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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
StepHypRef 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 ) )
63, 4, 5syl2an 285 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( A +e
B )  =  ( A  +  B ) )
76eqeq1d 2123 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( ( A +e B )  =  0  <->  ( A  +  B )  =  0 ) )
8 nn0ge0 8906 . . . . . . . . . . . . 13  |-  ( A  e.  NN0  ->  0  <_  A )
93, 8jca 302 . . . . . . . . . . . 12  |-  ( A  e.  NN0  ->  ( A  e.  RR  /\  0  <_  A ) )
10 nn0ge0 8906 . . . . . . . . . . . . 13  |-  ( B  e.  NN0  ->  0  <_  B )
114, 10jca 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 ) ) )
139, 11, 12syl2an 285 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( ( A  +  B )  =  0  <-> 
( A  =  0  /\  B  =  0 ) ) )
147, 13bitrd 187 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( ( A +e B )  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )
1514biimpd 143 . . . . . . . . 9  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( ( A +e B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) )
1615expcom 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 ) )
1817eqeq1d 2123 . . . . . . . . . . . 12  |-  ( B  = +oo  ->  (
( A +e
B )  =  0  <-> 
( A +e +oo )  =  0
) )
1918adantr 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 )
2320, 21, 223syl 17 . . . . . . . . . . . . 13  |-  ( A  e.  NN0  ->  ( A +e +oo )  = +oo )
2423adantl 273 . . . . . . . . . . . 12  |-  ( ( B  = +oo  /\  A  e.  NN0 )  -> 
( A +e +oo )  = +oo )
2524eqeq1d 2123 . . . . . . . . . . 11  |-  ( ( B  = +oo  /\  A  e.  NN0 )  -> 
( ( A +e +oo )  =  0  <-> +oo  =  0 ) )
2619, 25bitrd 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 )
2927, 28ax-mp 7 . . . . . . . . . . . 12  |-  0  =/= +oo
3029nesymi 2328 . . . . . . . . . . 11  |-  -. +oo  =  0
3130pm2.21i 618 . . . . . . . . . 10  |-  ( +oo  =  0  ->  ( A  =  0  /\  B  =  0 ) )
3226, 31syl6bi 162 . . . . . . . . 9  |-  ( ( B  = +oo  /\  A  e.  NN0 )  -> 
( ( A +e B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) )
3332ex 114 . . . . . . . 8  |-  ( B  = +oo  ->  ( A  e.  NN0  ->  (
( A +e
B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) ) )
3416, 33jaoi 688 . . . . . . 7  |-  ( ( B  e.  NN0  \/  B  = +oo )  ->  ( A  e.  NN0  ->  ( ( A +e B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) ) )
352, 34sylbi 120 . . . . . 6  |-  ( B  e. NN0*  ->  ( A  e. 
NN0  ->  ( ( A +e B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) ) )
3635com12 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 ) )
3837eqeq1d 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 )
4139, 40syl 14 . . . . . . . . 9  |-  ( B  e. NN0*  ->  ( +oo +e B )  = +oo )
4241eqeq1d 2123 . . . . . . . 8  |-  ( B  e. NN0*  ->  ( ( +oo +e B )  =  0  <-> +oo  =  0 ) )
4338, 42sylan9bb 455 . . . . . . 7  |-  ( ( A  = +oo  /\  B  e. NN0* )  ->  ( ( A +e
B )  =  0  <-> +oo  =  0 ) )
4443, 31syl6bi 162 . . . . . 6  |-  ( ( A  = +oo  /\  B  e. NN0* )  ->  ( ( A +e
B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) )
4544ex 114 . . . . 5  |-  ( A  = +oo  ->  ( B  e. NN0*  ->  ( ( A +e B )  =  0  -> 
( A  =  0  /\  B  =  0 ) ) ) )
4636, 45jaoi 688 . . . 4  |-  ( ( A  e.  NN0  \/  A  = +oo )  ->  ( B  e. NN0*  ->  ( ( A +e
B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) ) )
471, 46sylbi 120 . . 3  |-  ( A  e. NN0*  ->  ( B  e. NN0* 
->  ( ( A +e B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) ) )
4847imp 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 )
5250, 51ax-mp 7 . . 3  |-  ( 0 +e 0 )  =  0
5349, 52syl6eq 2163 . 2  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A +e B )  =  0 )
5448, 53impbid1 141 1  |-  ( ( A  e. NN0*  /\  B  e. NN0* )  ->  ( ( A +e B )  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )
Colors of variables: wff set class
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)
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