Theorem abs2difabs 11036

Description: Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

Assertion

Ref	Expression
abs2difabs	|- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( ( abs ` A ) - ( abs ` B ) ) ) <_ ( abs ` ( A - B ) ) )

Proof of Theorem abs2difabs

Step	Hyp	Ref	Expression
1		abs2dif 11034	. . . 4 |- ( ( B e. CC /\ A e. CC ) -> ( ( abs ` B ) - ( abs ` A ) ) <_ ( abs ` ( B - A ) ) )
2	1	ancoms 266	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` B ) - ( abs ` A ) ) <_ ( abs ` ( B - A ) ) )
3		abscl 10979	. . . . 5 |- ( A e. CC -> ( abs ` A ) e. RR )
4	3	recnd 7918	. . . 4 |- ( A e. CC -> ( abs ` A ) e. CC )
5		abscl 10979	. . . . 5 |- ( B e. CC -> ( abs ` B ) e. RR )
6	5	recnd 7918	. . . 4 |- ( B e. CC -> ( abs ` B ) e. CC )
7		negsubdi2 8148	. . . 4 |- ( ( ( abs ` A ) e. CC /\ ( abs ` B ) e. CC ) -> -u ( ( abs ` A ) - ( abs ` B ) ) = ( ( abs ` B ) - ( abs ` A ) ) )
8	4, 6, 7	syl2an 287	. . 3 |- ( ( A e. CC /\ B e. CC ) -> -u ( ( abs ` A ) - ( abs ` B ) ) = ( ( abs ` B ) - ( abs ` A ) ) )
9		abssub 11029	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A - B ) ) = ( abs ` ( B - A ) ) )
10	2, 8, 9	3brtr4d 4008	. 2 |- ( ( A e. CC /\ B e. CC ) -> -u ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) )
11		abs2dif 11034	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) )
12		resubcl 8153	. . . . 5 |- ( ( ( abs ` A ) e. RR /\ ( abs ` B ) e. RR ) -> ( ( abs ` A ) - ( abs ` B ) ) e. RR )
13	3, 5, 12	syl2an 287	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` A ) - ( abs ` B ) ) e. RR )
14		subcl 8088	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
15		abscl 10979	. . . . 5 |- ( ( A - B ) e. CC -> ( abs ` ( A - B ) ) e. RR )
16	14, 15	syl 14	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A - B ) ) e. RR )
17		absle 11017	. . . 4 |- ( ( ( ( abs ` A ) - ( abs ` B ) ) e. RR /\ ( abs ` ( A - B ) ) e. RR ) -> ( ( abs ` ( ( abs ` A ) - ( abs ` B ) ) ) <_ ( abs ` ( A - B ) ) <-> ( -u ( abs ` ( A - B ) ) <_ ( ( abs ` A ) - ( abs ` B ) ) /\ ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) ) ) )
18	13, 16, 17	syl2anc 409	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( ( abs ` A ) - ( abs ` B ) ) ) <_ ( abs ` ( A - B ) ) <-> ( -u ( abs ` ( A - B ) ) <_ ( ( abs ` A ) - ( abs ` B ) ) /\ ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) ) ) )
19		lenegcon1 8355	. . . . 5 |- ( ( ( ( abs ` A ) - ( abs ` B ) ) e. RR /\ ( abs ` ( A - B ) ) e. RR ) -> ( -u ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) <-> -u ( abs ` ( A - B ) ) <_ ( ( abs ` A ) - ( abs ` B ) ) ) )
20	13, 16, 19	syl2anc 409	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( -u ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) <-> -u ( abs ` ( A - B ) ) <_ ( ( abs ` A ) - ( abs ` B ) ) ) )
21	20	anbi1d 461	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( -u ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) /\ ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) ) <-> ( -u ( abs ` ( A - B ) ) <_ ( ( abs ` A ) - ( abs ` B ) ) /\ ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) ) ) )
22	18, 21	bitr4d 190	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( ( abs ` A ) - ( abs ` B ) ) ) <_ ( abs ` ( A - B ) ) <-> ( -u ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) /\ ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) ) ) )
23	10, 11, 22	mpbir2and 933	1 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( ( abs ` A ) - ( abs ` B ) ) ) <_ ( abs ` ( A - B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1342   e. wcel 2135   class class class wbr 3976   ` cfv 5182  (class class class)co 5836   CCcc 7742   RRcr 7743   <_ cle 7925   - cmin 8060   -ucneg 8061   abscabs 10925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861  ax-pre-mulext 7862  ax-arch 7863  ax-caucvg 7864

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-frec 6350  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-div 8560  df-inn 8849  df-2 8907  df-3 8908  df-4 8909  df-n0 9106  df-z 9183  df-uz 9458  df-rp 9581  df-seqfrec 10371  df-exp 10445  df-cj 10770  df-re 10771  df-im 10772  df-rsqrt 10926  df-abs 10927

This theorem is referenced by:  abs2difabsd  11127  abscn2  11242