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Theorem algcvgblem 12068
Description: Lemma for algcvgb 12069. (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 9292 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  e.  ZZ )
2 0z 9283 . . . . . . . . 9  |-  0  e.  ZZ
3 zdceq 9347 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
41, 2, 3sylancl 413 . . . . . . . 8  |-  ( N  e.  NN0  -> DECID  N  =  0
)
54dcned 2366 . . . . . . 7  |-  ( N  e.  NN0  -> DECID  N  =/=  0
)
6 imordc 898 . . . . . . 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 277 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  <->  ( -.  N  =/=  0  \/  N  <  M ) ) )
9 nn0z 9292 . . . . . . . . . . . . . 14  |-  ( M  e.  NN0  ->  M  e.  ZZ )
10 zltnle 9318 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  ZZ  /\  M  e.  ZZ )  ->  ( 0  <  M  <->  -.  M  <_  0 ) )
112, 9, 10sylancr 414 . . . . . . . . . . . . 13  |-  ( M  e.  NN0  ->  ( 0  <  M  <->  -.  M  <_  0 ) )
1211adantr 276 . . . . . . . . . . . 12  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 0  <  M  <->  -.  M  <_  0 ) )
13 nn0le0eq0 9223 . . . . . . . . . . . . . 14  |-  ( M  e.  NN0  ->  ( M  <_  0  <->  M  = 
0 ) )
1413notbid 668 . . . . . . . . . . . . 13  |-  ( M  e.  NN0  ->  ( -.  M  <_  0  <->  -.  M  =  0 ) )
1514adantr 276 . . . . . . . . . . . 12  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( -.  M  <_ 
0  <->  -.  M  = 
0 ) )
1612, 15bitrd 188 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 0  <  M  <->  -.  M  =  0 ) )
17 df-ne 2361 . . . . . . . . . . 11  |-  ( M  =/=  0  <->  -.  M  =  0 )
1816, 17bitr4di 198 . . . . . . . . . 10  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 0  <  M  <->  M  =/=  0 ) )
1918anbi2d 464 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  /\  0  <  M )  <->  ( -.  N  =/=  0  /\  M  =/=  0 ) ) )
201adantl 277 . . . . . . . . . . . . . 14  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  N  e.  ZZ )
2120, 2, 3sylancl 413 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> DECID  N  =  0 )
22 nnedc 2365 . . . . . . . . . . . . 13  |-  (DECID  N  =  0  ->  ( -.  N  =/=  0  <->  N  = 
0 ) )
2321, 22syl 14 . . . . . . . . . . . 12  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( -.  N  =/=  0  <->  N  =  0
) )
24 breq1 4021 . . . . . . . . . . . 12  |-  ( N  =  0  ->  ( N  <  M  <->  0  <  M ) )
2523, 24biimtrdi 163 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( -.  N  =/=  0  ->  ( N  <  M  <->  0  <  M
) ) )
26 biimpr 130 . . . . . . . . . . 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 254 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  /\  0  <  M )  ->  N  <  M ) )
2919, 28sylbird 170 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  /\  M  =/=  0 )  ->  N  <  M ) )
3029expd 258 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( -.  N  =/=  0  ->  ( M  =/=  0  ->  N  < 
M ) ) )
31 ax-1 6 . . . . . . 7  |-  ( N  <  M  ->  ( M  =/=  0  ->  N  <  M ) )
3230, 31jctir 313 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  ->  ( M  =/=  0  ->  N  <  M ) )  /\  ( N  <  M  -> 
( M  =/=  0  ->  N  <  M ) ) ) )
33 jaob 711 . . . . . 6  |-  ( ( ( -.  N  =/=  0  \/  N  < 
M )  ->  ( M  =/=  0  ->  N  <  M ) )  <->  ( ( -.  N  =/=  0  ->  ( M  =/=  0  ->  N  <  M ) )  /\  ( N  <  M  ->  ( M  =/=  0  ->  N  <  M ) ) ) )
3432, 33sylibr 134 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  \/  N  <  M )  ->  ( M  =/=  0  ->  N  <  M ) ) )
358, 34sylbid 150 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  ->  ( M  =/=  0  ->  N  < 
M ) ) )
36 nn0ge0 9220 . . . . . . . 8  |-  ( N  e.  NN0  ->  0  <_  N )
3736adantl 277 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
0  <_  N )
38 nn0re 9204 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  RR )
39 nn0re 9204 . . . . . . . 8  |-  ( M  e.  NN0  ->  M  e.  RR )
40 0re 7976 . . . . . . . . 9  |-  0  e.  RR
41 lelttr 8065 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  N  e.  RR  /\  M  e.  RR )  ->  (
( 0  <_  N  /\  N  <  M )  ->  0  <  M
) )
4240, 41mp3an1 1335 . . . . . . . 8  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  ( ( 0  <_  N  /\  N  <  M
)  ->  0  <  M ) )
4338, 39, 42syl2anr 290 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( 0  <_  N  /\  N  <  M
)  ->  0  <  M ) )
4437, 43mpand 429 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( N  <  M  ->  0  <  M ) )
4544, 18sylibd 149 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( N  <  M  ->  M  =/=  0 ) )
4645imim2d 54 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  ->  ( N  =/=  0  ->  M  =/=  0 ) ) )
4735, 46jcad 307 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  ->  ( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) ) ) )
48 pm3.34 346 . . 3  |-  ( ( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) )  -> 
( N  =/=  0  ->  N  <  M ) )
4947, 48impbid1 142 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  <->  ( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) ) ) )
50 con34bdc 872 . . . . 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 2361 . . . . 5  |-  ( N  =/=  0  <->  -.  N  =  0 )
5352, 17imbi12i 239 . . . 4  |-  ( ( N  =/=  0  ->  M  =/=  0 )  <->  ( -.  N  =  0  ->  -.  M  =  0 ) )
5451, 53bitr4di 198 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( M  =  0  ->  N  = 
0 )  <->  ( N  =/=  0  ->  M  =/=  0 ) ) )
5554anbi2d 464 . 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 191 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 104    <-> wb 105    \/ wo 709  DECID wdc 835    = wceq 1364    e. wcel 2160    =/= wne 2360   class class class wbr 4018   RRcr 7829   0cc0 7830    < clt 8011    <_ cle 8012   NN0cn0 9195   ZZcz 9272
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 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-addcom 7930  ax-addass 7932  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-0id 7938  ax-rnegex 7939  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946
This theorem depends on definitions:  df-bi 117  df-stab 832  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-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  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 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5233  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-inn 8939  df-n0 9196  df-z 9273
This theorem is referenced by:  algcvgb  12069
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