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Theorem lspsnne1 21034

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 2725	. . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
4		lspsnne1.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑊)
5		lspsnne1.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec)
6		lveclmod 21020	. . . . . 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 20890	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
10	7, 8, 9	syl2anc 582	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
11		lspsnne1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
12	11	eldifad 3956	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
13	2, 3, 4, 7, 10, 12	lspsnel5 20908	. . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌})))
14	13	notbid 317	. . 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 21033	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
17	16	necon3bbid 2967	. . 3 ⊢ (𝜑 → (¬ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
18	14, 17	bitrd 278	. 2 ⊢ (𝜑 → (¬ 𝑋 ∈ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
19	1, 18	mpbird 256	1 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∖ cdif 3941 ⊆ wss 3944 {csn 4630 ‘cfv 6549 Basecbs 17199 0_gc0g 17440 LModclmod 20772 LSubSpclss 20844 LSpanclspn 20884 LVecclvec 21016

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-sets 17152 df-slot 17170 df-ndx 17182 df-base 17200 df-ress 17229 df-plusg 17265 df-mulr 17266 df-0g 17442 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-grp 18917 df-minusg 18918 df-sbg 18919 df-cmn 19766 df-abl 19767 df-mgp 20104 df-rng 20122 df-ur 20151 df-ring 20204 df-oppr 20302 df-dvdsr 20325 df-unit 20326 df-invr 20356 df-drng 20655 df-lmod 20774 df-lss 20845 df-lsp 20885 df-lvec 21017

This theorem is referenced by: lspsnnecom 21036 lsatfixedN 38631 baerlem5amN 41339 baerlem5bmN 41340 baerlem5abmN 41341 mapdh6dN 41362 hdmaplem4 41397 hdmap1l6d 41436 hdmaprnlem3N 41473

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