Theorem add20 7697

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 501	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> 0 <_ A )
2		simplrl 502	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B e. RR )
3		simplll 500	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> A e. RR )
4		addge02 7696	. . . . . . . . . 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 3829	. . . . . . 7 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B <_ 0 )
9		simplrr 503	. . . . . . 7 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> 0 <_ B )
10		0red 7234	. . . . . . . 8 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> 0 e. RR )
11	2, 10	letri3d 7345	. . . . . . 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 886	. . . . . 6 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B = 0 )
13	12	oveq2d 5579	. . . . 5 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( A + B ) = ( A + 0 ) )
14	3	recnd 7261	. . . . . 6 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> A e. CC )
15	14	addid1d 7376	. . . . 5 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( A + 0 ) = A )
16	13, 7, 15	3eqtr3rd 2124	. . . 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 5572	. . 3 |- ( ( A = 0 /\ B = 0 ) -> ( A + B ) = ( 0 + 0 ) )
20		00id 7368	. . 3 |- ( 0 + 0 ) = 0
21	19, 20	syl6eq 2131	. 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 1285   e. wcel 1434   class class class wbr 3805  (class class class)co 5563   RRcr 7094   0cc0 7095   + caddc 7098   <_ cle 7268

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-i2m1 7195  ax-0id 7198  ax-rnegex 7199  ax-pre-ltirr 7202  ax-pre-apti 7205  ax-pre-ltadd 7206

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-xp 4397  df-cnv 4399  df-iota 4917  df-fv 4960  df-ov 5566  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273

This theorem is referenced by:  add20i  7712  sumsqeq0  9703