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Theorem xnn0xadd0 9899
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 9272 . . . 4  |-  ( A  e. NN0* 
<->  ( A  e.  NN0  \/  A  = +oo )
)
2 elxnn0 9272 . . . . . . 7  |-  ( B  e. NN0* 
<->  ( B  e.  NN0  \/  B  = +oo )
)
3 nn0re 9216 . . . . . . . . . . . . 13  |-  ( A  e.  NN0  ->  A  e.  RR )
4 nn0re 9216 . . . . . . . . . . . . 13  |-  ( B  e.  NN0  ->  B  e.  RR )
5 rexadd 9884 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )
63, 4, 5syl2an 289 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( A +e
B )  =  ( A  +  B ) )
76eqeq1d 2198 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( ( A +e B )  =  0  <->  ( A  +  B )  =  0 ) )
8 nn0ge0 9232 . . . . . . . . . . . . 13  |-  ( A  e.  NN0  ->  0  <_  A )
93, 8jca 306 . . . . . . . . . . . 12  |-  ( A  e.  NN0  ->  ( A  e.  RR  /\  0  <_  A ) )
10 nn0ge0 9232 . . . . . . . . . . . . 13  |-  ( B  e.  NN0  ->  0  <_  B )
114, 10jca 306 . . . . . . . . . . . 12  |-  ( B  e.  NN0  ->  ( B  e.  RR  /\  0  <_  B ) )
12 add20 8462 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  +  B )  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )
139, 11, 12syl2an 289 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( ( A  +  B )  =  0  <-> 
( A  =  0  /\  B  =  0 ) ) )
147, 13bitrd 188 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( ( A +e B )  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )
1514biimpd 144 . . . . . . . . 9  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( ( A +e B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) )
1615expcom 116 . . . . . . . 8  |-  ( B  e.  NN0  ->  ( A  e.  NN0  ->  ( ( A +e B )  =  0  -> 
( A  =  0  /\  B  =  0 ) ) ) )
17 oveq2 5905 . . . . . . . . . . . . 13  |-  ( B  = +oo  ->  ( A +e B )  =  ( A +e +oo ) )
1817eqeq1d 2198 . . . . . . . . . . . 12  |-  ( B  = +oo  ->  (
( A +e
B )  =  0  <-> 
( A +e +oo )  =  0
) )
1918adantr 276 . . . . . . . . . . 11  |-  ( ( B  = +oo  /\  A  e.  NN0 )  -> 
( ( A +e B )  =  0  <->  ( A +e +oo )  =  0 ) )
20 nn0xnn0 9274 . . . . . . . . . . . . . 14  |-  ( A  e.  NN0  ->  A  e. NN0*
)
21 xnn0xrnemnf 9282 . . . . . . . . . . . . . 14  |-  ( A  e. NN0*  ->  ( A  e. 
RR*  /\  A  =/= -oo ) )
22 xaddpnf1 9878 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
2320, 21, 223syl 17 . . . . . . . . . . . . 13  |-  ( A  e.  NN0  ->  ( A +e +oo )  = +oo )
2423adantl 277 . . . . . . . . . . . 12  |-  ( ( B  = +oo  /\  A  e.  NN0 )  -> 
( A +e +oo )  = +oo )
2524eqeq1d 2198 . . . . . . . . . . 11  |-  ( ( B  = +oo  /\  A  e.  NN0 )  -> 
( ( A +e +oo )  =  0  <-> +oo  =  0 ) )
2619, 25bitrd 188 . . . . . . . . . 10  |-  ( ( B  = +oo  /\  A  e.  NN0 )  -> 
( ( A +e B )  =  0  <-> +oo  =  0 ) )
27 0re 7988 . . . . . . . . . . . . 13  |-  0  e.  RR
28 renepnf 8036 . . . . . . . . . . . . 13  |-  ( 0  e.  RR  ->  0  =/= +oo )
2927, 28ax-mp 5 . . . . . . . . . . . 12  |-  0  =/= +oo
3029nesymi 2406 . . . . . . . . . . 11  |-  -. +oo  =  0
3130pm2.21i 647 . . . . . . . . . 10  |-  ( +oo  =  0  ->  ( A  =  0  /\  B  =  0 ) )
3226, 31biimtrdi 163 . . . . . . . . 9  |-  ( ( B  = +oo  /\  A  e.  NN0 )  -> 
( ( A +e B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) )
3332ex 115 . . . . . . . 8  |-  ( B  = +oo  ->  ( A  e.  NN0  ->  (
( A +e
B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) ) )
3416, 33jaoi 717 . . . . . . 7  |-  ( ( B  e.  NN0  \/  B  = +oo )  ->  ( A  e.  NN0  ->  ( ( A +e B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) ) )
352, 34sylbi 121 . . . . . 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 5904 . . . . . . . . 9  |-  ( A  = +oo  ->  ( A +e B )  =  ( +oo +e B ) )
3837eqeq1d 2198 . . . . . . . 8  |-  ( A  = +oo  ->  (
( A +e
B )  =  0  <-> 
( +oo +e B )  =  0 ) )
39 xnn0xrnemnf 9282 . . . . . . . . . 10  |-  ( B  e. NN0*  ->  ( B  e. 
RR*  /\  B  =/= -oo ) )
40 xaddpnf2 9879 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
4139, 40syl 14 . . . . . . . . 9  |-  ( B  e. NN0*  ->  ( +oo +e B )  = +oo )
4241eqeq1d 2198 . . . . . . . 8  |-  ( B  e. NN0*  ->  ( ( +oo +e B )  =  0  <-> +oo  =  0 ) )
4338, 42sylan9bb 462 . . . . . . 7  |-  ( ( A  = +oo  /\  B  e. NN0* )  ->  ( ( A +e
B )  =  0  <-> +oo  =  0 ) )
4443, 31biimtrdi 163 . . . . . 6  |-  ( ( A  = +oo  /\  B  e. NN0* )  ->  ( ( A +e
B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) )
4544ex 115 . . . . 5  |-  ( A  = +oo  ->  ( B  e. NN0*  ->  ( ( A +e B )  =  0  -> 
( A  =  0  /\  B  =  0 ) ) ) )
4636, 45jaoi 717 . . . 4  |-  ( ( A  e.  NN0  \/  A  = +oo )  ->  ( B  e. NN0*  ->  ( ( A +e
B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) ) )
471, 46sylbi 121 . . 3  |-  ( A  e. NN0*  ->  ( B  e. NN0* 
->  ( ( A +e B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) ) )
4847imp 124 . 2  |-  ( ( A  e. NN0*  /\  B  e. NN0* )  ->  ( ( A +e B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) )
49 oveq12 5906 . . 3  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A +e B )  =  ( 0 +e 0 ) )
50 0xr 8035 . . . 4  |-  0  e.  RR*
51 xaddid1 9894 . . . 4  |-  ( 0  e.  RR*  ->  ( 0 +e 0 )  =  0 )
5250, 51ax-mp 5 . . 3  |-  ( 0 +e 0 )  =  0
5349, 52eqtrdi 2238 . 2  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A +e B )  =  0 )
5448, 53impbid1 142 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 104    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2160    =/= wne 2360   class class class wbr 4018  (class class class)co 5897   RRcr 7841   0cc0 7842    + caddc 7845   +oocpnf 8020   -oocmnf 8021   RR*cxr 8022    <_ cle 8024   NN0cn0 9207  NN0*cxnn0 9270   +ecxad 9802
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-i2m1 7947  ax-0lt1 7948  ax-0id 7950  ax-rnegex 7951  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958
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 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-inn 8951  df-n0 9208  df-xnn0 9271  df-xadd 9805
This theorem is referenced by: (None)
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