Theorem xnn0xadd0 9803

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 9179	. . . 4 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
2		elxnn0 9179	. . . . . . 7 |- ( B e. NN0* <-> ( B e. NN0 \/ B = +oo ) )
3		nn0re 9123	. . . . . . . . . . . . 13 |- ( A e. NN0 -> A e. RR )
4		nn0re 9123	. . . . . . . . . . . . 13 |- ( B e. NN0 -> B e. RR )
5		rexadd 9788	. . . . . . . . . . . . 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 2174	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A +e B ) = 0 <-> ( A + B ) = 0 ) )
8		nn0ge0 9139	. . . . . . . . . . . . 13 |- ( A e. NN0 -> 0 <_ A )
9	3, 8	jca 304	. . . . . . . . . . . 12 |- ( A e. NN0 -> ( A e. RR /\ 0 <_ A ) )
10		nn0ge0 9139	. . . . . . . . . . . . 13 |- ( B e. NN0 -> 0 <_ B )
11	4, 10	jca 304	. . . . . . . . . . . 12 |- ( B e. NN0 -> ( B e. RR /\ 0 <_ B ) )
12		add20 8372	. . . . . . . . . . . 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 5850	. . . . . . . . . . . . 13 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
18	17	eqeq1d 2174	. . . . . . . . . . . 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 9181	. . . . . . . . . . . . . 14 |- ( A e. NN0 -> A e. NN0* )
21		xnn0xrnemnf 9189	. . . . . . . . . . . . . 14 |- ( A e. NN0* -> ( A e. RR* /\ A =/= -oo ) )
22		xaddpnf1 9782	. . . . . . . . . . . . . 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 2174	. . . . . . . . . . 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 7899	. . . . . . . . . . . . 13 |- 0 e. RR
28		renepnf 7946	. . . . . . . . . . . . 13 |- ( 0 e. RR -> 0 =/= +oo )
29	27, 28	ax-mp 5	. . . . . . . . . . . 12 |- 0 =/= +oo
30	29	nesymi 2382	. . . . . . . . . . 11 |- -. +oo = 0
31	30	pm2.21i 636	. . . . . . . . . 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 706	. . . . . . 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 5849	. . . . . . . . 9 |- ( A = +oo -> ( A +e B ) = ( +oo +e B ) )
38	37	eqeq1d 2174	. . . . . . . 8 |- ( A = +oo -> ( ( A +e B ) = 0 <-> ( +oo +e B ) = 0 ) )
39		xnn0xrnemnf 9189	. . . . . . . . . 10 |- ( B e. NN0* -> ( B e. RR* /\ B =/= -oo ) )
40		xaddpnf2 9783	. . . . . . . . . 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 2174	. . . . . . . 8 |- ( B e. NN0* -> ( ( +oo +e B ) = 0 <-> +oo = 0 ) )
43	38, 42	sylan9bb 458	. . . . . . 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 706	. . . 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 5851	. . 3 |- ( ( A = 0 /\ B = 0 ) -> ( A +e B ) = ( 0 +e 0 ) )
50		0xr 7945	. . . 4 |- 0 e. RR*
51		xaddid1 9798	. . . 4 |- ( 0 e. RR* -> ( 0 +e 0 ) = 0 )
52	50, 51	ax-mp 5	. . 3 |- ( 0 +e 0 ) = 0
53	49, 52	eqtrdi 2215	. 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 698   = wceq 1343   e. wcel 2136   =/= wne 2336   class class class wbr 3982  (class class class)co 5842   RRcr 7752   0cc0 7753   + caddc 7756   +oocpnf 7930   -oocmnf 7931   RR*cxr 7932   <_ cle 7934   NN0cn0 9114  NN0*cxnn0 9177   +ecxad 9706

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-inn 8858  df-n0 9115  df-xnn0 9178  df-xadd 9709

This theorem is referenced by: (None)