Theorem add20 7578

Description: Two nonnegative numbers are zero iff their sum is zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
add20	|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )

Proof of Theorem add20

Step	Hyp	Ref	Expression
1		simpllr 500	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> 0 <_ A )
2		simplrl 501	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B e. RR )
3		simplll 499	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> A e. RR )
4		addge02 7577	. . . . . . . . . 10 |- ( ( B e. RR /\ A e. RR ) -> ( 0 <_ A <-> B <_ ( A + B ) ) )
5	2, 3, 4	syl2anc 403	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( 0 <_ A <-> B <_ ( A + B ) ) )
6	1, 5	mpbid 145	. . . . . . . 8 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B <_ ( A + B ) )
7		simpr 108	. . . . . . . 8 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( A + B ) = 0 )
8	6, 7	breqtrd 3809	. . . . . . 7 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B <_ 0 )
9		simplrr 502	. . . . . . 7 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> 0 <_ B )
10		0red 7120	. . . . . . . 8 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> 0 e. RR )
11	2, 10	letri3d 7226	. . . . . . 7 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( B = 0 <-> ( B <_ 0 /\ 0 <_ B ) ) )
12	8, 9, 11	mpbir2and 885	. . . . . 6 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B = 0 )
13	12	oveq2d 5548	. . . . 5 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( A + B ) = ( A + 0 ) )
14	3	recnd 7147	. . . . . 6 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> A e. CC )
15	14	addid1d 7257	. . . . 5 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( A + 0 ) = A )
16	13, 7, 15	3eqtr3rd 2122	. . . 4 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> A = 0 )
17	16, 12	jca 300	. . 3 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( A = 0 /\ B = 0 ) )
18	17	ex 113	. 2 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A + B ) = 0 -> ( A = 0 /\ B = 0 ) ) )
19		oveq12 5541	. . 3 |- ( ( A = 0 /\ B = 0 ) -> ( A + B ) = ( 0 + 0 ) )
20		00id 7249	. . 3 |- ( 0 + 0 ) = 0
21	19, 20	syl6eq 2129	. 2 |- ( ( A = 0 /\ B = 0 ) -> ( A + B ) = 0 )
22	18, 21	impbid1 140	1 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   class class class wbr 3785  (class class class)co 5532   RRcr 6980   0cc0 6981   + caddc 6984   <_ cle 7154

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-i2m1 7081  ax-0id 7084  ax-rnegex 7085  ax-pre-ltirr 7088  ax-pre-apti 7091  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-iota 4887  df-fv 4930  df-ov 5535  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159

This theorem is referenced by:  add20i  7593  sumsqeq0  9554