Theorem caovdilem

Description: Lemma used by real number construction. (Contributed by NM, 26-Aug-1995)

	Ref	Expression
Hypotheses	caovdir.1	⊢ 𝐴 ∈ V
	caovdir.2	⊢ 𝐵 ∈ V
	caovdir.3	⊢ 𝐶 ∈ V
	caovdir.com	⊢ ( 𝑥 𝐺 𝑦 ) = ( 𝑦 𝐺 𝑥 )
	caovdir.distr	⊢ ( 𝑥 𝐺 ( 𝑦 𝐹 𝑧 ) ) = ( ( 𝑥 𝐺 𝑦 ) 𝐹 ( 𝑥 𝐺 𝑧 ) )
	caovdl.4	⊢ 𝐷 ∈ V
	caovdl.5	⊢ 𝐻 ∈ V
	caovdl.ass	⊢ ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) )
Assertion	caovdilem	⊢ ( ( ( 𝐴 𝐺 𝐶 ) 𝐹 ( 𝐵 𝐺 𝐷 ) ) 𝐺 𝐻 ) = ( ( 𝐴 𝐺 ( 𝐶 𝐺 𝐻 ) ) 𝐹 ( 𝐵 𝐺 ( 𝐷 𝐺 𝐻 ) ) )

Proof

Step	Hyp	Ref	Expression
1		caovdir.1	⊢ 𝐴 ∈ V
2		caovdir.2	⊢ 𝐵 ∈ V
3		caovdir.3	⊢ 𝐶 ∈ V
4		caovdir.com	⊢ ( 𝑥 𝐺 𝑦 ) = ( 𝑦 𝐺 𝑥 )
5		caovdir.distr	⊢ ( 𝑥 𝐺 ( 𝑦 𝐹 𝑧 ) ) = ( ( 𝑥 𝐺 𝑦 ) 𝐹 ( 𝑥 𝐺 𝑧 ) )
6		caovdl.4	⊢ 𝐷 ∈ V
7		caovdl.5	⊢ 𝐻 ∈ V
8		caovdl.ass	⊢ ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) )
9		ovex	⊢ ( 𝐴 𝐺 𝐶 ) ∈ V
10		ovex	⊢ ( 𝐵 𝐺 𝐷 ) ∈ V
11	9 10 7 4 5	caovdir	⊢ ( ( ( 𝐴 𝐺 𝐶 ) 𝐹 ( 𝐵 𝐺 𝐷 ) ) 𝐺 𝐻 ) = ( ( ( 𝐴 𝐺 𝐶 ) 𝐺 𝐻 ) 𝐹 ( ( 𝐵 𝐺 𝐷 ) 𝐺 𝐻 ) )
12	1 3 7 8	caovass	⊢ ( ( 𝐴 𝐺 𝐶 ) 𝐺 𝐻 ) = ( 𝐴 𝐺 ( 𝐶 𝐺 𝐻 ) )
13	2 6 7 8	caovass	⊢ ( ( 𝐵 𝐺 𝐷 ) 𝐺 𝐻 ) = ( 𝐵 𝐺 ( 𝐷 𝐺 𝐻 ) )
14	12 13	oveq12i	⊢ ( ( ( 𝐴 𝐺 𝐶 ) 𝐺 𝐻 ) 𝐹 ( ( 𝐵 𝐺 𝐷 ) 𝐺 𝐻 ) ) = ( ( 𝐴 𝐺 ( 𝐶 𝐺 𝐻 ) ) 𝐹 ( 𝐵 𝐺 ( 𝐷 𝐺 𝐻 ) ) )
15	11 14	eqtri	⊢ ( ( ( 𝐴 𝐺 𝐶 ) 𝐹 ( 𝐵 𝐺 𝐷 ) ) 𝐺 𝐻 ) = ( ( 𝐴 𝐺 ( 𝐶 𝐺 𝐻 ) ) 𝐹 ( 𝐵 𝐺 ( 𝐷 𝐺 𝐻 ) ) )