Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdindp.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdindp.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdindp.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdindp.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdindp.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdindp.d | ⊢ 𝐷 = (Base‘𝐶) |
mapdindp.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdindp.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdindp.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
mapdindp.mx | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
mapdn0.o | ⊢ 0 = (0g‘𝑈) |
mapdn0.z | ⊢ 𝑍 = (0g‘𝐶) |
mapdn0.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
Ref | Expression |
---|---|
mapdn0 | ⊢ (𝜑 → 𝐹 ∈ (𝐷 ∖ {𝑍})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdindp.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
2 | mapdn0.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
3 | eldifsni 4708 | . . . 4 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 ) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
5 | mapdindp.mx | . . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
6 | sneq 4563 | . . . . . . . . 9 ⊢ (𝐹 = 𝑍 → {𝐹} = {𝑍}) | |
7 | 6 | fveq2d 6660 | . . . . . . . 8 ⊢ (𝐹 = 𝑍 → (𝐽‘{𝐹}) = (𝐽‘{𝑍})) |
8 | 5, 7 | sylan9eq 2876 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 = 𝑍) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑍})) |
9 | mapdindp.h | . . . . . . . . . 10 ⊢ 𝐻 = (LHyp‘𝐾) | |
10 | mapdindp.m | . . . . . . . . . 10 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
11 | mapdindp.u | . . . . . . . . . 10 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
12 | mapdn0.o | . . . . . . . . . 10 ⊢ 0 = (0g‘𝑈) | |
13 | mapdindp.c | . . . . . . . . . 10 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
14 | mapdn0.z | . . . . . . . . . 10 ⊢ 𝑍 = (0g‘𝐶) | |
15 | mapdindp.k | . . . . . . . . . 10 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
16 | 9, 10, 11, 12, 13, 14, 15 | mapd0 38833 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀‘{ 0 }) = {𝑍}) |
17 | 9, 13, 15 | lcdlmod 38760 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ LMod) |
18 | mapdindp.j | . . . . . . . . . . 11 ⊢ 𝐽 = (LSpan‘𝐶) | |
19 | 14, 18 | lspsn0 19763 | . . . . . . . . . 10 ⊢ (𝐶 ∈ LMod → (𝐽‘{𝑍}) = {𝑍}) |
20 | 17, 19 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (𝐽‘{𝑍}) = {𝑍}) |
21 | 16, 20 | eqtr4d 2859 | . . . . . . . 8 ⊢ (𝜑 → (𝑀‘{ 0 }) = (𝐽‘{𝑍})) |
22 | 21 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 = 𝑍) → (𝑀‘{ 0 }) = (𝐽‘{𝑍})) |
23 | 8, 22 | eqtr4d 2859 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐹 = 𝑍) → (𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 })) |
24 | 23 | ex 415 | . . . . 5 ⊢ (𝜑 → (𝐹 = 𝑍 → (𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 }))) |
25 | eqid 2821 | . . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
26 | 9, 11, 15 | dvhlmod 38278 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
27 | 2 | eldifad 3936 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
28 | mapdindp.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑈) | |
29 | mapdindp.n | . . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑈) | |
30 | 28, 25, 29 | lspsncl 19732 | . . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
31 | 26, 27, 30 | syl2anc 586 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
32 | 12, 25 | lsssn0 19702 | . . . . . . . 8 ⊢ (𝑈 ∈ LMod → { 0 } ∈ (LSubSp‘𝑈)) |
33 | 26, 32 | syl 17 | . . . . . . 7 ⊢ (𝜑 → { 0 } ∈ (LSubSp‘𝑈)) |
34 | 9, 11, 25, 10, 15, 31, 33 | mapd11 38807 | . . . . . 6 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 }) ↔ (𝑁‘{𝑋}) = { 0 })) |
35 | 28, 12, 29 | lspsneq0 19767 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 )) |
36 | 26, 27, 35 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 )) |
37 | 34, 36 | bitrd 281 | . . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 }) ↔ 𝑋 = 0 )) |
38 | 24, 37 | sylibd 241 | . . . 4 ⊢ (𝜑 → (𝐹 = 𝑍 → 𝑋 = 0 )) |
39 | 38 | necon3d 3037 | . . 3 ⊢ (𝜑 → (𝑋 ≠ 0 → 𝐹 ≠ 𝑍)) |
40 | 4, 39 | mpd 15 | . 2 ⊢ (𝜑 → 𝐹 ≠ 𝑍) |
41 | eldifsn 4705 | . 2 ⊢ (𝐹 ∈ (𝐷 ∖ {𝑍}) ↔ (𝐹 ∈ 𝐷 ∧ 𝐹 ≠ 𝑍)) | |
42 | 1, 40, 41 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐷 ∖ {𝑍})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3921 {csn 4553 ‘cfv 6341 Basecbs 16466 0gc0g 16696 LModclmod 19617 LSubSpclss 19686 LSpanclspn 19726 HLchlt 36518 LHypclh 37152 DVecHcdvh 38246 LCDualclcd 38754 mapdcmpd 38792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-riotaBAD 36121 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-iin 4908 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-of 7395 df-om 7567 df-1st 7675 df-2nd 7676 df-tpos 7878 df-undef 7925 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-map 8394 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-n0 11885 df-z 11969 df-uz 12231 df-fz 12883 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-mulr 16562 df-sca 16564 df-vsca 16565 df-0g 16698 df-mre 16840 df-mrc 16841 df-acs 16843 df-proset 17521 df-poset 17539 df-plt 17551 df-lub 17567 df-glb 17568 df-join 17569 df-meet 17570 df-p0 17632 df-p1 17633 df-lat 17639 df-clat 17701 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-submnd 17940 df-grp 18089 df-minusg 18090 df-sbg 18091 df-subg 18259 df-cntz 18430 df-oppg 18457 df-lsm 18744 df-cmn 18891 df-abl 18892 df-mgp 19223 df-ur 19235 df-ring 19282 df-oppr 19356 df-dvdsr 19374 df-unit 19375 df-invr 19405 df-dvr 19416 df-drng 19487 df-lmod 19619 df-lss 19687 df-lsp 19727 df-lvec 19858 df-lsatoms 36144 df-lshyp 36145 df-lcv 36187 df-lfl 36226 df-lkr 36254 df-ldual 36292 df-oposet 36344 df-ol 36346 df-oml 36347 df-covers 36434 df-ats 36435 df-atl 36466 df-cvlat 36490 df-hlat 36519 df-llines 36666 df-lplanes 36667 df-lvols 36668 df-lines 36669 df-psubsp 36671 df-pmap 36672 df-padd 36964 df-lhyp 37156 df-laut 37157 df-ldil 37272 df-ltrn 37273 df-trl 37327 df-tgrp 37911 df-tendo 37923 df-edring 37925 df-dveca 38171 df-disoa 38197 df-dvech 38247 df-dib 38307 df-dic 38341 df-dih 38397 df-doch 38516 df-djh 38563 df-lcdual 38755 df-mapd 38793 |
This theorem is referenced by: mapdheq4lem 38899 mapdh6lem1N 38901 mapdh6lem2N 38902 hdmap1l6lem1 38975 hdmap1l6lem2 38976 |
Copyright terms: Public domain | W3C validator |