mapdn0 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdn0		Structured version Visualization version GIF version

Theorem mapdn0 37745

Description: Transfer nonzero property from domain to range of projectivity mapd. (Contributed by NM, 12-Apr-2015.)

**Hypotheses**
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 = (0_g‘𝑈)
mapdn0.z	⊢ 𝑍 = (0_g‘𝐶)
mapdn0.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))

**Assertion**
Ref	Expression
mapdn0	⊢ (𝜑 → 𝐹 ∈ (𝐷 ∖ {𝑍}))

**Proof of Theorem mapdn0**
Step	Hyp	Ref	Expression
1		mapdindp.f	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
2		mapdn0.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
3		eldifsni 4541	. . . 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 4408	. . . . . . . . 9 ⊢ (𝐹 = 𝑍 → {𝐹} = {𝑍})
7	6	fveq2d 6438	. . . . . . . 8 ⊢ (𝐹 = 𝑍 → (𝐽‘{𝐹}) = (𝐽‘{𝑍}))
8	5, 7	sylan9eq 2882	. . . . . . 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 = (0_g‘𝑈)
13		mapdindp.c	. . . . . . . . . 10 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
14		mapdn0.z	. . . . . . . . . 10 ⊢ 𝑍 = (0_g‘𝐶)
15		mapdindp.k	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16	9, 10, 11, 12, 13, 14, 15	mapd0 37741	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘{ 0 }) = {𝑍})
17	9, 13, 15	lcdlmod 37668	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ LMod)
18		mapdindp.j	. . . . . . . . . . 11 ⊢ 𝐽 = (LSpan‘𝐶)
19	14, 18	lspsn0 19368	. . . . . . . . . 10 ⊢ (𝐶 ∈ LMod → (𝐽‘{𝑍}) = {𝑍})
20	17, 19	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐽‘{𝑍}) = {𝑍})
21	16, 20	eqtr4d 2865	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘{ 0 }) = (𝐽‘{𝑍}))
22	21	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 = 𝑍) → (𝑀‘{ 0 }) = (𝐽‘{𝑍}))
23	8, 22	eqtr4d 2865	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 = 𝑍) → (𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 }))
24	23	ex 403	. . . . 5 ⊢ (𝜑 → (𝐹 = 𝑍 → (𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 })))
25		eqid 2826	. . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
26	9, 11, 15	dvhlmod 37186	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
27	2	eldifad 3811	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
28		mapdindp.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑈)
29		mapdindp.n	. . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑈)
30	28, 25, 29	lspsncl 19337	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
31	26, 27, 30	syl2anc 581	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
32	12, 25	lsssn0 19305	. . . . . . . 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 37715	. . . . . 6 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 }) ↔ (𝑁‘{𝑋}) = { 0 }))
35	28, 12, 29	lspsneq0 19372	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 ))
36	26, 27, 35	syl2anc 581	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 ))
37	34, 36	bitrd 271	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 }) ↔ 𝑋 = 0 ))
38	24, 37	sylibd 231	. . . 4 ⊢ (𝜑 → (𝐹 = 𝑍 → 𝑋 = 0 ))
39	38	necon3d 3021	. . 3 ⊢ (𝜑 → (𝑋 ≠ 0 → 𝐹 ≠ 𝑍))
40	4, 39	mpd 15	. 2 ⊢ (𝜑 → 𝐹 ≠ 𝑍)
41		eldifsn 4537	. 2 ⊢ (𝐹 ∈ (𝐷 ∖ {𝑍}) ↔ (𝐹 ∈ 𝐷 ∧ 𝐹 ≠ 𝑍))
42	1, 40, 41	sylanbrc 580	1 ⊢ (𝜑 → 𝐹 ∈ (𝐷 ∖ {𝑍}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 ∖ cdif 3796 {csn 4398 ‘cfv 6124 Basecbs 16223 0_gc0g 16454 LModclmod 19220 LSubSpclss 19289 LSpanclspn 19331 HLchlt 35426 LHypclh 36060 DVecHcdvh 37154 LCDualclcd 37662 mapdcmpd 37700

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-riotaBAD 35029

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-iin 4744 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-of 7158 df-om 7328 df-1st 7429 df-2nd 7430 df-tpos 7618 df-undef 7665 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-map 8125 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-n0 11620 df-z 11706 df-uz 11970 df-fz 12621 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-sca 16322 df-vsca 16323 df-0g 16456 df-mre 16600 df-mrc 16601 df-acs 16603 df-proset 17282 df-poset 17300 df-plt 17312 df-lub 17328 df-glb 17329 df-join 17330 df-meet 17331 df-p0 17393 df-p1 17394 df-lat 17400 df-clat 17462 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-submnd 17690 df-grp 17780 df-minusg 17781 df-sbg 17782 df-subg 17943 df-cntz 18101 df-oppg 18127 df-lsm 18403 df-cmn 18549 df-abl 18550 df-mgp 18845 df-ur 18857 df-ring 18904 df-oppr 18978 df-dvdsr 18996 df-unit 18997 df-invr 19027 df-dvr 19038 df-drng 19106 df-lmod 19222 df-lss 19290 df-lsp 19332 df-lvec 19463 df-lsatoms 35052 df-lshyp 35053 df-lcv 35095 df-lfl 35134 df-lkr 35162 df-ldual 35200 df-oposet 35252 df-ol 35254 df-oml 35255 df-covers 35342 df-ats 35343 df-atl 35374 df-cvlat 35398 df-hlat 35427 df-llines 35574 df-lplanes 35575 df-lvols 35576 df-lines 35577 df-psubsp 35579 df-pmap 35580 df-padd 35872 df-lhyp 36064 df-laut 36065 df-ldil 36180 df-ltrn 36181 df-trl 36235 df-tgrp 36819 df-tendo 36831 df-edring 36833 df-dveca 37079 df-disoa 37105 df-dvech 37155 df-dib 37215 df-dic 37249 df-dih 37305 df-doch 37424 df-djh 37471 df-lcdual 37663 df-mapd 37701

This theorem is referenced by: mapdheq4lem 37807 mapdh6lem1N 37809 mapdh6lem2N 37810 hdmap1l6lem1 37883 hdmap1l6lem2 37884

W3C validator