mapdn0 - Mathbox for Norm Megill

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Theorem mapdn0 41670

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 4757	. . . 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 4602	. . . . . . . . 9 ⊢ (𝐹 = 𝑍 → {𝐹} = {𝑍})
7	6	fveq2d 6865	. . . . . . . 8 ⊢ (𝐹 = 𝑍 → (𝐽‘{𝐹}) = (𝐽‘{𝑍}))
8	5, 7	sylan9eq 2785	. . . . . . 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 41666	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘{ 0 }) = {𝑍})
17	9, 13, 15	lcdlmod 41593	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ LMod)
18		mapdindp.j	. . . . . . . . . . 11 ⊢ 𝐽 = (LSpan‘𝐶)
19	14, 18	lspsn0 20921	. . . . . . . . . 10 ⊢ (𝐶 ∈ LMod → (𝐽‘{𝑍}) = {𝑍})
20	17, 19	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐽‘{𝑍}) = {𝑍})
21	16, 20	eqtr4d 2768	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘{ 0 }) = (𝐽‘{𝑍}))
22	21	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 = 𝑍) → (𝑀‘{ 0 }) = (𝐽‘{𝑍}))
23	8, 22	eqtr4d 2768	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 = 𝑍) → (𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 }))
24	23	ex 412	. . . . 5 ⊢ (𝜑 → (𝐹 = 𝑍 → (𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 })))
25		eqid 2730	. . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
26	9, 11, 15	dvhlmod 41111	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
27	2	eldifad 3929	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
28		mapdindp.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑈)
29		mapdindp.n	. . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑈)
30	28, 25, 29	lspsncl 20890	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
31	26, 27, 30	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
32	12, 25	lsssn0 20861	. . . . . . . 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 41640	. . . . . 6 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 }) ↔ (𝑁‘{𝑋}) = { 0 }))
35	28, 12, 29	lspsneq0 20925	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 ))
36	26, 27, 35	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 ))
37	34, 36	bitrd 279	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 }) ↔ 𝑋 = 0 ))
38	24, 37	sylibd 239	. . . 4 ⊢ (𝜑 → (𝐹 = 𝑍 → 𝑋 = 0 ))
39	38	necon3d 2947	. . 3 ⊢ (𝜑 → (𝑋 ≠ 0 → 𝐹 ≠ 𝑍))
40	4, 39	mpd 15	. 2 ⊢ (𝜑 → 𝐹 ≠ 𝑍)
41		eldifsn 4753	. 2 ⊢ (𝐹 ∈ (𝐷 ∖ {𝑍}) ↔ (𝐹 ∈ 𝐷 ∧ 𝐹 ≠ 𝑍))
42	1, 40, 41	sylanbrc 583	1 ⊢ (𝜑 → 𝐹 ∈ (𝐷 ∖ {𝑍}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∖ cdif 3914 {csn 4592 ‘cfv 6514 Basecbs 17186 0_gc0g 17409 LModclmod 20773 LSubSpclss 20844 LSpanclspn 20884 HLchlt 39350 LHypclh 39985 DVecHcdvh 41079 LCDualclcd 41587 mapdcmpd 41625

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-riotaBAD 38953

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-tpos 8208 df-undef 8255 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-0g 17411 df-mre 17554 df-mrc 17555 df-acs 17557 df-proset 18262 df-poset 18281 df-plt 18296 df-lub 18312 df-glb 18313 df-join 18314 df-meet 18315 df-p0 18391 df-p1 18392 df-lat 18398 df-clat 18465 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-cntz 19256 df-oppg 19285 df-lsm 19573 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-nzr 20429 df-rlreg 20610 df-domn 20611 df-drng 20647 df-lmod 20775 df-lss 20845 df-lsp 20885 df-lvec 21017 df-lsatoms 38976 df-lshyp 38977 df-lcv 39019 df-lfl 39058 df-lkr 39086 df-ldual 39124 df-oposet 39176 df-ol 39178 df-oml 39179 df-covers 39266 df-ats 39267 df-atl 39298 df-cvlat 39322 df-hlat 39351 df-llines 39499 df-lplanes 39500 df-lvols 39501 df-lines 39502 df-psubsp 39504 df-pmap 39505 df-padd 39797 df-lhyp 39989 df-laut 39990 df-ldil 40105 df-ltrn 40106 df-trl 40160 df-tgrp 40744 df-tendo 40756 df-edring 40758 df-dveca 41004 df-disoa 41030 df-dvech 41080 df-dib 41140 df-dic 41174 df-dih 41230 df-doch 41349 df-djh 41396 df-lcdual 41588 df-mapd 41626

This theorem is referenced by: mapdheq4lem 41732 mapdh6lem1N 41734 mapdh6lem2N 41735 hdmap1l6lem1 41808 hdmap1l6lem2 41809

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