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Theorem algcvgblem 10575
Description: Lemma for algcvgb 10576. (Contributed by Paul Chapman, 31-Mar-2011.)
Assertion
Ref Expression
algcvgblem  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  <->  ( ( M  =/=  0  ->  N  <  M )  /\  ( M  =  0  ->  N  =  0 ) ) ) )

Proof of Theorem algcvgblem
StepHypRef Expression
1 nn0z 8452 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  e.  ZZ )
2 0z 8443 . . . . . . . . 9  |-  0  e.  ZZ
3 zdceq 8504 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
41, 2, 3sylancl 404 . . . . . . . 8  |-  ( N  e.  NN0  -> DECID  N  =  0
)
54dcned 2252 . . . . . . 7  |-  ( N  e.  NN0  -> DECID  N  =/=  0
)
6 imordc 830 . . . . . . 7  |-  (DECID  N  =/=  0  ->  ( ( N  =/=  0  ->  N  <  M )  <->  ( -.  N  =/=  0  \/  N  <  M ) ) )
75, 6syl 14 . . . . . 6  |-  ( N  e.  NN0  ->  ( ( N  =/=  0  ->  N  <  M )  <->  ( -.  N  =/=  0  \/  N  <  M ) ) )
87adantl 271 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  <->  ( -.  N  =/=  0  \/  N  <  M ) ) )
9 nn0z 8452 . . . . . . . . . . . . . 14  |-  ( M  e.  NN0  ->  M  e.  ZZ )
10 zltnle 8478 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  ZZ  /\  M  e.  ZZ )  ->  ( 0  <  M  <->  -.  M  <_  0 ) )
112, 9, 10sylancr 405 . . . . . . . . . . . . 13  |-  ( M  e.  NN0  ->  ( 0  <  M  <->  -.  M  <_  0 ) )
1211adantr 270 . . . . . . . . . . . 12  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 0  <  M  <->  -.  M  <_  0 ) )
13 nn0le0eq0 8383 . . . . . . . . . . . . . 14  |-  ( M  e.  NN0  ->  ( M  <_  0  <->  M  = 
0 ) )
1413notbid 625 . . . . . . . . . . . . 13  |-  ( M  e.  NN0  ->  ( -.  M  <_  0  <->  -.  M  =  0 ) )
1514adantr 270 . . . . . . . . . . . 12  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( -.  M  <_ 
0  <->  -.  M  = 
0 ) )
1612, 15bitrd 186 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 0  <  M  <->  -.  M  =  0 ) )
17 df-ne 2247 . . . . . . . . . . 11  |-  ( M  =/=  0  <->  -.  M  =  0 )
1816, 17syl6bbr 196 . . . . . . . . . 10  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 0  <  M  <->  M  =/=  0 ) )
1918anbi2d 452 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  /\  0  <  M )  <->  ( -.  N  =/=  0  /\  M  =/=  0 ) ) )
201adantl 271 . . . . . . . . . . . . . 14  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  N  e.  ZZ )
2120, 2, 3sylancl 404 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> DECID  N  =  0 )
22 nnedc 2251 . . . . . . . . . . . . 13  |-  (DECID  N  =  0  ->  ( -.  N  =/=  0  <->  N  = 
0 ) )
2321, 22syl 14 . . . . . . . . . . . 12  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( -.  N  =/=  0  <->  N  =  0
) )
24 breq1 3796 . . . . . . . . . . . 12  |-  ( N  =  0  ->  ( N  <  M  <->  0  <  M ) )
2523, 24syl6bi 161 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( -.  N  =/=  0  ->  ( N  <  M  <->  0  <  M
) ) )
26 bi2 128 . . . . . . . . . . 11  |-  ( ( N  <  M  <->  0  <  M )  ->  ( 0  <  M  ->  N  <  M ) )
2725, 26syl6 33 . . . . . . . . . 10  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( -.  N  =/=  0  ->  ( 0  <  M  ->  N  <  M ) ) )
2827impd 251 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  /\  0  <  M )  ->  N  <  M ) )
2919, 28sylbird 168 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  /\  M  =/=  0 )  ->  N  <  M ) )
3029expd 254 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( -.  N  =/=  0  ->  ( M  =/=  0  ->  N  < 
M ) ) )
31 ax-1 5 . . . . . . 7  |-  ( N  <  M  ->  ( M  =/=  0  ->  N  <  M ) )
3230, 31jctir 306 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  ->  ( M  =/=  0  ->  N  <  M ) )  /\  ( N  <  M  -> 
( M  =/=  0  ->  N  <  M ) ) ) )
33 jaob 664 . . . . . 6  |-  ( ( ( -.  N  =/=  0  \/  N  < 
M )  ->  ( M  =/=  0  ->  N  <  M ) )  <->  ( ( -.  N  =/=  0  ->  ( M  =/=  0  ->  N  <  M ) )  /\  ( N  <  M  ->  ( M  =/=  0  ->  N  <  M ) ) ) )
3432, 33sylibr 132 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  \/  N  <  M )  ->  ( M  =/=  0  ->  N  <  M ) ) )
358, 34sylbid 148 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  ->  ( M  =/=  0  ->  N  < 
M ) ) )
36 nn0ge0 8380 . . . . . . . 8  |-  ( N  e.  NN0  ->  0  <_  N )
3736adantl 271 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
0  <_  N )
38 nn0re 8364 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  RR )
39 nn0re 8364 . . . . . . . 8  |-  ( M  e.  NN0  ->  M  e.  RR )
40 0re 7181 . . . . . . . . 9  |-  0  e.  RR
41 lelttr 7266 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  N  e.  RR  /\  M  e.  RR )  ->  (
( 0  <_  N  /\  N  <  M )  ->  0  <  M
) )
4240, 41mp3an1 1256 . . . . . . . 8  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  ( ( 0  <_  N  /\  N  <  M
)  ->  0  <  M ) )
4338, 39, 42syl2anr 284 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( 0  <_  N  /\  N  <  M
)  ->  0  <  M ) )
4437, 43mpand 420 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( N  <  M  ->  0  <  M ) )
4544, 18sylibd 147 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( N  <  M  ->  M  =/=  0 ) )
4645imim2d 53 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  ->  ( N  =/=  0  ->  M  =/=  0 ) ) )
4735, 46jcad 301 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  ->  ( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) ) ) )
48 pm3.34 338 . . 3  |-  ( ( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) )  -> 
( N  =/=  0  ->  N  <  M ) )
4947, 48impbid1 140 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  <->  ( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) ) ) )
50 con34bdc 799 . . . . 5  |-  (DECID  N  =  0  ->  ( ( M  =  0  ->  N  =  0 )  <->  ( -.  N  =  0  ->  -.  M  =  0 ) ) )
5121, 50syl 14 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( M  =  0  ->  N  = 
0 )  <->  ( -.  N  =  0  ->  -.  M  =  0 ) ) )
52 df-ne 2247 . . . . 5  |-  ( N  =/=  0  <->  -.  N  =  0 )
5352, 17imbi12i 237 . . . 4  |-  ( ( N  =/=  0  ->  M  =/=  0 )  <->  ( -.  N  =  0  ->  -.  M  =  0 ) )
5451, 53syl6bbr 196 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( M  =  0  ->  N  = 
0 )  <->  ( N  =/=  0  ->  M  =/=  0 ) ) )
5554anbi2d 452 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( ( M  =/=  0  ->  N  <  M )  /\  ( M  =  0  ->  N  =  0 ) )  <-> 
( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) ) ) )
5649, 55bitr4d 189 1  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  <->  ( ( M  =/=  0  ->  N  <  M )  /\  ( M  =  0  ->  N  =  0 ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662  DECID wdc 776    = wceq 1285    e. wcel 1434    =/= wne 2246   class class class wbr 3793   RRcr 7042   0cc0 7043    < clt 7215    <_ cle 7216   NN0cn0 8355   ZZcz 8432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433
This theorem is referenced by:  algcvgb  10576
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