mapdn0 - Mathbox for Norm Megill

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Theorem mapdn0 42257

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 4749	. . . 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 4591	. . . . . . . . 9 ⊢ (𝐹 = 𝑍 → {𝐹} = {𝑍})
7	6	fveq2d 6867	. . . . . . . 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 42253	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘{ 0 }) = {𝑍})
17	9, 13, 15	lcdlmod 42180	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ LMod)
18		mapdindp.j	. . . . . . . . . . 11 ⊢ 𝐽 = (LSpan‘𝐶)
19	14, 18	lspsn0 21055	. . . . . . . . . 10 ⊢ (𝐶 ∈ LMod → (𝐽‘{𝑍}) = {𝑍})
20	17, 19	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐽‘{𝑍}) = {𝑍})
21	16, 20	eqtr4d 2799	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘{ 0 }) = (𝐽‘{𝑍}))
22	21	adantr 484	. . . . . . 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 416	. . . . 5 ⊢ (𝜑 → (𝐹 = 𝑍 → (𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 })))
25		eqid 2761	. . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
26	9, 11, 15	dvhlmod 41698	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
27	2	eldifad 3916	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
28		mapdindp.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑈)
29		mapdindp.n	. . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑈)
30	28, 25, 29	lspsncl 21024	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
31	26, 27, 30	syl2anc 593	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
32	12, 25	lsssn0 20995	. . . . . . . 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 42227	. . . . . 6 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) = (𝑀‘{ 0 }) ↔ (𝑁‘{𝑋}) = { 0 }))
35	28, 12, 29	lspsneq0 21059	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 ))
36	26, 27, 35	syl2anc 593	. . . . . 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 2977	. . 3 ⊢ (𝜑 → (𝑋 ≠ 0 → 𝐹 ≠ 𝑍))
40	4, 39	mpd 15	. 2 ⊢ (𝜑 → 𝐹 ≠ 𝑍)
41		eldifsn 4745	. 2 ⊢ (𝐹 ∈ (𝐷 ∖ {𝑍}) ↔ (𝐹 ∈ 𝐷 ∧ 𝐹 ≠ 𝑍))
42	1, 40, 41	sylanbrc 592	1 ⊢ (𝜑 → 𝐹 ∈ (𝐷 ∖ {𝑍}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∖ cdif 3901 {csn 4581 ‘cfv 6517 Basecbs 17228 0_gc0g 17451 LModclmod 20907 LSubSpclss 20978 LSpanclspn 21018 HLchlt 39938 LHypclh 40572 DVecHcdvh 41666 LCDualclcd 42174 mapdcmpd 42212

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-riotaBAD 39541

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-tpos 8201 df-undef 8248 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8673 df-map 8805 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-n0 12479 df-z 12566 df-uz 12837 df-fz 13510 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-0g 17453 df-mre 17597 df-mrc 17598 df-acs 17600 df-proset 18309 df-poset 18328 df-plt 18343 df-lub 18359 df-glb 18360 df-join 18361 df-meet 18362 df-p0 18438 df-p1 18439 df-lat 18447 df-clat 18514 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-submnd 18801 df-grp 18961 df-minusg 18962 df-sbg 18963 df-subg 19148 df-cntz 19340 df-oppg 19369 df-lsm 19659 df-cmn 19805 df-abl 19806 df-mgp 20170 df-rng 20182 df-ur 20211 df-ring 20264 df-oppr 20365 df-dvdsr 20385 df-unit 20386 df-invr 20416 df-dvr 20429 df-nzr 20542 df-rlreg 20723 df-domn 20724 df-drng 20760 df-lmod 20909 df-lss 20979 df-lsp 21019 df-lvec 21150 df-lsatoms 39564 df-lshyp 39565 df-lcv 39607 df-lfl 39646 df-lkr 39674 df-ldual 39712 df-oposet 39764 df-ol 39766 df-oml 39767 df-covers 39854 df-ats 39855 df-atl 39886 df-cvlat 39910 df-hlat 39939 df-llines 40086 df-lplanes 40087 df-lvols 40088 df-lines 40089 df-psubsp 40091 df-pmap 40092 df-padd 40384 df-lhyp 40576 df-laut 40577 df-ldil 40692 df-ltrn 40693 df-trl 40747 df-tgrp 41331 df-tendo 41343 df-edring 41345 df-dveca 41591 df-disoa 41617 df-dvech 41667 df-dib 41727 df-dic 41761 df-dih 41817 df-doch 41936 df-djh 41983 df-lcdual 42175 df-mapd 42213

This theorem is referenced by: mapdheq4lem 42319 mapdh6lem1N 42321 mapdh6lem2N 42322 hdmap1l6lem1 42395 hdmap1l6lem2 42396

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