Metamath Proof Explorer

Theorem lukshefth1

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

Ref Expression
Assertion lukshefth1 
|- ( ( ( ( ta -/\ ps ) -/\ ( ( ph -/\ ta ) -/\ ( ph -/\ ta ) ) ) -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ( ph -/\ ( ps -/\ ch ) ) )

Proof

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