Theorem dnibndlem1 32468

Description: Lemma for dnibnd 32481. (Contributed by Asger C. Ipsen, 4-Apr-2021.)

Hypotheses

Ref	Expression
dnibndlem1.1	|- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
dnibndlem1.2	|- ( ph -> A e. RR )
dnibndlem1.3	|- ( ph -> B e. RR )

Assertion

Ref	Expression
dnibndlem1	|- ( ph -> ( ( abs ` ( ( T ` B ) - ( T ` A ) ) ) <_ S <-> ( abs ` ( ( abs ` ( ( |_ ` ( B + ( 1 / 2 ) ) ) - B ) ) - ( abs ` ( ( |_ ` ( A + ( 1 / 2 ) ) ) - A ) ) ) ) <_ S ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   S( x)   T( x)

Proof of Theorem dnibndlem1

Step	Hyp	Ref	Expression
1		dnibndlem1.3	. . . . 5 |- ( ph -> B e. RR )
2		dnibndlem1.1	. . . . . 6 |- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
3	2	dnival 32461	. . . . 5 |- ( B e. RR -> ( T ` B ) = ( abs ` ( ( |_ ` ( B + ( 1 / 2 ) ) ) - B ) ) )
4	1, 3	syl 17	. . . 4 |- ( ph -> ( T ` B ) = ( abs ` ( ( |_ ` ( B + ( 1 / 2 ) ) ) - B ) ) )
5		dnibndlem1.2	. . . . 5 |- ( ph -> A e. RR )
6	2	dnival 32461	. . . . 5 |- ( A e. RR -> ( T ` A ) = ( abs ` ( ( |_ ` ( A + ( 1 / 2 ) ) ) - A ) ) )
7	5, 6	syl 17	. . . 4 |- ( ph -> ( T ` A ) = ( abs ` ( ( |_ ` ( A + ( 1 / 2 ) ) ) - A ) ) )
8	4, 7	oveq12d 6668	. . 3 |- ( ph -> ( ( T ` B ) - ( T ` A ) ) = ( ( abs ` ( ( |_ ` ( B + ( 1 / 2 ) ) ) - B ) ) - ( abs ` ( ( |_ ` ( A + ( 1 / 2 ) ) ) - A ) ) ) )
9	8	fveq2d 6195	. 2 |- ( ph -> ( abs ` ( ( T ` B ) - ( T ` A ) ) ) = ( abs ` ( ( abs ` ( ( |_ ` ( B + ( 1 / 2 ) ) ) - B ) ) - ( abs ` ( ( |_ ` ( A + ( 1 / 2 ) ) ) - A ) ) ) ) )
10	9	breq1d 4663	1 |- ( ph -> ( ( abs ` ( ( T ` B ) - ( T ` A ) ) ) <_ S <-> ( abs ` ( ( abs ` ( ( |_ ` ( B + ( 1 / 2 ) ) ) - B ) ) - ( abs ` ( ( |_ ` ( A + ( 1 / 2 ) ) ) - A ) ) ) ) <_ S ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   + caddc 9939   <_ cle 10075   - cmin 10266   / cdiv 10684   2c2 11070   |_cfl 12591   abscabs 13974

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  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-sbc 3436  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-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by:  dnibndlem2  32469  dnibndlem9  32476  dnibndlem12  32479