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Theorem algcvgblem 13023
Description: Lemma for algcvgb 13024. (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 imor 402 . . . . 5  |-  ( ( N  =/=  0  ->  N  <  M )  <->  ( -.  N  =/=  0  \/  N  <  M ) )
2 0re 9047 . . . . . . . . . . . . 13  |-  0  e.  RR
3 nn0re 10186 . . . . . . . . . . . . . 14  |-  ( M  e.  NN0  ->  M  e.  RR )
43adantr 452 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  M  e.  RR )
5 ltnle 9111 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  M  e.  RR )  ->  ( 0  <  M  <->  -.  M  <_  0 ) )
62, 4, 5sylancr 645 . . . . . . . . . . . 12  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 0  <  M  <->  -.  M  <_  0 ) )
7 nn0le0eq0 10206 . . . . . . . . . . . . . 14  |-  ( M  e.  NN0  ->  ( M  <_  0  <->  M  = 
0 ) )
87notbid 286 . . . . . . . . . . . . 13  |-  ( M  e.  NN0  ->  ( -.  M  <_  0  <->  -.  M  =  0 ) )
98adantr 452 . . . . . . . . . . . 12  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( -.  M  <_ 
0  <->  -.  M  = 
0 ) )
106, 9bitrd 245 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 0  <  M  <->  -.  M  =  0 ) )
11 df-ne 2569 . . . . . . . . . . 11  |-  ( M  =/=  0  <->  -.  M  =  0 )
1210, 11syl6bbr 255 . . . . . . . . . 10  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 0  <  M  <->  M  =/=  0 ) )
1312anbi2d 685 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  /\  0  <  M )  <->  ( -.  N  =/=  0  /\  M  =/=  0 ) ) )
14 nne 2571 . . . . . . . . . . 11  |-  ( -.  N  =/=  0  <->  N  =  0 )
15 breq1 4175 . . . . . . . . . . 11  |-  ( N  =  0  ->  ( N  <  M  <->  0  <  M ) )
1614, 15sylbi 188 . . . . . . . . . 10  |-  ( -.  N  =/=  0  -> 
( N  <  M  <->  0  <  M ) )
1716biimpar 472 . . . . . . . . 9  |-  ( ( -.  N  =/=  0  /\  0  <  M )  ->  N  <  M
)
1813, 17syl6bir 221 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  /\  M  =/=  0 )  ->  N  <  M ) )
1918exp3a 426 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( -.  N  =/=  0  ->  ( M  =/=  0  ->  N  < 
M ) ) )
20 ax-1 5 . . . . . . 7  |-  ( N  <  M  ->  ( M  =/=  0  ->  N  <  M ) )
2119, 20jctir 525 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  ->  ( M  =/=  0  ->  N  <  M ) )  /\  ( N  <  M  -> 
( M  =/=  0  ->  N  <  M ) ) ) )
22 jaob 759 . . . . . 6  |-  ( ( ( -.  N  =/=  0  \/  N  < 
M )  ->  ( M  =/=  0  ->  N  <  M ) )  <->  ( ( -.  N  =/=  0  ->  ( M  =/=  0  ->  N  <  M ) )  /\  ( N  <  M  ->  ( M  =/=  0  ->  N  <  M ) ) ) )
2321, 22sylibr 204 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  \/  N  <  M )  ->  ( M  =/=  0  ->  N  <  M ) ) )
241, 23syl5bi 209 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  ->  ( M  =/=  0  ->  N  < 
M ) ) )
25 nn0ge0 10203 . . . . . . . 8  |-  ( N  e.  NN0  ->  0  <_  N )
2625adantl 453 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
0  <_  N )
27 nn0re 10186 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  RR )
28 lelttr 9121 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  N  e.  RR  /\  M  e.  RR )  ->  (
( 0  <_  N  /\  N  <  M )  ->  0  <  M
) )
292, 28mp3an1 1266 . . . . . . . 8  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  ( ( 0  <_  N  /\  N  <  M
)  ->  0  <  M ) )
3027, 3, 29syl2anr 465 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( 0  <_  N  /\  N  <  M
)  ->  0  <  M ) )
3126, 30mpand 657 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( N  <  M  ->  0  <  M ) )
3231, 12sylibd 206 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( N  <  M  ->  M  =/=  0 ) )
3332imim2d 50 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  ->  ( N  =/=  0  ->  M  =/=  0 ) ) )
3424, 33jcad 520 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  ->  ( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) ) ) )
35 pm3.34 570 . . 3  |-  ( ( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) )  -> 
( N  =/=  0  ->  N  <  M ) )
3634, 35impbid1 195 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  <->  ( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) ) ) )
37 con34b 284 . . . 4  |-  ( ( M  =  0  ->  N  =  0 )  <-> 
( -.  N  =  0  ->  -.  M  =  0 ) )
38 df-ne 2569 . . . . 5  |-  ( N  =/=  0  <->  -.  N  =  0 )
3938, 11imbi12i 317 . . . 4  |-  ( ( N  =/=  0  ->  M  =/=  0 )  <->  ( -.  N  =  0  ->  -.  M  =  0 ) )
4037, 39bitr4i 244 . . 3  |-  ( ( M  =  0  ->  N  =  0 )  <-> 
( N  =/=  0  ->  M  =/=  0 ) )
4140anbi2i 676 . 2  |-  ( ( ( M  =/=  0  ->  N  <  M )  /\  ( M  =  0  ->  N  = 
0 ) )  <->  ( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) ) )
4236, 41syl6bbr 255 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    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   RRcr 8945   0cc0 8946    < clt 9076    <_ cle 9077   NN0cn0 10177
This theorem is referenced by:  algcvgb  13024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178
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