Metamath Proof Explorer

Theorem lukshefth2

Description: Lemma for renicax . (Contributed by NM, 31-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion lukshefth2 
|- ( ( ta -/\ th ) -/\ ( ( th -/\ ta ) -/\ ( th -/\ ta ) ) )

Proof

Step Hyp Ref Expression
1 lukshef-ax1 
 |-  ( ( ps -/\ ( ch -/\ ph ) ) -/\ ( ( th -/\ ( th -/\ th ) ) -/\ ( ( th -/\ ch ) -/\ ( ( ps -/\ th ) -/\ ( ps -/\ th ) ) ) ) )
2 lukshef-ax1 
 |-  ( ( ( ps -/\ ( ch -/\ ph ) ) -/\ ( ( th -/\ ( th -/\ th ) ) -/\ ( ( th -/\ ch ) -/\ ( ( ps -/\ th ) -/\ ( ps -/\ th ) ) ) ) ) -/\ ( ( th -/\ ( th -/\ th ) ) -/\ ( ( th -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ( ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) -/\ ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) ) ) ) )
3 1 2 nic-mp 
 |-  ( ( th -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ( ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) -/\ ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) ) )
4 lukshefth1 
 |-  ( ( ( ( ta -/\ ph ) -/\ ( ( ph -/\ ta ) -/\ ( ph -/\ ta ) ) ) -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ( ph -/\ ( ph -/\ ph ) ) )
5 lukshef-ax1 
 |-  ( ( ( th -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ( ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) -/\ ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) ) ) -/\ ( ( ph -/\ ( ph -/\ ph ) ) -/\ ( ( ph -/\ ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) ) -/\ ( ( ( th -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ph ) -/\ ( ( th -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ph ) ) ) ) )
6 lukshef-ax1 
 |-  ( ( ( ( th -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ( ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) -/\ ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) ) ) -/\ ( ( ph -/\ ( ph -/\ ph ) ) -/\ ( ( ph -/\ ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) ) -/\ ( ( ( th -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ph ) -/\ ( ( th -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ph ) ) ) ) ) -/\ ( ( ( ( ( ta -/\ ph ) -/\ ( ( ph -/\ ta ) -/\ ( ph -/\ ta ) ) ) -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ( ( ( ( ta -/\ ph ) -/\ ( ( ph -/\ ta ) -/\ ( ph -/\ ta ) ) ) -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ( ( ( ta -/\ ph ) -/\ ( ( ph -/\ ta ) -/\ ( ph -/\ ta ) ) ) -/\ ( th -/\ ( th -/\ th ) ) ) ) ) -/\ ( ( ( ( ( ta -/\ ph ) -/\ ( ( ph -/\ ta ) -/\ ( ph -/\ ta ) ) ) -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ( ph -/\ ( ph -/\ ph ) ) ) -/\ ( ( ( ( th -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ( ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) -/\ ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) ) ) -/\ ( ( ( ta -/\ ph ) -/\ ( ( ph -/\ ta ) -/\ ( ph -/\ ta ) ) ) -/\ ( th -/\ ( th -/\ th ) ) ) ) -/\ ( ( ( th -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ( ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) -/\ ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) ) ) -/\ ( ( ( ta -/\ ph ) -/\ ( ( ph -/\ ta ) -/\ ( ph -/\ ta ) ) ) -/\ ( th -/\ ( th -/\ th ) ) ) ) ) ) ) )
7 5 6 nic-mp 
 |-  ( ( ( ( ( ta -/\ ph ) -/\ ( ( ph -/\ ta ) -/\ ( ph -/\ ta ) ) ) -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ( ph -/\ ( ph -/\ ph ) ) ) -/\ ( ( ( ( th -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ( ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) -/\ ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) ) ) -/\ ( ( ( ta -/\ ph ) -/\ ( ( ph -/\ ta ) -/\ ( ph -/\ ta ) ) ) -/\ ( th -/\ ( th -/\ th ) ) ) ) -/\ ( ( ( th -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ( ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) -/\ ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) ) ) -/\ ( ( ( ta -/\ ph ) -/\ ( ( ph -/\ ta ) -/\ ( ph -/\ ta ) ) ) -/\ ( th -/\ ( th -/\ th ) ) ) ) ) )
8 4 7 nic-mp 
 |-  ( ( ( th -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ( ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) -/\ ( ( ps -/\ ( ch -/\ ph ) ) -/\ th ) ) ) -/\ ( ( ( ta -/\ ph ) -/\ ( ( ph -/\ ta ) -/\ ( ph -/\ ta ) ) ) -/\ ( th -/\ ( th -/\ th ) ) ) )
9 3 8 nic-mp 
 |-  ( th -/\ ( th -/\ th ) )
10 lukshef-ax1 
 |-  ( ( th -/\ ( th -/\ th ) ) -/\ ( ( ta -/\ ( ta -/\ ta ) ) -/\ ( ( ta -/\ th ) -/\ ( ( th -/\ ta ) -/\ ( th -/\ ta ) ) ) ) )
11 9 10 nic-mp 
 |-  ( ( ta -/\ th ) -/\ ( ( th -/\ ta ) -/\ ( th -/\ ta ) ) )