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Theorem lspsnne1 21049

Description: Two ways to express that vectors have different spans. (Contributed by NM, 28-May-2015.)

**Hypotheses**
Ref	Expression
lspsnne1.v	⊢ 𝑉 = (Base‘𝑊)
lspsnne1.o	⊢ 0 = (0_g‘𝑊)
lspsnne1.n	⊢ 𝑁 = (LSpan‘𝑊)
lspsnne1.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lspsnne1.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lspsnne1.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
lspsnne1.e	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))

**Assertion**
Ref	Expression
lspsnne1	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))

**Proof of Theorem lspsnne1**
Step	Hyp	Ref	Expression
1		lspsnne1.e	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
2		lspsnne1.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
3		eqid 2731	. . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
4		lspsnne1.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑊)
5		lspsnne1.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec)
6		lveclmod 21035	. . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod)
8		lspsnne1.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
9	2, 3, 4	lspsncl 20905	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
10	7, 8, 9	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
11		lspsnne1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
12	11	eldifad 3909	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
13	2, 3, 4, 7, 10, 12	ellspsn5b 20923	. . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌})))
14	13	notbid 318	. . 3 ⊢ (𝜑 → (¬ 𝑋 ∈ (𝑁‘{𝑌}) ↔ ¬ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌})))
15		lspsnne1.o	. . . . 5 ⊢ 0 = (0_g‘𝑊)
16	2, 15, 4, 5, 11, 8	lspsncmp 21048	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
17	16	necon3bbid 2965	. . 3 ⊢ (𝜑 → (¬ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
18	14, 17	bitrd 279	. 2 ⊢ (𝜑 → (¬ 𝑋 ∈ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
19	1, 18	mpbird 257	1 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∖ cdif 3894 ⊆ wss 3897 {csn 4571 ‘cfv 6476 Basecbs 17115 0_gc0g 17338 LModclmod 20788 LSubSpclss 20859 LSpanclspn 20899 LVecclvec 21031

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-tpos 8151 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-nn 12121 df-2 12183 df-3 12184 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-0g 17340 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-grp 18844 df-minusg 18845 df-sbg 18846 df-cmn 19689 df-abl 19690 df-mgp 20054 df-rng 20066 df-ur 20095 df-ring 20148 df-oppr 20250 df-dvdsr 20270 df-unit 20271 df-invr 20301 df-drng 20641 df-lmod 20790 df-lss 20860 df-lsp 20900 df-lvec 21032

This theorem is referenced by: lspsnnecom 21051 lsatfixedN 39048 baerlem5amN 41755 baerlem5bmN 41756 baerlem5abmN 41757 mapdh6dN 41778 hdmaplem4 41813 hdmap1l6d 41852 hdmaprnlem3N 41889

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