mapdn0 - Mathbox for Norm Megill

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Theorem mapdn0 42694

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 4753	. . . 4 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 )
4	2, 3	syl 18	. . 3 ⊢ (𝜑 → 𝑋 ≠ 0 )
5		mapdindp.mx	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
6		sneq 4594	. . . . . . . . 9 ⊢ (𝐹 = 𝑍 → {𝐹} = {𝑍})
7	6	fveq2d 6881	. . . . . . . 8 ⊢ (𝐹 = 𝑍 → (𝐽‘{𝐹}) = (𝐽‘{𝑍}))
8	5, 7	sylan9eq 2816	. . . . . . 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 42690	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘{ 0 }) = {𝑍})
17	9, 13, 15	lcdlmod 42617	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ LMod)
18		mapdindp.j	. . . . . . . . . . 11 ⊢ 𝐽 = (LSpan‘𝐶)
19	14, 18	lspsn0 21263	. . . . . . . . . 10 ⊢ (𝐶 ∈ LMod → (𝐽‘{𝑍}) = {𝑍})
20	17, 19	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐽‘{𝑍}) = {𝑍})
21	16, 20	eqtr4d 2799	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘{ 0 }) = (𝐽‘{𝑍}))
22	21	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 = 𝑍) → (𝑀‘{ 0 }) = (𝐽‘{𝑍}))
23	8, 22	eqtr4d 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 = 𝑍) → (𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 }))
24	23	ex 418	. . . . 5 ⊢ (𝜑 → (𝐹 = 𝑍 → (𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 })))
25		eqid 2761	. . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
26	9, 11, 15	dvhlmod 42135	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
27	2	eldifad 3911	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
28		mapdindp.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑈)
29		mapdindp.n	. . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑈)
30	28, 25, 29	lspsncl 21232	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
31	26, 27, 30	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
32	12, 25	lsssn0 21203	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → { 0 } ∈ (LSubSp‘𝑈))
33	26, 32	syl 18	. . . . . . 7 ⊢ (𝜑 → { 0 } ∈ (LSubSp‘𝑈))
34	9, 11, 25, 10, 15, 31, 33	mapd11 42664	. . . . . 6 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 }) ↔ (𝑁‘{𝑋}) = { 0 }))
35	28, 12, 29	lspsneq0 21267	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 ))
36	26, 27, 35	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 ))
37	34, 36	bitrd 282	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 }) ↔ 𝑋 = 0 ))
38	24, 37	sylibd 242	. . . 4 ⊢ (𝜑 → (𝐹 = 𝑍 → 𝑋 = 0 ))
39	38	necon3d 2977	. . 3 ⊢ (𝜑 → (𝑋 ≠ 0 → 𝐹 ≠ 𝑍))
40	4, 39	mpd 16	. 2 ⊢ (𝜑 → 𝐹 ≠ 𝑍)
41		eldifsn 4748	. 2 ⊢ (𝐹 ∈ (𝐷 ∖ {𝑍}) ↔ (𝐹 ∈ 𝐷 ∧ 𝐹 ≠ 𝑍))
42	1, 40, 41	sylanbrc 595	1 ⊢ (𝜑 → 𝐹 ∈ (𝐷 ∖ {𝑍}))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∖ cdif 3896 {csn 4584 ‘cfv 6531 Basecbs 17367 0_gc0g 17590 LModclmod 21115 LSubSpclss 21186 LSpanclspn 21226 HLchlt 40375 LHypclh 41009 DVecHcdvh 42103 LCDualclcd 42611 mapdcmpd 42649

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-riotaBAD 39978

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-tpos 8227 df-undef 8274 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-n0 12588 df-z 12675 df-uz 12947 df-fz 13621 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-sca 17424 df-vsca 17425 df-0g 17592 df-mre 17736 df-mrc 17737 df-acs 17739 df-proset 18448 df-poset 18467 df-plt 18482 df-lub 18498 df-glb 18499 df-join 18500 df-meet 18501 df-p0 18577 df-p1 18578 df-lat 18586 df-clat 18653 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-submnd 18959 df-grp 19127 df-minusg 19128 df-sbg 19129 df-subg 19313 df-cntz 19511 df-oppg 19540 df-lsm 19830 df-cmn 19976 df-abl 19977 df-mgp 20341 df-rng 20355 df-ur 20388 df-ring 20441 df-oppr 20547 df-dvdsr 20567 df-unit 20568 df-invr 20598 df-dvr 20611 df-nzr 20743 df-rlreg 20926 df-domn 20927 df-drng 20962 df-lmod 21117 df-lss 21187 df-lsp 21227 df-lvec 21358 df-lsatoms 40001 df-lshyp 40002 df-lcv 40044 df-lfl 40083 df-lkr 40111 df-ldual 40149 df-oposet 40201 df-ol 40203 df-oml 40204 df-covers 40291 df-ats 40292 df-atl 40323 df-cvlat 40347 df-hlat 40376 df-llines 40523 df-lplanes 40524 df-lvols 40525 df-lines 40526 df-psubsp 40528 df-pmap 40529 df-padd 40821 df-lhyp 41013 df-laut 41014 df-ldil 41129 df-ltrn 41130 df-trl 41184 df-tgrp 41768 df-tendo 41780 df-edring 41782 df-dveca 42028 df-disoa 42054 df-dvech 42104 df-dib 42164 df-dic 42198 df-dih 42254 df-doch 42373 df-djh 42420 df-lcdual 42612 df-mapd 42650

This theorem is used by: mapdheq4lem 42756 mapdh6lem1N 42758 mapdh6lem2N 42759 hdmap1l6lem1 42832 hdmap1l6lem2 42833

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