# Metamath Proof Explorer

## Theorem mapdindp3

Description: Vector independence lemma. (Contributed by NM, 29-Apr-2015)

Ref Expression
Hypotheses mapdindp1.v
`|- V = ( Base ` W )`
mapdindp1.p
`|- .+ = ( +g ` W )`
mapdindp1.o
`|- .0. = ( 0g ` W )`
mapdindp1.n
`|- N = ( LSpan ` W )`
mapdindp1.w
`|- ( ph -> W e. LVec )`
mapdindp1.x
`|- ( ph -> X e. ( V \ { .0. } ) )`
mapdindp1.y
`|- ( ph -> Y e. ( V \ { .0. } ) )`
mapdindp1.z
`|- ( ph -> Z e. ( V \ { .0. } ) )`
mapdindp1.W
`|- ( ph -> w e. ( V \ { .0. } ) )`
mapdindp1.e
`|- ( ph -> ( N ` { Y } ) = ( N ` { Z } ) )`
mapdindp1.ne
`|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )`
mapdindp1.f
`|- ( ph -> -. w e. ( N ` { X , Y } ) )`
Assertion mapdindp3
`|- ( ph -> ( N ` { X } ) =/= ( N ` { ( w .+ Y ) } ) )`

### Proof

Step Hyp Ref Expression
1 mapdindp1.v
` |-  V = ( Base ` W )`
2 mapdindp1.p
` |-  .+ = ( +g ` W )`
3 mapdindp1.o
` |-  .0. = ( 0g ` W )`
4 mapdindp1.n
` |-  N = ( LSpan ` W )`
5 mapdindp1.w
` |-  ( ph -> W e. LVec )`
6 mapdindp1.x
` |-  ( ph -> X e. ( V \ { .0. } ) )`
7 mapdindp1.y
` |-  ( ph -> Y e. ( V \ { .0. } ) )`
8 mapdindp1.z
` |-  ( ph -> Z e. ( V \ { .0. } ) )`
9 mapdindp1.W
` |-  ( ph -> w e. ( V \ { .0. } ) )`
10 mapdindp1.e
` |-  ( ph -> ( N ` { Y } ) = ( N ` { Z } ) )`
11 mapdindp1.ne
` |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )`
12 mapdindp1.f
` |-  ( ph -> -. w e. ( N ` { X , Y } ) )`
13 lveclmod
` |-  ( W e. LVec -> W e. LMod )`
14 5 13 syl
` |-  ( ph -> W e. LMod )`
` |-  ( ph -> w e. V )`
` |-  ( ph -> Y e. V )`
` |-  ( ( W e. LMod /\ w e. V /\ Y e. V ) -> ( N ` { ( w .+ Y ) } ) C_ ( N ` { w , Y } ) )`
18 14 15 16 17 syl3anc
` |-  ( ph -> ( N ` { ( w .+ Y ) } ) C_ ( N ` { w , Y } ) )`
19 1 3 4 5 6 16 15 11 12 lspindp1
` |-  ( ph -> ( ( N ` { w } ) =/= ( N ` { Y } ) /\ -. X e. ( N ` { w , Y } ) ) )`
20 19 simprd
` |-  ( ph -> -. X e. ( N ` { w , Y } ) )`
21 18 20 ssneldd
` |-  ( ph -> -. X e. ( N ` { ( w .+ Y ) } ) )`
` |-  ( ph -> X e. V )`
` |-  ( ( W e. LMod /\ X e. V ) -> X e. ( N ` { X } ) )`
` |-  ( ph -> X e. ( N ` { X } ) )`
` |-  ( ( N ` { X } ) = ( N ` { ( w .+ Y ) } ) -> ( X e. ( N ` { X } ) <-> X e. ( N ` { ( w .+ Y ) } ) ) )`
` |-  ( ph -> ( ( N ` { X } ) = ( N ` { ( w .+ Y ) } ) -> X e. ( N ` { ( w .+ Y ) } ) ) )`
` |-  ( ph -> ( -. X e. ( N ` { ( w .+ Y ) } ) -> ( N ` { X } ) =/= ( N ` { ( w .+ Y ) } ) ) )`
` |-  ( ph -> ( N ` { X } ) =/= ( N ` { ( w .+ Y ) } ) )`