Theorem xnn0xadd0 9942

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 9314	. . . 4 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
2		elxnn0 9314	. . . . . . 7 |- ( B e. NN0* <-> ( B e. NN0 \/ B = +oo ) )
3		nn0re 9258	. . . . . . . . . . . . 13 |- ( A e. NN0 -> A e. RR )
4		nn0re 9258	. . . . . . . . . . . . 13 |- ( B e. NN0 -> B e. RR )
5		rexadd 9927	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
6	3, 4, 5	syl2an 289	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A +e B ) = ( A + B ) )
7	6	eqeq1d 2205	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A +e B ) = 0 <-> ( A + B ) = 0 ) )
8		nn0ge0 9274	. . . . . . . . . . . . 13 |- ( A e. NN0 -> 0 <_ A )
9	3, 8	jca 306	. . . . . . . . . . . 12 |- ( A e. NN0 -> ( A e. RR /\ 0 <_ A ) )
10		nn0ge0 9274	. . . . . . . . . . . . 13 |- ( B e. NN0 -> 0 <_ B )
11	4, 10	jca 306	. . . . . . . . . . . 12 |- ( B e. NN0 -> ( B e. RR /\ 0 <_ B ) )
12		add20 8501	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
13	9, 11, 12	syl2an 289	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
14	7, 13	bitrd 188	. . . . . . . . . 10 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A +e B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
15	14	biimpd 144	. . . . . . . . 9 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) )
16	15	expcom 116	. . . . . . . 8 |- ( B e. NN0 -> ( A e. NN0 -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) ) )
17		oveq2 5930	. . . . . . . . . . . . 13 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
18	17	eqeq1d 2205	. . . . . . . . . . . 12 |- ( B = +oo -> ( ( A +e B ) = 0 <-> ( A +e +oo ) = 0 ) )
19	18	adantr 276	. . . . . . . . . . 11 |- ( ( B = +oo /\ A e. NN0 ) -> ( ( A +e B ) = 0 <-> ( A +e +oo ) = 0 ) )
20		nn0xnn0 9316	. . . . . . . . . . . . . 14 |- ( A e. NN0 -> A e. NN0* )
21		xnn0xrnemnf 9324	. . . . . . . . . . . . . 14 |- ( A e. NN0* -> ( A e. RR* /\ A =/= -oo ) )
22		xaddpnf1 9921	. . . . . . . . . . . . . 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 277	. . . . . . . . . . . 12 |- ( ( B = +oo /\ A e. NN0 ) -> ( A +e +oo ) = +oo )
25	24	eqeq1d 2205	. . . . . . . . . . 11 |- ( ( B = +oo /\ A e. NN0 ) -> ( ( A +e +oo ) = 0 <-> +oo = 0 ) )
26	19, 25	bitrd 188	. . . . . . . . . 10 |- ( ( B = +oo /\ A e. NN0 ) -> ( ( A +e B ) = 0 <-> +oo = 0 ) )
27		0re 8026	. . . . . . . . . . . . 13 |- 0 e. RR
28		renepnf 8074	. . . . . . . . . . . . 13 |- ( 0 e. RR -> 0 =/= +oo )
29	27, 28	ax-mp 5	. . . . . . . . . . . 12 |- 0 =/= +oo
30	29	nesymi 2413	. . . . . . . . . . 11 |- -. +oo = 0
31	30	pm2.21i 647	. . . . . . . . . 10 |- ( +oo = 0 -> ( A = 0 /\ B = 0 ) )
32	26, 31	biimtrdi 163	. . . . . . . . 9 |- ( ( B = +oo /\ A e. NN0 ) -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) )
33	32	ex 115	. . . . . . . 8 |- ( B = +oo -> ( A e. NN0 -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) ) )
34	16, 33	jaoi 717	. . . . . . 7 |- ( ( B e. NN0 \/ B = +oo ) -> ( A e. NN0 -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) ) )
35	2, 34	sylbi 121	. . . . . 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 5929	. . . . . . . . 9 |- ( A = +oo -> ( A +e B ) = ( +oo +e B ) )
38	37	eqeq1d 2205	. . . . . . . 8 |- ( A = +oo -> ( ( A +e B ) = 0 <-> ( +oo +e B ) = 0 ) )
39		xnn0xrnemnf 9324	. . . . . . . . . 10 |- ( B e. NN0* -> ( B e. RR* /\ B =/= -oo ) )
40		xaddpnf2 9922	. . . . . . . . . 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 2205	. . . . . . . 8 |- ( B e. NN0* -> ( ( +oo +e B ) = 0 <-> +oo = 0 ) )
43	38, 42	sylan9bb 462	. . . . . . 7 |- ( ( A = +oo /\ B e. NN0* ) -> ( ( A +e B ) = 0 <-> +oo = 0 ) )
44	43, 31	biimtrdi 163	. . . . . 6 |- ( ( A = +oo /\ B e. NN0* ) -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) )
45	44	ex 115	. . . . 5 |- ( A = +oo -> ( B e. NN0* -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) ) )
46	36, 45	jaoi 717	. . . 4 |- ( ( A e. NN0 \/ A = +oo ) -> ( B e. NN0* -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) ) )
47	1, 46	sylbi 121	. . 3 |- ( A e. NN0* -> ( B e. NN0* -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) ) )
48	47	imp 124	. 2 |- ( ( A e. NN0* /\ B e. NN0* ) -> ( ( A +e B ) = 0 -> ( A = 0 /\ B = 0 ) ) )
49		oveq12 5931	. . 3 |- ( ( A = 0 /\ B = 0 ) -> ( A +e B ) = ( 0 +e 0 ) )
50		0xr 8073	. . . 4 |- 0 e. RR*
51		xaddid1 9937	. . . 4 |- ( 0 e. RR* -> ( 0 +e 0 ) = 0 )
52	50, 51	ax-mp 5	. . 3 |- ( 0 +e 0 ) = 0
53	49, 52	eqtrdi 2245	. 2 |- ( ( A = 0 /\ B = 0 ) -> ( A +e B ) = 0 )
54	48, 53	impbid1 142	1 |- ( ( A e. NN0* /\ B e. NN0* ) -> ( ( A +e B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2167   =/= wne 2367   class class class wbr 4033  (class class class)co 5922   RRcr 7878   0cc0 7879   + caddc 7882   +oocpnf 8058   -oocmnf 8059   RR*cxr 8060   <_ cle 8062   NN0cn0 9249  NN0*cxnn0 9312   +ecxad 9845

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-inn 8991  df-n0 9250  df-xnn0 9313  df-xadd 9848

This theorem is referenced by: (None)