Theorem xnn0xadd0 10101

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 9466	. . . 4 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
2		elxnn0 9466	. . . . . . 7 |- ( B e. NN0* <-> ( B e. NN0 \/ B = +oo ) )
3		nn0re 9410	. . . . . . . . . . . . 13 |- ( A e. NN0 -> A e. RR )
4		nn0re 9410	. . . . . . . . . . . . 13 |- ( B e. NN0 -> B e. RR )
5		rexadd 10086	. . . . . . . . . . . . 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 2240	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A +e B ) = 0 <-> ( A + B ) = 0 ) )
8		nn0ge0 9426	. . . . . . . . . . . . 13 |- ( A e. NN0 -> 0 <_ A )
9	3, 8	jca 306	. . . . . . . . . . . 12 |- ( A e. NN0 -> ( A e. RR /\ 0 <_ A ) )
10		nn0ge0 9426	. . . . . . . . . . . . 13 |- ( B e. NN0 -> 0 <_ B )
11	4, 10	jca 306	. . . . . . . . . . . 12 |- ( B e. NN0 -> ( B e. RR /\ 0 <_ B ) )
12		add20 8653	. . . . . . . . . . . 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 6025	. . . . . . . . . . . . 13 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
18	17	eqeq1d 2240	. . . . . . . . . . . 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 9468	. . . . . . . . . . . . . 14 |- ( A e. NN0 -> A e. NN0* )
21		xnn0xrnemnf 9476	. . . . . . . . . . . . . 14 |- ( A e. NN0* -> ( A e. RR* /\ A =/= -oo ) )
22		xaddpnf1 10080	. . . . . . . . . . . . . 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 2240	. . . . . . . . . . 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 8178	. . . . . . . . . . . . 13 |- 0 e. RR
28		renepnf 8226	. . . . . . . . . . . . 13 |- ( 0 e. RR -> 0 =/= +oo )
29	27, 28	ax-mp 5	. . . . . . . . . . . 12 |- 0 =/= +oo
30	29	nesymi 2448	. . . . . . . . . . 11 |- -. +oo = 0
31	30	pm2.21i 651	. . . . . . . . . 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 723	. . . . . . 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 6024	. . . . . . . . 9 |- ( A = +oo -> ( A +e B ) = ( +oo +e B ) )
38	37	eqeq1d 2240	. . . . . . . 8 |- ( A = +oo -> ( ( A +e B ) = 0 <-> ( +oo +e B ) = 0 ) )
39		xnn0xrnemnf 9476	. . . . . . . . . 10 |- ( B e. NN0* -> ( B e. RR* /\ B =/= -oo ) )
40		xaddpnf2 10081	. . . . . . . . . 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 2240	. . . . . . . 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 723	. . . 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 6026	. . 3 |- ( ( A = 0 /\ B = 0 ) -> ( A +e B ) = ( 0 +e 0 ) )
50		0xr 8225	. . . 4 |- 0 e. RR*
51		xaddid1 10096	. . . 4 |- ( 0 e. RR* -> ( 0 +e 0 ) = 0 )
52	50, 51	ax-mp 5	. . 3 |- ( 0 +e 0 ) = 0
53	49, 52	eqtrdi 2280	. 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 715   = wceq 1397   e. wcel 2202   =/= wne 2402   class class class wbr 4088  (class class class)co 6017   RRcr 8030   0cc0 8031   + caddc 8034   +oocpnf 8210   -oocmnf 8211   RR*cxr 8212   <_ cle 8214   NN0cn0 9401  NN0*cxnn0 9464   +ecxad 10004

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-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  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-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-inn 9143  df-n0 9402  df-xnn0 9465  df-xadd 10007

This theorem is referenced by: (None)