Theorem abs2difabs 11534

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 11532	. . . 4 |- ( ( B e. CC /\ A e. CC ) -> ( ( abs ` B ) - ( abs ` A ) ) <_ ( abs ` ( B - A ) ) )
2	1	ancoms 268	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` B ) - ( abs ` A ) ) <_ ( abs ` ( B - A ) ) )
3		abscl 11477	. . . . 5 |- ( A e. CC -> ( abs ` A ) e. RR )
4	3	recnd 8136	. . . 4 |- ( A e. CC -> ( abs ` A ) e. CC )
5		abscl 11477	. . . . 5 |- ( B e. CC -> ( abs ` B ) e. RR )
6	5	recnd 8136	. . . 4 |- ( B e. CC -> ( abs ` B ) e. CC )
7		negsubdi2 8366	. . . 4 |- ( ( ( abs ` A ) e. CC /\ ( abs ` B ) e. CC ) -> -u ( ( abs ` A ) - ( abs ` B ) ) = ( ( abs ` B ) - ( abs ` A ) ) )
8	4, 6, 7	syl2an 289	. . 3 |- ( ( A e. CC /\ B e. CC ) -> -u ( ( abs ` A ) - ( abs ` B ) ) = ( ( abs ` B ) - ( abs ` A ) ) )
9		abssub 11527	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A - B ) ) = ( abs ` ( B - A ) ) )
10	2, 8, 9	3brtr4d 4091	. 2 |- ( ( A e. CC /\ B e. CC ) -> -u ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) )
11		abs2dif 11532	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) )
12		resubcl 8371	. . . . 5 |- ( ( ( abs ` A ) e. RR /\ ( abs ` B ) e. RR ) -> ( ( abs ` A ) - ( abs ` B ) ) e. RR )
13	3, 5, 12	syl2an 289	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` A ) - ( abs ` B ) ) e. RR )
14		subcl 8306	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
15		abscl 11477	. . . . 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 11515	. . . 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 411	. . 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 8574	. . . . 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 411	. . . 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 465	. . 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 191	. 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 947	1 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( ( abs ` A ) - ( abs ` B ) ) ) <_ ( abs ` ( A - B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   CCcc 7958   RRcr 7959   <_ cle 8143   - cmin 8278   -ucneg 8279   abscabs 11423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-rp 9811  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425

This theorem is referenced by:  abs2difabsd  11625  abscn2  11741