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Theorem uzaddcl 9393
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 5782 . . . . 5  |-  ( j  =  0  ->  ( N  +  j )  =  ( N  + 
0 ) )
21eleq1d 2208 . . . 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 5782 . . . . 5  |-  ( j  =  k  ->  ( N  +  j )  =  ( N  +  k ) )
54eleq1d 2208 . . . 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 5782 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( N  +  j )  =  ( N  +  ( k  +  1 ) ) )
87eleq1d 2208 . . . 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 5782 . . . . 5  |-  ( j  =  K  ->  ( N  +  j )  =  ( N  +  K ) )
1110eleq1d 2208 . . . 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 9349 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  CC )
1413addid1d 7923 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  0 )  =  N )
1514eleq1d 2208 . . . 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 8999 . . . . . . . 8  |-  ( k  e.  NN0  ->  k  e.  CC )
18 ax-1cn 7725 . . . . . . . . 9  |-  1  e.  CC
19 addass 7762 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  (
( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
2018, 19mp3an3 1304 . . . . . . . 8  |-  ( ( N  e.  CC  /\  k  e.  CC )  ->  ( ( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
2113, 17, 20syl2anr 288 . . . . . . 7  |-  ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  -> 
( ( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
2221adantr 274 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  /\  ( N  +  k
)  e.  ( ZZ>= `  M ) )  -> 
( ( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
23 peano2uz 9390 . . . . . . 7  |-  ( ( N  +  k )  e.  ( ZZ>= `  M
)  ->  ( ( N  +  k )  +  1 )  e.  ( ZZ>= `  M )
)
2423adantl 275 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  /\  ( N  +  k
)  e.  ( ZZ>= `  M ) )  -> 
( ( N  +  k )  +  1 )  e.  ( ZZ>= `  M ) )
2522, 24eqeltrrd 2217 . . . . 5  |-  ( ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  /\  ( N  +  k
)  e.  ( ZZ>= `  M ) )  -> 
( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M ) )
2625exp31 361 . . . 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 9177 . 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 1331    e. wcel 1480   ` cfv 5123  (class class class)co 5774   CCcc 7630   0cc0 7632   1c1 7633    + caddc 7635   NN0cn0 8989   ZZ>=cuz 9338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-ltadd 7748
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-n0 8990  df-z 9067  df-uz 9339
This theorem is referenced by:  elfz0add  9912  zpnn0elfzo  9996  mertenslemi1  11316  eftlub  11408
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