Theorem xnn0xadd0 9650

Description: The sum of two extended nonnegative integers is 0 iff each of the two extended nonnegative integers is 0. (Contributed by AV, 14-Dec-2020.)

Assertion

Ref	Expression
xnn0xadd0	|- ( ( A e. NN0* /\ B e. NN0* ) -> ( ( A +e B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )

Proof of Theorem xnn0xadd0

Step	Hyp	Ref	Expression
1		elxnn0 9042	. . . 4 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
2		elxnn0 9042	. . . . . . 7 |- ( B e. NN0* <-> ( B e. NN0 \/ B = +oo ) )
3		nn0re 8986	. . . . . . . . . . . . 13 |- ( A e. NN0 -> A e. RR )
4		nn0re 8986	. . . . . . . . . . . . 13 |- ( B e. NN0 -> B e. RR )
5		rexadd 9635	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
6	3, 4, 5	syl2an 287	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A +e B ) = ( A + B ) )
7	6	eqeq1d 2148	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A +e B ) = 0 <-> ( A + B ) = 0 ) )
8		nn0ge0 9002	. . . . . . . . . . . . 13 |- ( A e. NN0 -> 0 <_ A )
9	3, 8	jca 304	. . . . . . . . . . . 12 |- ( A e. NN0 -> ( A e. RR /\ 0 <_ A ) )
10		nn0ge0 9002	. . . . . . . . . . . . 13 |- ( B e. NN0 -> 0 <_ B )
11	4, 10	jca 304	. . . . . . . . . . . 12 |- ( B e. NN0 -> ( B e. RR /\ 0 <_ B ) )
12		add20 8236	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
13	9, 11, 12	syl2an 287	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
14	7, 13	bitrd 187	. . . . . . . . . 10 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A +e B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
15	14	biimpd 143	. . . . . . . . 9 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) )
16	15	expcom 115	. . . . . . . 8 |- ( B e. NN0 -> ( A e. NN0 -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) ) )
17		oveq2 5782	. . . . . . . . . . . . 13 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
18	17	eqeq1d 2148	. . . . . . . . . . . 12 |- ( B = +oo -> ( ( A +e B ) = 0 <-> ( A +e +oo ) = 0 ) )
19	18	adantr 274	. . . . . . . . . . 11 |- ( ( B = +oo /\ A e. NN0 ) -> ( ( A +e B ) = 0 <-> ( A +e +oo ) = 0 ) )
20		nn0xnn0 9044	. . . . . . . . . . . . . 14 |- ( A e. NN0 -> A e. NN0* )
21		xnn0xrnemnf 9052	. . . . . . . . . . . . . 14 |- ( A e. NN0* -> ( A e. RR* /\ A =/= -oo ) )
22		xaddpnf1 9629	. . . . . . . . . . . . . 14 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
23	20, 21, 22	3syl 17	. . . . . . . . . . . . 13 |- ( A e. NN0 -> ( A +e +oo ) = +oo )
24	23	adantl 275	. . . . . . . . . . . 12 |- ( ( B = +oo /\ A e. NN0 ) -> ( A +e +oo ) = +oo )
25	24	eqeq1d 2148	. . . . . . . . . . 11 |- ( ( B = +oo /\ A e. NN0 ) -> ( ( A +e +oo ) = 0 <-> +oo = 0 ) )
26	19, 25	bitrd 187	. . . . . . . . . 10 |- ( ( B = +oo /\ A e. NN0 ) -> ( ( A +e B ) = 0 <-> +oo = 0 ) )
27		0re 7766	. . . . . . . . . . . . 13 |- 0 e. RR
28		renepnf 7813	. . . . . . . . . . . . 13 |- ( 0 e. RR -> 0 =/= +oo )
29	27, 28	ax-mp 5	. . . . . . . . . . . 12 |- 0 =/= +oo
30	29	nesymi 2354	. . . . . . . . . . 11 |- -. +oo = 0
31	30	pm2.21i 635	. . . . . . . . . 10 |- ( +oo = 0 -> ( A = 0 /\ B = 0 ) )
32	26, 31	syl6bi 162	. . . . . . . . 9 |- ( ( B = +oo /\ A e. NN0 ) -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) )
33	32	ex 114	. . . . . . . 8 |- ( B = +oo -> ( A e. NN0 -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) ) )
34	16, 33	jaoi 705	. . . . . . 7 |- ( ( B e. NN0 \/ B = +oo ) -> ( A e. NN0 -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) ) )
35	2, 34	sylbi 120	. . . . . 6 |- ( B e. NN0* -> ( A e. NN0 -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) ) )
36	35	com12 30	. . . . 5 |- ( A e. NN0 -> ( B e. NN0* -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) ) )
37		oveq1 5781	. . . . . . . . 9 |- ( A = +oo -> ( A +e B ) = ( +oo +e B ) )
38	37	eqeq1d 2148	. . . . . . . 8 |- ( A = +oo -> ( ( A +e B ) = 0 <-> ( +oo +e B ) = 0 ) )
39		xnn0xrnemnf 9052	. . . . . . . . . 10 |- ( B e. NN0* -> ( B e. RR* /\ B =/= -oo ) )
40		xaddpnf2 9630	. . . . . . . . . 10 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
41	39, 40	syl 14	. . . . . . . . 9 |- ( B e. NN0* -> ( +oo +e B ) = +oo )
42	41	eqeq1d 2148	. . . . . . . 8 |- ( B e. NN0* -> ( ( +oo +e B ) = 0 <-> +oo = 0 ) )
43	38, 42	sylan9bb 457	. . . . . . 7 |- ( ( A = +oo /\ B e. NN0* ) -> ( ( A +e B ) = 0 <-> +oo = 0 ) )
44	43, 31	syl6bi 162	. . . . . 6 |- ( ( A = +oo /\ B e. NN0* ) -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) )
45	44	ex 114	. . . . 5 |- ( A = +oo -> ( B e. NN0* -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) ) )
46	36, 45	jaoi 705	. . . 4 |- ( ( A e. NN0 \/ A = +oo ) -> ( B e. NN0* -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) ) )
47	1, 46	sylbi 120	. . 3 |- ( A e. NN0* -> ( B e. NN0* -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) ) )
48	47	imp 123	. 2 |- ( ( A e. NN0* /\ B e. NN0* ) -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) )
49		oveq12 5783	. . 3 |- ( ( A = 0 /\ B = 0 ) -> ( A +e B ) = ( 0 +e 0 ) )
50		0xr 7812	. . . 4 |- 0 e. RR*
51		xaddid1 9645	. . . 4 |- ( 0 e. RR* -> ( 0 +e 0 ) = 0 )
52	50, 51	ax-mp 5	. . 3 |- ( 0 +e 0 ) = 0
53	49, 52	syl6eq 2188	. 2 |- ( ( A = 0 /\ B = 0 ) -> ( A +e B ) = 0 )
54	48, 53	impbid1 141	1 |- ( ( A e. NN0* /\ B e. NN0* ) -> ( ( A +e B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   = wceq 1331   e. wcel 1480   =/= wne 2308   class class class wbr 3929  (class class class)co 5774   RRcr 7619   0cc0 7620   + caddc 7623   +oocpnf 7797   -oocmnf 7798   RR*cxr 7799   <_ cle 7801   NN0cn0 8977  NN0*cxnn0 9040   +ecxad 9557

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-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  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-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-inn 8721  df-n0 8978  df-xnn0 9041  df-xadd 9560

This theorem is referenced by: (None)