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Theorem zaddcllempos 8469
Description: Lemma for zaddcl 8472. Special case in which  N is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
Assertion
Ref Expression
zaddcllempos  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  +  N
)  e.  ZZ )

Proof of Theorem zaddcllempos
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5551 . . . . 5  |-  ( x  =  1  ->  ( M  +  x )  =  ( M  + 
1 ) )
21eleq1d 2148 . . . 4  |-  ( x  =  1  ->  (
( M  +  x
)  e.  ZZ  <->  ( M  +  1 )  e.  ZZ ) )
32imbi2d 228 . . 3  |-  ( x  =  1  ->  (
( M  e.  ZZ  ->  ( M  +  x
)  e.  ZZ )  <-> 
( M  e.  ZZ  ->  ( M  +  1 )  e.  ZZ ) ) )
4 oveq2 5551 . . . . 5  |-  ( x  =  y  ->  ( M  +  x )  =  ( M  +  y ) )
54eleq1d 2148 . . . 4  |-  ( x  =  y  ->  (
( M  +  x
)  e.  ZZ  <->  ( M  +  y )  e.  ZZ ) )
65imbi2d 228 . . 3  |-  ( x  =  y  ->  (
( M  e.  ZZ  ->  ( M  +  x
)  e.  ZZ )  <-> 
( M  e.  ZZ  ->  ( M  +  y )  e.  ZZ ) ) )
7 oveq2 5551 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( M  +  x )  =  ( M  +  ( y  +  1 ) ) )
87eleq1d 2148 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
( M  +  x
)  e.  ZZ  <->  ( M  +  ( y  +  1 ) )  e.  ZZ ) )
98imbi2d 228 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( M  e.  ZZ  ->  ( M  +  x
)  e.  ZZ )  <-> 
( M  e.  ZZ  ->  ( M  +  ( y  +  1 ) )  e.  ZZ ) ) )
10 oveq2 5551 . . . . 5  |-  ( x  =  N  ->  ( M  +  x )  =  ( M  +  N ) )
1110eleq1d 2148 . . . 4  |-  ( x  =  N  ->  (
( M  +  x
)  e.  ZZ  <->  ( M  +  N )  e.  ZZ ) )
1211imbi2d 228 . . 3  |-  ( x  =  N  ->  (
( M  e.  ZZ  ->  ( M  +  x
)  e.  ZZ )  <-> 
( M  e.  ZZ  ->  ( M  +  N
)  e.  ZZ ) ) )
13 peano2z 8468 . . 3  |-  ( M  e.  ZZ  ->  ( M  +  1 )  e.  ZZ )
14 peano2z 8468 . . . . . 6  |-  ( ( M  +  y )  e.  ZZ  ->  (
( M  +  y )  +  1 )  e.  ZZ )
15 zcn 8437 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  M  e.  CC )
1615adantl 271 . . . . . . . 8  |-  ( ( y  e.  NN  /\  M  e.  ZZ )  ->  M  e.  CC )
17 nncn 8114 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  e.  CC )
1817adantr 270 . . . . . . . 8  |-  ( ( y  e.  NN  /\  M  e.  ZZ )  ->  y  e.  CC )
19 1cnd 7197 . . . . . . . 8  |-  ( ( y  e.  NN  /\  M  e.  ZZ )  ->  1  e.  CC )
2016, 18, 19addassd 7203 . . . . . . 7  |-  ( ( y  e.  NN  /\  M  e.  ZZ )  ->  ( ( M  +  y )  +  1 )  =  ( M  +  ( y  +  1 ) ) )
2120eleq1d 2148 . . . . . 6  |-  ( ( y  e.  NN  /\  M  e.  ZZ )  ->  ( ( ( M  +  y )  +  1 )  e.  ZZ  <->  ( M  +  ( y  +  1 ) )  e.  ZZ ) )
2214, 21syl5ib 152 . . . . 5  |-  ( ( y  e.  NN  /\  M  e.  ZZ )  ->  ( ( M  +  y )  e.  ZZ  ->  ( M  +  ( y  +  1 ) )  e.  ZZ ) )
2322ex 113 . . . 4  |-  ( y  e.  NN  ->  ( M  e.  ZZ  ->  ( ( M  +  y )  e.  ZZ  ->  ( M  +  ( y  +  1 ) )  e.  ZZ ) ) )
2423a2d 26 . . 3  |-  ( y  e.  NN  ->  (
( M  e.  ZZ  ->  ( M  +  y )  e.  ZZ )  ->  ( M  e.  ZZ  ->  ( M  +  ( y  +  1 ) )  e.  ZZ ) ) )
253, 6, 9, 12, 13, 24nnind 8122 . 2  |-  ( N  e.  NN  ->  ( M  e.  ZZ  ->  ( M  +  N )  e.  ZZ ) )
2625impcom 123 1  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  +  N
)  e.  ZZ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434  (class class class)co 5543   CCcc 7041   1c1 7044    + caddc 7046   NNcn 8106   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-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-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-0id 7146  ax-rnegex 7147  ax-cnre 7149
This theorem depends on definitions:  df-bi 115  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-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-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433
This theorem is referenced by:  zaddcl  8472
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