Metamath Proof Explorer


Theorem cdleme31snd

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 1-Apr-2013)

Ref Expression
Hypotheses cdleme31snd.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
cdleme31snd.n 𝑁 = ( ( 𝑣 𝑉 ) ( 𝑃 ( ( 𝑄 𝑣 ) 𝑊 ) ) )
cdleme31snd.e 𝐸 = ( ( 𝑂 𝑈 ) ( 𝑄 ( ( 𝑃 𝑂 ) 𝑊 ) ) )
cdleme31snd.o 𝑂 = ( ( 𝑆 𝑉 ) ( 𝑃 ( ( 𝑄 𝑆 ) 𝑊 ) ) )
Assertion cdleme31snd ( 𝑆𝐴 𝑆 / 𝑣 𝑁 / 𝑡 𝐷 = 𝐸 )

Proof

Step Hyp Ref Expression
1 cdleme31snd.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
2 cdleme31snd.n 𝑁 = ( ( 𝑣 𝑉 ) ( 𝑃 ( ( 𝑄 𝑣 ) 𝑊 ) ) )
3 cdleme31snd.e 𝐸 = ( ( 𝑂 𝑈 ) ( 𝑄 ( ( 𝑃 𝑂 ) 𝑊 ) ) )
4 cdleme31snd.o 𝑂 = ( ( 𝑆 𝑉 ) ( 𝑃 ( ( 𝑄 𝑆 ) 𝑊 ) ) )
5 csbnestgw ( 𝑆𝐴 𝑆 / 𝑣 𝑁 / 𝑡 𝐷 = 𝑆 / 𝑣 𝑁 / 𝑡 𝐷 )
6 2 4 cdleme31sc ( 𝑆𝐴 𝑆 / 𝑣 𝑁 = 𝑂 )
7 6 csbeq1d ( 𝑆𝐴 𝑆 / 𝑣 𝑁 / 𝑡 𝐷 = 𝑂 / 𝑡 𝐷 )
8 4 ovexi 𝑂 ∈ V
9 1 3 cdleme31sc ( 𝑂 ∈ V → 𝑂 / 𝑡 𝐷 = 𝐸 )
10 8 9 ax-mp 𝑂 / 𝑡 𝐷 = 𝐸
11 7 10 eqtrdi ( 𝑆𝐴 𝑆 / 𝑣 𝑁 / 𝑡 𝐷 = 𝐸 )
12 5 11 eqtrd ( 𝑆𝐴 𝑆 / 𝑣 𝑁 / 𝑡 𝐷 = 𝐸 )