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Theorem uzaddcl 8755
Description: Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.)
Assertion
Ref Expression
uzaddcl  |-  ( ( N  e.  ( ZZ>= `  M )  /\  K  e.  NN0 )  ->  ( N  +  K )  e.  ( ZZ>= `  M )
)

Proof of Theorem uzaddcl
Dummy variables  j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5551 . . . . 5  |-  ( j  =  0  ->  ( N  +  j )  =  ( N  + 
0 ) )
21eleq1d 2148 . . . 4  |-  ( j  =  0  ->  (
( N  +  j )  e.  ( ZZ>= `  M )  <->  ( N  +  0 )  e.  ( ZZ>= `  M )
) )
32imbi2d 228 . . 3  |-  ( j  =  0  ->  (
( N  e.  (
ZZ>= `  M )  -> 
( N  +  j )  e.  ( ZZ>= `  M ) )  <->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  0 )  e.  ( ZZ>= `  M ) ) ) )
4 oveq2 5551 . . . . 5  |-  ( j  =  k  ->  ( N  +  j )  =  ( N  +  k ) )
54eleq1d 2148 . . . 4  |-  ( j  =  k  ->  (
( N  +  j )  e.  ( ZZ>= `  M )  <->  ( N  +  k )  e.  ( ZZ>= `  M )
) )
65imbi2d 228 . . 3  |-  ( j  =  k  ->  (
( N  e.  (
ZZ>= `  M )  -> 
( N  +  j )  e.  ( ZZ>= `  M ) )  <->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  k )  e.  ( ZZ>= `  M ) ) ) )
7 oveq2 5551 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( N  +  j )  =  ( N  +  ( k  +  1 ) ) )
87eleq1d 2148 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( N  +  j )  e.  ( ZZ>= `  M )  <->  ( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M )
) )
98imbi2d 228 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( N  e.  (
ZZ>= `  M )  -> 
( N  +  j )  e.  ( ZZ>= `  M ) )  <->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M ) ) ) )
10 oveq2 5551 . . . . 5  |-  ( j  =  K  ->  ( N  +  j )  =  ( N  +  K ) )
1110eleq1d 2148 . . . 4  |-  ( j  =  K  ->  (
( N  +  j )  e.  ( ZZ>= `  M )  <->  ( N  +  K )  e.  (
ZZ>= `  M ) ) )
1211imbi2d 228 . . 3  |-  ( j  =  K  ->  (
( N  e.  (
ZZ>= `  M )  -> 
( N  +  j )  e.  ( ZZ>= `  M ) )  <->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  K
)  e.  ( ZZ>= `  M ) ) ) )
13 eluzelcn 8711 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  CC )
1413addid1d 7324 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  0 )  =  N )
1514eleq1d 2148 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( N  +  0 )  e.  ( ZZ>= `  M
)  <->  N  e.  ( ZZ>=
`  M ) ) )
1615ibir 175 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  0 )  e.  ( ZZ>= `  M )
)
17 nn0cn 8365 . . . . . . . 8  |-  ( k  e.  NN0  ->  k  e.  CC )
18 ax-1cn 7131 . . . . . . . . 9  |-  1  e.  CC
19 addass 7165 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  (
( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
2018, 19mp3an3 1258 . . . . . . . 8  |-  ( ( N  e.  CC  /\  k  e.  CC )  ->  ( ( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
2113, 17, 20syl2anr 284 . . . . . . 7  |-  ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  -> 
( ( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
2221adantr 270 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  /\  ( N  +  k
)  e.  ( ZZ>= `  M ) )  -> 
( ( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
23 peano2uz 8752 . . . . . . 7  |-  ( ( N  +  k )  e.  ( ZZ>= `  M
)  ->  ( ( N  +  k )  +  1 )  e.  ( ZZ>= `  M )
)
2423adantl 271 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  /\  ( N  +  k
)  e.  ( ZZ>= `  M ) )  -> 
( ( N  +  k )  +  1 )  e.  ( ZZ>= `  M ) )
2522, 24eqeltrrd 2157 . . . . 5  |-  ( ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  /\  ( N  +  k
)  e.  ( ZZ>= `  M ) )  -> 
( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M ) )
2625exp31 356 . . . 4  |-  ( k  e.  NN0  ->  ( N  e.  ( ZZ>= `  M
)  ->  ( ( N  +  k )  e.  ( ZZ>= `  M )  ->  ( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M ) ) ) )
2726a2d 26 . . 3  |-  ( k  e.  NN0  ->  ( ( N  e.  ( ZZ>= `  M )  ->  ( N  +  k )  e.  ( ZZ>= `  M )
)  ->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M ) ) ) )
283, 6, 9, 12, 16, 27nn0ind 8542 . 2  |-  ( K  e.  NN0  ->  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  K )  e.  (
ZZ>= `  M ) ) )
2928impcom 123 1  |-  ( ( N  e.  ( ZZ>= `  M )  /\  K  e.  NN0 )  ->  ( N  +  K )  e.  ( ZZ>= `  M )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   ` cfv 4932  (class class class)co 5543   CCcc 7041   0cc0 7043   1c1 7044    + caddc 7046   NN0cn0 8355   ZZ>=cuz 8700
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-13 1445  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-un 4196  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-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154
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-nel 2341  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-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701
This theorem is referenced by:  elfz0add  9211  zpnn0elfzo  9293
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