Theorem lnophmlem1 28875

Description: Lemma for lnophmi 28877. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lnophmlem.1	|- A e. ~H
lnophmlem.2	|- B e. ~H
lnophmlem.3	|- T e. LinOp
lnophmlem.4	|- A. x e. ~H ( x .ih ( T ` x ) ) e. RR

Assertion

Ref	Expression
lnophmlem1	|- ( A .ih ( T ` A ) ) e. RR

Distinct variable groups:   x, A   x, B   x, T

Proof of Theorem lnophmlem1

Step	Hyp	Ref	Expression
1		lnophmlem.1	. 2 |- A e. ~H
2		lnophmlem.4	. 2 |- A. x e. ~H ( x .ih ( T ` x ) ) e. RR
3		id 22	. . . . 5 |- ( x = A -> x = A )
4		fveq2 6191	. . . . 5 |- ( x = A -> ( T ` x ) = ( T ` A ) )
5	3, 4	oveq12d 6668	. . . 4 |- ( x = A -> ( x .ih ( T ` x ) ) = ( A .ih ( T ` A ) ) )
6	5	eleq1d 2686	. . 3 |- ( x = A -> ( ( x .ih ( T ` x ) ) e. RR <-> ( A .ih ( T ` A ) ) e. RR ) )
7	6	rspcv 3305	. 2 |- ( A e. ~H -> ( A. x e. ~H ( x .ih ( T ` x ) ) e. RR -> ( A .ih ( T ` A ) ) e. RR ) )
8	1, 2, 7	mp2 9	1 |- ( A .ih ( T ` A ) ) e. RR

Syntax hints:   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   RRcr 9935   ~Hchil 27776   .ih csp 27779   LinOpclo 27804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  lnophmlem2  28876