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Theorem uzaddcl 9231
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 5714 . . . . 5  |-  ( j  =  0  ->  ( N  +  j )  =  ( N  + 
0 ) )
21eleq1d 2168 . . . 4  |-  ( j  =  0  ->  (
( N  +  j )  e.  ( ZZ>= `  M )  <->  ( N  +  0 )  e.  ( ZZ>= `  M )
) )
32imbi2d 229 . . 3  |-  ( j  =  0  ->  (
( N  e.  (
ZZ>= `  M )  -> 
( N  +  j )  e.  ( ZZ>= `  M ) )  <->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  0 )  e.  ( ZZ>= `  M ) ) ) )
4 oveq2 5714 . . . . 5  |-  ( j  =  k  ->  ( N  +  j )  =  ( N  +  k ) )
54eleq1d 2168 . . . 4  |-  ( j  =  k  ->  (
( N  +  j )  e.  ( ZZ>= `  M )  <->  ( N  +  k )  e.  ( ZZ>= `  M )
) )
65imbi2d 229 . . 3  |-  ( j  =  k  ->  (
( N  e.  (
ZZ>= `  M )  -> 
( N  +  j )  e.  ( ZZ>= `  M ) )  <->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  k )  e.  ( ZZ>= `  M ) ) ) )
7 oveq2 5714 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( N  +  j )  =  ( N  +  ( k  +  1 ) ) )
87eleq1d 2168 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( N  +  j )  e.  ( ZZ>= `  M )  <->  ( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M )
) )
98imbi2d 229 . . 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 5714 . . . . 5  |-  ( j  =  K  ->  ( N  +  j )  =  ( N  +  K ) )
1110eleq1d 2168 . . . 4  |-  ( j  =  K  ->  (
( N  +  j )  e.  ( ZZ>= `  M )  <->  ( N  +  K )  e.  (
ZZ>= `  M ) ) )
1211imbi2d 229 . . 3  |-  ( j  =  K  ->  (
( N  e.  (
ZZ>= `  M )  -> 
( N  +  j )  e.  ( ZZ>= `  M ) )  <->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  K
)  e.  ( ZZ>= `  M ) ) ) )
13 eluzelcn 9187 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  CC )
1413addid1d 7782 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  0 )  =  N )
1514eleq1d 2168 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( N  +  0 )  e.  ( ZZ>= `  M
)  <->  N  e.  ( ZZ>=
`  M ) ) )
1615ibir 176 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  0 )  e.  ( ZZ>= `  M )
)
17 nn0cn 8839 . . . . . . . 8  |-  ( k  e.  NN0  ->  k  e.  CC )
18 ax-1cn 7588 . . . . . . . . 9  |-  1  e.  CC
19 addass 7622 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  (
( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
2018, 19mp3an3 1272 . . . . . . . 8  |-  ( ( N  e.  CC  /\  k  e.  CC )  ->  ( ( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
2113, 17, 20syl2anr 286 . . . . . . 7  |-  ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  -> 
( ( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
2221adantr 272 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  /\  ( N  +  k
)  e.  ( ZZ>= `  M ) )  -> 
( ( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
23 peano2uz 9228 . . . . . . 7  |-  ( ( N  +  k )  e.  ( ZZ>= `  M
)  ->  ( ( N  +  k )  +  1 )  e.  ( ZZ>= `  M )
)
2423adantl 273 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  /\  ( N  +  k
)  e.  ( ZZ>= `  M ) )  -> 
( ( N  +  k )  +  1 )  e.  ( ZZ>= `  M ) )
2522, 24eqeltrrd 2177 . . . . 5  |-  ( ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  /\  ( N  +  k
)  e.  ( ZZ>= `  M ) )  -> 
( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M ) )
2625exp31 359 . . . 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 9017 . 2  |-  ( K  e.  NN0  ->  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  K )  e.  (
ZZ>= `  M ) ) )
2928impcom 124 1  |-  ( ( N  e.  ( ZZ>= `  M )  /\  K  e.  NN0 )  ->  ( N  +  K )  e.  ( ZZ>= `  M )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1299    e. wcel 1448   ` cfv 5059  (class class class)co 5706   CCcc 7498   0cc0 7500   1c1 7501    + caddc 7503   NN0cn0 8829   ZZ>=cuz 9176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-ltadd 7611
This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-n0 8830  df-z 8907  df-uz 9177
This theorem is referenced by:  elfz0add  9741  zpnn0elfzo  9825  mertenslemi1  11143  eftlub  11194
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