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Metamath Proof Explorer

Theorem pexmidlem1N

Description: Lemma for pexmidN . Holland's proof implicitly requires q =/= r , which we prove here. (Contributed by NM, 2-Feb-2012) (New usage is discouraged.)

Ref Expression
Hypotheses pexmidlem.l 
|- .<_ = ( le ` K )
pexmidlem.j 
|- .\/ = ( join ` K )
pexmidlem.a 
|- A = ( Atoms ` K )
pexmidlem.p 
|- .+ = ( +P ` K )
pexmidlem.o 
|- ._|_ = ( _|_P ` K )
pexmidlem.m 
|- M = ( X .+ { p } )
Assertion pexmidlem1N 
|- ( ( ( K e. HL /\ X C_ A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) ) ) -> q =/= r )

Proof

Step Hyp Ref Expression
1 pexmidlem.l 
 |-  .<_ = ( le ` K )
2 pexmidlem.j 
 |-  .\/ = ( join ` K )
3 pexmidlem.a 
 |-  A = ( Atoms ` K )
4 pexmidlem.p 
 |-  .+ = ( +P ` K )
5 pexmidlem.o 
 |-  ._|_ = ( _|_P ` K )
6 pexmidlem.m 
 |-  M = ( X .+ { p } )
7 n0i 
 |-  ( r e. ( X i^i ( ._|_ ` X ) ) -> -. ( X i^i ( ._|_ ` X ) ) = (/) )
8 3 5 pnonsingN 
 |-  ( ( K e. HL /\ X C_ A ) -> ( X i^i ( ._|_ ` X ) ) = (/) )
9 8 adantr 
 |-  ( ( ( K e. HL /\ X C_ A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) ) ) -> ( X i^i ( ._|_ ` X ) ) = (/) )
10 7 9 nsyl3 
 |-  ( ( ( K e. HL /\ X C_ A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) ) ) -> -. r e. ( X i^i ( ._|_ ` X ) ) )
11 simprr 
 |-  ( ( ( K e. HL /\ X C_ A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) ) ) -> q e. ( ._|_ ` X ) )
12 eleq1w 
 |-  ( q = r -> ( q e. ( ._|_ ` X ) <-> r e. ( ._|_ ` X ) ) )
13 11 12 syl5ibcom 
 |-  ( ( ( K e. HL /\ X C_ A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) ) ) -> ( q = r -> r e. ( ._|_ ` X ) ) )
14 simprl 
 |-  ( ( ( K e. HL /\ X C_ A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) ) ) -> r e. X )
15 13 14 jctild 
 |-  ( ( ( K e. HL /\ X C_ A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) ) ) -> ( q = r -> ( r e. X /\ r e. ( ._|_ ` X ) ) ) )
16 elin 
 |-  ( r e. ( X i^i ( ._|_ ` X ) ) <-> ( r e. X /\ r e. ( ._|_ ` X ) ) )
17 15 16 imbitrrdi 
 |-  ( ( ( K e. HL /\ X C_ A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) ) ) -> ( q = r -> r e. ( X i^i ( ._|_ ` X ) ) ) )
18 17 necon3bd 
 |-  ( ( ( K e. HL /\ X C_ A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) ) ) -> ( -. r e. ( X i^i ( ._|_ ` X ) ) -> q =/= r ) )
19 10 18 mpd 
 |-  ( ( ( K e. HL /\ X C_ A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) ) ) -> q =/= r )