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Theorem algcvgblem 11657
Description: Lemma for algcvgb 11658. (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 9042 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  e.  ZZ )
2 0z 9033 . . . . . . . . 9  |-  0  e.  ZZ
3 zdceq 9094 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
41, 2, 3sylancl 409 . . . . . . . 8  |-  ( N  e.  NN0  -> DECID  N  =  0
)
54dcned 2291 . . . . . . 7  |-  ( N  e.  NN0  -> DECID  N  =/=  0
)
6 imordc 867 . . . . . . 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 275 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  <->  ( -.  N  =/=  0  \/  N  <  M ) ) )
9 nn0z 9042 . . . . . . . . . . . . . 14  |-  ( M  e.  NN0  ->  M  e.  ZZ )
10 zltnle 9068 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  ZZ  /\  M  e.  ZZ )  ->  ( 0  <  M  <->  -.  M  <_  0 ) )
112, 9, 10sylancr 410 . . . . . . . . . . . . 13  |-  ( M  e.  NN0  ->  ( 0  <  M  <->  -.  M  <_  0 ) )
1211adantr 274 . . . . . . . . . . . 12  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 0  <  M  <->  -.  M  <_  0 ) )
13 nn0le0eq0 8973 . . . . . . . . . . . . . 14  |-  ( M  e.  NN0  ->  ( M  <_  0  <->  M  = 
0 ) )
1413notbid 641 . . . . . . . . . . . . 13  |-  ( M  e.  NN0  ->  ( -.  M  <_  0  <->  -.  M  =  0 ) )
1514adantr 274 . . . . . . . . . . . 12  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( -.  M  <_ 
0  <->  -.  M  = 
0 ) )
1612, 15bitrd 187 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 0  <  M  <->  -.  M  =  0 ) )
17 df-ne 2286 . . . . . . . . . . 11  |-  ( M  =/=  0  <->  -.  M  =  0 )
1816, 17syl6bbr 197 . . . . . . . . . 10  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 0  <  M  <->  M  =/=  0 ) )
1918anbi2d 459 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  /\  0  <  M )  <->  ( -.  N  =/=  0  /\  M  =/=  0 ) ) )
201adantl 275 . . . . . . . . . . . . . 14  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  N  e.  ZZ )
2120, 2, 3sylancl 409 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> DECID  N  =  0 )
22 nnedc 2290 . . . . . . . . . . . . 13  |-  (DECID  N  =  0  ->  ( -.  N  =/=  0  <->  N  = 
0 ) )
2321, 22syl 14 . . . . . . . . . . . 12  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( -.  N  =/=  0  <->  N  =  0
) )
24 breq1 3902 . . . . . . . . . . . 12  |-  ( N  =  0  ->  ( N  <  M  <->  0  <  M ) )
2523, 24syl6bi 162 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( -.  N  =/=  0  ->  ( N  <  M  <->  0  <  M
) ) )
26 bi2 129 . . . . . . . . . . 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 252 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  /\  0  <  M )  ->  N  <  M ) )
2919, 28sylbird 169 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  /\  M  =/=  0 )  ->  N  <  M ) )
3029expd 256 . . . . . . 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 311 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  ->  ( M  =/=  0  ->  N  <  M ) )  /\  ( N  <  M  -> 
( M  =/=  0  ->  N  <  M ) ) ) )
33 jaob 684 . . . . . 6  |-  ( ( ( -.  N  =/=  0  \/  N  < 
M )  ->  ( M  =/=  0  ->  N  <  M ) )  <->  ( ( -.  N  =/=  0  ->  ( M  =/=  0  ->  N  <  M ) )  /\  ( N  <  M  ->  ( M  =/=  0  ->  N  <  M ) ) ) )
3432, 33sylibr 133 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  \/  N  <  M )  ->  ( M  =/=  0  ->  N  <  M ) ) )
358, 34sylbid 149 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  ->  ( M  =/=  0  ->  N  < 
M ) ) )
36 nn0ge0 8970 . . . . . . . 8  |-  ( N  e.  NN0  ->  0  <_  N )
3736adantl 275 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
0  <_  N )
38 nn0re 8954 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  RR )
39 nn0re 8954 . . . . . . . 8  |-  ( M  e.  NN0  ->  M  e.  RR )
40 0re 7734 . . . . . . . . 9  |-  0  e.  RR
41 lelttr 7820 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  N  e.  RR  /\  M  e.  RR )  ->  (
( 0  <_  N  /\  N  <  M )  ->  0  <  M
) )
4240, 41mp3an1 1287 . . . . . . . 8  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  ( ( 0  <_  N  /\  N  <  M
)  ->  0  <  M ) )
4338, 39, 42syl2anr 288 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( 0  <_  N  /\  N  <  M
)  ->  0  <  M ) )
4437, 43mpand 425 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( N  <  M  ->  0  <  M ) )
4544, 18sylibd 148 . . . . 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 305 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  ->  ( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) ) ) )
48 pm3.34 343 . . 3  |-  ( ( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) )  -> 
( N  =/=  0  ->  N  <  M ) )
4947, 48impbid1 141 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  <->  ( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) ) ) )
50 con34bdc 841 . . . . 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 2286 . . . . 5  |-  ( N  =/=  0  <->  -.  N  =  0 )
5352, 17imbi12i 238 . . . 4  |-  ( ( N  =/=  0  ->  M  =/=  0 )  <->  ( -.  N  =  0  ->  -.  M  =  0 ) )
5451, 53syl6bbr 197 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( M  =  0  ->  N  = 
0 )  <->  ( N  =/=  0  ->  M  =/=  0 ) ) )
5554anbi2d 459 . 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 190 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 103    <-> wb 104    \/ wo 682  DECID wdc 804    = wceq 1316    e. wcel 1465    =/= wne 2285   class class class wbr 3899   RRcr 7587   0cc0 7588    < clt 7768    <_ cle 7769   NN0cn0 8945   ZZcz 9022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-stab 801  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8689  df-n0 8946  df-z 9023
This theorem is referenced by:  algcvgb  11658
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