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Theorem algcvgblem 14290
Description: Lemma for algcvgb 14291. (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 410 . . . . 5  |-  ( ( N  =/=  0  ->  N  <  M )  <->  ( -.  N  =/=  0  \/  N  <  M ) )
2 0re 9585 . . . . . . . . . . . . 13  |-  0  e.  RR
3 nn0re 10800 . . . . . . . . . . . . . 14  |-  ( M  e.  NN0  ->  M  e.  RR )
43adantr 463 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  M  e.  RR )
5 ltnle 9653 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  M  e.  RR )  ->  ( 0  <  M  <->  -.  M  <_  0 ) )
62, 4, 5sylancr 661 . . . . . . . . . . . 12  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 0  <  M  <->  -.  M  <_  0 ) )
7 nn0le0eq0 10820 . . . . . . . . . . . . . 14  |-  ( M  e.  NN0  ->  ( M  <_  0  <->  M  = 
0 ) )
87notbid 292 . . . . . . . . . . . . 13  |-  ( M  e.  NN0  ->  ( -.  M  <_  0  <->  -.  M  =  0 ) )
98adantr 463 . . . . . . . . . . . 12  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( -.  M  <_ 
0  <->  -.  M  = 
0 ) )
106, 9bitrd 253 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 0  <  M  <->  -.  M  =  0 ) )
11 df-ne 2651 . . . . . . . . . . 11  |-  ( M  =/=  0  <->  -.  M  =  0 )
1210, 11syl6bbr 263 . . . . . . . . . 10  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( 0  <  M  <->  M  =/=  0 ) )
1312anbi2d 701 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  /\  0  <  M )  <->  ( -.  N  =/=  0  /\  M  =/=  0 ) ) )
14 nne 2655 . . . . . . . . . . 11  |-  ( -.  N  =/=  0  <->  N  =  0 )
15 breq1 4442 . . . . . . . . . . 11  |-  ( N  =  0  ->  ( N  <  M  <->  0  <  M ) )
1614, 15sylbi 195 . . . . . . . . . 10  |-  ( -.  N  =/=  0  -> 
( N  <  M  <->  0  <  M ) )
1716biimpar 483 . . . . . . . . 9  |-  ( ( -.  N  =/=  0  /\  0  <  M )  ->  N  <  M
)
1813, 17syl6bir 229 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  /\  M  =/=  0 )  ->  N  <  M ) )
1918expd 434 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( -.  N  =/=  0  ->  ( M  =/=  0  ->  N  < 
M ) ) )
20 ax-1 6 . . . . . . 7  |-  ( N  <  M  ->  ( M  =/=  0  ->  N  <  M ) )
2119, 20jctir 536 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  ->  ( M  =/=  0  ->  N  <  M ) )  /\  ( N  <  M  -> 
( M  =/=  0  ->  N  <  M ) ) ) )
22 jaob 781 . . . . . 6  |-  ( ( ( -.  N  =/=  0  \/  N  < 
M )  ->  ( M  =/=  0  ->  N  <  M ) )  <->  ( ( -.  N  =/=  0  ->  ( M  =/=  0  ->  N  <  M ) )  /\  ( N  <  M  ->  ( M  =/=  0  ->  N  <  M ) ) ) )
2321, 22sylibr 212 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( -.  N  =/=  0  \/  N  <  M )  ->  ( M  =/=  0  ->  N  <  M ) ) )
241, 23syl5bi 217 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  ->  ( M  =/=  0  ->  N  < 
M ) ) )
25 nn0ge0 10817 . . . . . . . 8  |-  ( N  e.  NN0  ->  0  <_  N )
2625adantl 464 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
0  <_  N )
27 nn0re 10800 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  RR )
28 lelttr 9664 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  N  e.  RR  /\  M  e.  RR )  ->  (
( 0  <_  N  /\  N  <  M )  ->  0  <  M
) )
292, 28mp3an1 1309 . . . . . . . 8  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  ( ( 0  <_  N  /\  N  <  M
)  ->  0  <  M ) )
3027, 3, 29syl2anr 476 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( 0  <_  N  /\  N  <  M
)  ->  0  <  M ) )
3126, 30mpand 673 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( N  <  M  ->  0  <  M ) )
3231, 12sylibd 214 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( N  <  M  ->  M  =/=  0 ) )
3332imim2d 52 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  ->  ( N  =/=  0  ->  M  =/=  0 ) ) )
3424, 33jcad 531 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  ->  ( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) ) ) )
35 pm3.34 584 . . 3  |-  ( ( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) )  -> 
( N  =/=  0  ->  N  <  M ) )
3634, 35impbid1 203 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  <->  ( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) ) ) )
37 con34b 290 . . . 4  |-  ( ( M  =  0  ->  N  =  0 )  <-> 
( -.  N  =  0  ->  -.  M  =  0 ) )
38 df-ne 2651 . . . . 5  |-  ( N  =/=  0  <->  -.  N  =  0 )
3938, 11imbi12i 324 . . . 4  |-  ( ( N  =/=  0  ->  M  =/=  0 )  <->  ( -.  N  =  0  ->  -.  M  =  0 ) )
4037, 39bitr4i 252 . . 3  |-  ( ( M  =  0  ->  N  =  0 )  <-> 
( N  =/=  0  ->  M  =/=  0 ) )
4140anbi2i 692 . 2  |-  ( ( ( M  =/=  0  ->  N  <  M )  /\  ( M  =  0  ->  N  = 
0 ) )  <->  ( ( M  =/=  0  ->  N  <  M )  /\  ( N  =/=  0  ->  M  =/=  0 ) ) )
4236, 41syl6bbr 263 1  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  =/=  0  ->  N  <  M )  <->  ( ( M  =/=  0  ->  N  <  M )  /\  ( M  =  0  ->  N  =  0 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   RRcr 9480   0cc0 9481    < clt 9617    <_ cle 9618   NN0cn0 10791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792
This theorem is referenced by:  algcvgb  14291
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