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Theorem abstri 11848
Description: Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by NM, 7-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
Assertion
Ref Expression
abstri |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A + B ) ) <_ ( ( abs ` A ) + ( abs ` B ) ) )
Proof of Theorem abstri
StepHypRef Expression
1 2re 9353 . . . . . 6 |- 2 e. RR
21a1i 9 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> 2 e. RR )
3 simpl 109 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
4 simpr 110 . . . . . . . 8 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
54cjcld 11684 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( * ` B ) e. CC )
63, 5mulcld 8336 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A x. ( * ` B ) ) e. CC )
76recld 11682 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A x. ( * ` B ) ) ) e. RR )
82, 7remulcld 8346 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) e. RR )
9 abscl 11795 . . . . . . 7 |- ( A e. CC -> ( abs ` A ) e. RR )
103, 9syl 14 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` A ) e. RR )
11 abscl 11795 . . . . . . 7 |- ( B e. CC -> ( abs ` B ) e. RR )
124, 11syl 14 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` B ) e. RR )
1310, 12remulcld 8346 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` A ) x. ( abs ` B ) ) e. RR )
142, 13remulcld 8346 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( ( abs ` A ) x. ( abs ` B ) ) ) e. RR )
1510resqcld 11115 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` A ) ^ 2 ) e. RR )
1612resqcld 11115 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` B ) ^ 2 ) e. RR )
1715, 16readdcld 8345 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) e. RR )
18 releabs 11840 . . . . . . 7 |- ( ( A x. ( * ` B ) ) e. CC -> ( Re ` ( A x. ( * ` B ) ) ) <_ ( abs ` ( A x. ( * ` B ) ) ) )
196, 18syl 14 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A x. ( * ` B ) ) ) <_ ( abs ` ( A x. ( * ` B ) ) ) )
20 absmul 11813 . . . . . . . 8 |- ( ( A e. CC /\ ( * ` B ) e. CC ) -> ( abs ` ( A x. ( * ` B ) ) ) = ( ( abs ` A ) x. ( abs ` ( * ` B ) ) ) )
213, 5, 20syl2anc 415 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A x. ( * ` B ) ) ) = ( ( abs ` A ) x. ( abs ` ( * ` B ) ) ) )
22 abscj 11796 . . . . . . . . 9 |- ( B e. CC -> ( abs ` ( * ` B ) ) = ( abs ` B ) )
234, 22syl 14 . . . . . . . 8 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( * ` B ) ) = ( abs ` B ) )
2423oveq2d 6091 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` A ) x. ( abs ` ( * ` B ) ) ) = ( ( abs ` A ) x. ( abs ` B ) ) )
2521, 24eqtrd 2271 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A x. ( * ` B ) ) ) = ( ( abs ` A ) x. ( abs ` B ) ) )
2619, 25breqtrd 4151 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A x. ( * ` B ) ) ) <_ ( ( abs ` A ) x. ( abs ` B ) ) )
27 2rp 10038 . . . . . . 7 |- 2 e. RR+
2827a1i 9 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> 2 e. RR+ )
297, 13, 28lemul2d 10121 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( Re ` ( A x. ( * ` B ) ) ) <_ ( ( abs ` A ) x. ( abs ` B ) ) <-> ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) <_ ( 2 x. ( ( abs ` A ) x. ( abs ` B ) ) ) ) )
3026, 29mpbid 147 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) <_ ( 2 x. ( ( abs ` A ) x. ( abs ` B ) ) ) )
318, 14, 17, 30leadd2dd 8878 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) + ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) <_ ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) + ( 2 x. ( ( abs ` A ) x. ( abs ` B ) ) ) ) )
32 sqabsadd 11799 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A + B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) + ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) )
3310recnd 8344 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` A ) e. CC )
3412recnd 8344 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` B ) e. CC )
35 binom2 11066 . . . . 5 |- ( ( ( abs ` A ) e. CC /\ ( abs ` B ) e. CC ) -> ( ( ( abs ` A ) + ( abs ` B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( 2 x. ( ( abs ` A ) x. ( abs ` B ) ) ) ) + ( ( abs ` B ) ^ 2 ) ) )
3633, 34, 35syl2anc 415 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( abs ` A ) + ( abs ` B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( 2 x. ( ( abs ` A ) x. ( abs ` B ) ) ) ) + ( ( abs ` B ) ^ 2 ) ) )
3715recnd 8344 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` A ) ^ 2 ) e. CC )
3814recnd 8344 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( ( abs ` A ) x. ( abs ` B ) ) ) e. CC )
3916recnd 8344 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` B ) ^ 2 ) e. CC )
4037, 38, 39add32d 8484 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( abs ` A ) ^ 2 ) + ( 2 x. ( ( abs ` A ) x. ( abs ` B ) ) ) ) + ( ( abs ` B ) ^ 2 ) ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) + ( 2 x. ( ( abs ` A ) x. ( abs ` B ) ) ) ) )
4136, 40eqtrd 2271 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( abs ` A ) + ( abs ` B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) + ( 2 x. ( ( abs ` A ) x. ( abs ` B ) ) ) ) )
4231, 32, 413brtr4d 4157 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A + B ) ) ^ 2 ) <_ ( ( ( abs ` A ) + ( abs ` B ) ) ^ 2 ) )
43 addcl 8294 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
44 abscl 11795 . . . 4 |- ( ( A + B ) e. CC -> ( abs ` ( A + B ) ) e. RR )
4543, 44syl 14 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A + B ) ) e. RR )
4610, 12readdcld 8345 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` A ) + ( abs ` B ) ) e. RR )
47 absge0 11804 . . . 4 |- ( ( A + B ) e. CC -> 0 <_ ( abs ` ( A + B ) ) )
4843, 47syl 14 . . 3 |- ( ( A e. CC /\ B e. CC ) -> 0 <_ ( abs ` ( A + B ) ) )
49 absge0 11804 . . . . 5 |- ( A e. CC -> 0 <_ ( abs ` A ) )
503, 49syl 14 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> 0 <_ ( abs ` A ) )
51 absge0 11804 . . . . 5 |- ( B e. CC -> 0 <_ ( abs ` B ) )
524, 51syl 14 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> 0 <_ ( abs ` B ) )
5310, 12, 50, 52addge0d 8840 . . 3 |- ( ( A e. CC /\ B e. CC ) -> 0 <_ ( ( abs ` A ) + ( abs ` B ) ) )
5445, 46, 48, 53le2sqd 11121 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A + B ) ) <_ ( ( abs ` A ) + ( abs ` B ) ) <-> ( ( abs ` ( A + B ) ) ^ 2 ) <_ ( ( ( abs ` A ) + ( abs ` B ) ) ^ 2 ) ) )
5542, 54mpbird 167 1 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A + B ) ) <_ ( ( abs ` A ) + ( abs ` B ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   + caddc 8172   x. cmul 8174   <_ cle 8351   2c2 9334   RR+crp 10033   ^cexp 10953   *ccj 11582   Recre 11583   abscabs 11741
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289
This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-rp 10034  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743
This theorem is referenced by:  abs3dif  11849  abs2dif2  11851  abstrii  11899  abstrid  11940
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