Theorem lnophmlem1 9879

Description: Lemma for lnophm 9881.

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 59	. . . . 5 |- (x = A -> x = A)
4		fveq2 3715	. . . . 5 |- (x = A -> (T` x) = (T` A))
5	3, 4	opreq12d 3969	. . . 4 |- (x = A -> (x .ih (T` x)) = (A .ih (T` A)))
6	5	eleq1d 1537	. . 3 |- (x = A -> ((x .ih (T` x)) e. RR <-> (A .ih (T` A)) e. RR))
7	6	rcla4v 1869	. 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 43	1 |- (A .ih (T` A)) e. RR

Syntax hints:   = wceq 954   e. wcel 956  A.wral 1642  ` cfv 3177  (class class class)co 3954  RRcr 5213  H~chil 8727   .ih csp 8732  LinOpclo 8755

This theorem is referenced by:  lnophmlem2 9880

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fv 3193  df-opr 3956

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