41 | sylanbrc 582 | 1 β’ (π β πΉ β (π· β {π})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 β wne 2935 β cdif 3941 {csn 4624 βcfv 6542 Basecbs 17173 0gc0g 17414 LModclmod 20736 LSubSpclss 20808 LSpanclspn 20848 HLchlt 38811 LHypclh 39446 DVecHcdvh 40540 LCDualclcd 41048 mapdcmpd 41086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-riotaBAD 38414 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-n0 12497 df-z 12583 df-uz 12847 df-fz 13511 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-0g 17416 df-mre 17559 df-mrc 17560 df-acs 17562 df-proset 18280 df-poset 18298 df-plt 18315 df-lub 18331 df-glb 18332 df-join 18333 df-meet 18334 df-p0 18410 df-p1 18411 df-lat 18417 df-clat 18484 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-subg 19071 df-cntz 19261 df-oppg 19290 df-lsm 19584 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-oppr 20266 df-dvdsr 20289 df-unit 20290 df-invr 20320 df-dvr 20333 df-drng 20619 df-lmod 20738 df-lss 20809 df-lsp 20849 df-lvec 20981 df-lsatoms 38437 df-lshyp 38438 df-lcv 38480 df-lfl 38519 df-lkr 38547 df-ldual 38585 df-oposet 38637 df-ol 38639 df-oml 38640 df-covers 38727 df-ats 38728 df-atl 38759 df-cvlat 38783 df-hlat 38812 df-llines 38960 df-lplanes 38961 df-lvols 38962 df-lines 38963 df-psubsp 38965 df-pmap 38966 df-padd 39258 df-lhyp 39450 df-laut 39451 df-ldil 39566 df-ltrn 39567 df-trl 39621 df-tgrp 40205 df-tendo 40217 df-edring 40219 df-dveca 40465 df-disoa 40491 df-dvech 40541 df-dib 40601 df-dic 40635 df-dih 40691 df-doch 40810 df-djh 40857 df-lcdual 41049 df-mapd 41087 |
This theorem is referenced by: mapdheq4lem 41193 mapdh6lem1N 41195 mapdh6lem2N 41196 hdmap1l6lem1 41269 hdmap1l6lem2 41270 |
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