Theorem add20 8653

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 536	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> 0 <_ A )
2		simplrl 537	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B e. RR )
3		simplll 535	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> A e. RR )
4		addge02 8652	. . . . . . . . . 10 |- ( ( B e. RR /\ A e. RR ) -> ( 0 <_ A <-> B <_ ( A + B ) ) )
5	2, 3, 4	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( 0 <_ A <-> B <_ ( A + B ) ) )
6	1, 5	mpbid 147	. . . . . . . 8 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B <_ ( A + B ) )
7		simpr 110	. . . . . . . 8 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( A + B ) = 0 )
8	6, 7	breqtrd 4114	. . . . . . 7 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B <_ 0 )
9		simplrr 538	. . . . . . 7 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> 0 <_ B )
10		0red 8179	. . . . . . . 8 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> 0 e. RR )
11	2, 10	letri3d 8294	. . . . . . 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 952	. . . . . 6 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> B = 0 )
13	12	oveq2d 6033	. . . . 5 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( A + B ) = ( A + 0 ) )
14	3	recnd 8207	. . . . . 6 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> A e. CC )
15	14	addridd 8327	. . . . 5 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( A + 0 ) = A )
16	13, 7, 15	3eqtr3rd 2273	. . . 4 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> A = 0 )
17	16, 12	jca 306	. . 3 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) /\ ( A + B ) = 0 ) -> ( A = 0 /\ B = 0 ) )
18	17	ex 115	. 2 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A + B ) = 0 -> ( A = 0 /\ B = 0 ) ) )
19		oveq12 6026	. . 3 |- ( ( A = 0 /\ B = 0 ) -> ( A + B ) = ( 0 + 0 ) )
20		00id 8319	. . 3 |- ( 0 + 0 ) = 0
21	19, 20	eqtrdi 2280	. 2 |- ( ( A = 0 /\ B = 0 ) -> ( A + B ) = 0 )
22	18, 21	impbid1 142	1 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   RRcr 8030   0cc0 8031   + caddc 8034   <_ cle 8214

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-apti 8146  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-iota 5286  df-fv 5334  df-ov 6020  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219

This theorem is referenced by:  add20i  8671  xnn0xadd0  10101  sumsqeq0  10879  ccat0  11172  4sqlem15  12977  4sqlem16  12978  2sqlem7  15849  vtxd0nedgbfi  16149