Theorem add20 8332

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 524	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> 0 <_ A )
2		simplrl 525	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B e. RR )
3		simplll 523	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> A e. RR )
4		addge02 8331	. . . . . . . . . 10 |- ( ( B e. RR /\ A e. RR ) -> ( 0 <_ A <-> B <_ ( A + B ) ) )
5	2, 3, 4	syl2anc 409	. . . . . . . . 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 3990	. . . . . . 7 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B <_ 0 )
9		simplrr 526	. . . . . . 7 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> 0 <_ B )
10		0red 7862	. . . . . . . 8 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> 0 e. RR )
11	2, 10	letri3d 7975	. . . . . . 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 929	. . . . . 6 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B = 0 )
13	12	oveq2d 5834	. . . . 5 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( A + B ) = ( A + 0 ) )
14	3	recnd 7889	. . . . . 6 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> A e. CC )
15	14	addid1d 8007	. . . . 5 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( A + 0 ) = A )
16	13, 7, 15	3eqtr3rd 2199	. . . 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 5827	. . 3 |- ( ( A = 0 /\ B = 0 ) -> ( A + B ) = ( 0 + 0 ) )
20		00id 7999	. . 3 |- ( 0 + 0 ) = 0
21	19, 20	eqtrdi 2206	. 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 1335   e. wcel 2128   class class class wbr 3965  (class class class)co 5818   RRcr 7714   0cc0 7715   + caddc 7718   <_ cle 7896

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-addcom 7815  ax-addass 7817  ax-i2m1 7820  ax-0id 7823  ax-rnegex 7824  ax-pre-ltirr 7827  ax-pre-apti 7830  ax-pre-ltadd 7831

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4589  df-cnv 4591  df-iota 5132  df-fv 5175  df-ov 5821  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901

This theorem is referenced by:  add20i  8350  xnn0xadd0  9753  sumsqeq0  10479