Theorem xnn0xadd0 9543

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 8946	. . . 4 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
2		elxnn0 8946	. . . . . . 7 |- ( B e. NN0* <-> ( B e. NN0 \/ B = +oo ) )
3		nn0re 8890	. . . . . . . . . . . . 13 |- ( A e. NN0 -> A e. RR )
4		nn0re 8890	. . . . . . . . . . . . 13 |- ( B e. NN0 -> B e. RR )
5		rexadd 9528	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
6	3, 4, 5	syl2an 285	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A +e B ) = ( A + B ) )
7	6	eqeq1d 2123	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A +e B ) = 0 <-> ( A + B ) = 0 ) )
8		nn0ge0 8906	. . . . . . . . . . . . 13 |- ( A e. NN0 -> 0 <_ A )
9	3, 8	jca 302	. . . . . . . . . . . 12 |- ( A e. NN0 -> ( A e. RR /\ 0 <_ A ) )
10		nn0ge0 8906	. . . . . . . . . . . . 13 |- ( B e. NN0 -> 0 <_ B )
11	4, 10	jca 302	. . . . . . . . . . . 12 |- ( B e. NN0 -> ( B e. RR /\ 0 <_ B ) )
12		add20 8155	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
13	9, 11, 12	syl2an 285	. . . . . . . . . . 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 5736	. . . . . . . . . . . . 13 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
18	17	eqeq1d 2123	. . . . . . . . . . . 12 |- ( B = +oo -> ( ( A +e B ) = 0 <-> ( A +e +oo ) = 0 ) )
19	18	adantr 272	. . . . . . . . . . 11 |- ( ( B = +oo /\ A e. NN0 ) -> ( ( A +e B ) = 0 <-> ( A +e +oo ) = 0 ) )
20		nn0xnn0 8948	. . . . . . . . . . . . . 14 |- ( A e. NN0 -> A e. NN0* )
21		xnn0xrnemnf 8956	. . . . . . . . . . . . . 14 |- ( A e. NN0* -> ( A e. RR* /\ A =/= -oo ) )
22		xaddpnf1 9522	. . . . . . . . . . . . . 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 273	. . . . . . . . . . . 12 |- ( ( B = +oo /\ A e. NN0 ) -> ( A +e +oo ) = +oo )
25	24	eqeq1d 2123	. . . . . . . . . . 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 7690	. . . . . . . . . . . . 13 |- 0 e. RR
28		renepnf 7737	. . . . . . . . . . . . 13 |- ( 0 e. RR -> 0 =/= +oo )
29	27, 28	ax-mp 7	. . . . . . . . . . . 12 |- 0 =/= +oo
30	29	nesymi 2328	. . . . . . . . . . 11 |- -. +oo = 0
31	30	pm2.21i 618	. . . . . . . . . 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 688	. . . . . . 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 5735	. . . . . . . . 9 |- ( A = +oo -> ( A +e B ) = ( +oo +e B ) )
38	37	eqeq1d 2123	. . . . . . . 8 |- ( A = +oo -> ( ( A +e B ) = 0 <-> ( +oo +e B ) = 0 ) )
39		xnn0xrnemnf 8956	. . . . . . . . . 10 |- ( B e. NN0* -> ( B e. RR* /\ B =/= -oo ) )
40		xaddpnf2 9523	. . . . . . . . . 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 2123	. . . . . . . 8 |- ( B e. NN0* -> ( ( +oo +e B ) = 0 <-> +oo = 0 ) )
43	38, 42	sylan9bb 455	. . . . . . 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 688	. . . 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 5737	. . 3 |- ( ( A = 0 /\ B = 0 ) -> ( A +e B ) = ( 0 +e 0 ) )
50		0xr 7736	. . . 4 |- 0 e. RR*
51		xaddid1 9538	. . . 4 |- ( 0 e. RR* -> ( 0 +e 0 ) = 0 )
52	50, 51	ax-mp 7	. . 3 |- ( 0 +e 0 ) = 0
53	49, 52	syl6eq 2163	. 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 680   = wceq 1314   e. wcel 1463   =/= wne 2282   class class class wbr 3895  (class class class)co 5728   RRcr 7546   0cc0 7547   + caddc 7550   +oocpnf 7721   -oocmnf 7722   RR*cxr 7723   <_ cle 7725   NN0cn0 8881  NN0*cxnn0 8944   +ecxad 9450

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-addcom 7645  ax-addass 7647  ax-i2m1 7650  ax-0lt1 7651  ax-0id 7653  ax-rnegex 7654  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-inn 8631  df-n0 8882  df-xnn0 8945  df-xadd 9453

This theorem is referenced by: (None)