Theorem add20 8236

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 523	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> 0 <_ A )
2		simplrl 524	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B e. RR )
3		simplll 522	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> A e. RR )
4		addge02 8235	. . . . . . . . . 10 |- ( ( B e. RR /\ A e. RR ) -> ( 0 <_ A <-> B <_ ( A + B ) ) )
5	2, 3, 4	syl2anc 408	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( 0 <_ A <-> B <_ ( A + B ) ) )
6	1, 5	mpbid 146	. . . . . . . 8 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B <_ ( A + B ) )
7		simpr 109	. . . . . . . 8 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( A + B ) = 0 )
8	6, 7	breqtrd 3954	. . . . . . 7 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B <_ 0 )
9		simplrr 525	. . . . . . 7 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> 0 <_ B )
10		0red 7767	. . . . . . . 8 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> 0 e. RR )
11	2, 10	letri3d 7879	. . . . . . 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 928	. . . . . 6 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B = 0 )
13	12	oveq2d 5790	. . . . 5 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( A + B ) = ( A + 0 ) )
14	3	recnd 7794	. . . . . 6 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> A e. CC )
15	14	addid1d 7911	. . . . 5 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( A + 0 ) = A )
16	13, 7, 15	3eqtr3rd 2181	. . . 4 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> A = 0 )
17	16, 12	jca 304	. . 3 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( A = 0 /\ B = 0 ) )
18	17	ex 114	. 2 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A + B ) = 0 -> ( A = 0 /\ B = 0 ) ) )
19		oveq12 5783	. . 3 |- ( ( A = 0 /\ B = 0 ) -> ( A + B ) = ( 0 + 0 ) )
20		00id 7903	. . 3 |- ( 0 + 0 ) = 0
21	19, 20	syl6eq 2188	. 2 |- ( ( A = 0 /\ B = 0 ) -> ( A + B ) = 0 )
22	18, 21	impbid1 141	1 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   class class class wbr 3929  (class class class)co 5774   RRcr 7619   0cc0 7620   + caddc 7623   <_ cle 7801

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729  ax-pre-ltirr 7732  ax-pre-apti 7735  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-iota 5088  df-fv 5131  df-ov 5777  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806

This theorem is referenced by:  add20i  8254  xnn0xadd0  9650  sumsqeq0  10371