Theorem tcphsub

Description: The subtraction operation of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Thierry Arnoux, 30-Jun-2019)

	Ref	Expression
Hypotheses	tcphval.n	⊢ 𝐺 = ( toℂPreHil ‘ 𝑊 )
	tcphsub.v	⊢ − = ( -g ‘ 𝑊 )
Assertion	tcphsub	⊢ − = ( -g ‘ 𝐺 )

Proof

Step	Hyp	Ref	Expression
1		tcphval.n	⊢ 𝐺 = ( toℂPreHil ‘ 𝑊 )
2		tcphsub.v	⊢ − = ( -g ‘ 𝑊 )
3		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
4	1 3	tcphbas	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝐺 )
5	4	a1i	⊢ ( ⊤ → ( Base ‘ 𝑊 ) = ( Base ‘ 𝐺 ) )
6		eqid	⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 )
7	1 6	tchplusg	⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝐺 )
8	7	a1i	⊢ ( ⊤ → ( +g ‘ 𝑊 ) = ( +g ‘ 𝐺 ) )
9	5 8	grpsubpropd	⊢ ( ⊤ → ( -g ‘ 𝑊 ) = ( -g ‘ 𝐺 ) )
10	9	mptru	⊢ ( -g ‘ 𝑊 ) = ( -g ‘ 𝐺 )
11	2 10	eqtri	⊢ − = ( -g ‘ 𝐺 )