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Theorem uzaddcl 10909
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 eluzelz 10868 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
21zcnd 10746 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  CC )
3 nn0cn 10587 . . . . . . . 8  |-  ( k  e.  NN0  ->  k  e.  CC )
4 ax-1cn 9338 . . . . . . . . 9  |-  1  e.  CC
5 addass 9367 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  (
( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
64, 5mp3an3 1303 . . . . . . . 8  |-  ( ( N  e.  CC  /\  k  e.  CC )  ->  ( ( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
72, 3, 6syl2anr 478 . . . . . . 7  |-  ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  -> 
( ( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
87adantr 465 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  /\  ( N  +  k
)  e.  ( ZZ>= `  M ) )  -> 
( ( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
9 peano2uz 10906 . . . . . . 7  |-  ( ( N  +  k )  e.  ( ZZ>= `  M
)  ->  ( ( N  +  k )  +  1 )  e.  ( ZZ>= `  M )
)
109adantl 466 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  /\  ( N  +  k
)  e.  ( ZZ>= `  M ) )  -> 
( ( N  +  k )  +  1 )  e.  ( ZZ>= `  M ) )
118, 10eqeltrrd 2516 . . . . 5  |-  ( ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  /\  ( N  +  k
)  e.  ( ZZ>= `  M ) )  -> 
( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M ) )
1211exp31 604 . . . 4  |-  ( k  e.  NN0  ->  ( N  e.  ( ZZ>= `  M
)  ->  ( ( N  +  k )  e.  ( ZZ>= `  M )  ->  ( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M ) ) ) )
1312a2d 26 . . 3  |-  ( k  e.  NN0  ->  ( ( N  e.  ( ZZ>= `  M )  ->  ( N  +  k )  e.  ( ZZ>= `  M )
)  ->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M ) ) ) )
142addid1d 9567 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  0 )  =  N )
1514eleq1d 2507 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( N  +  0 )  e.  ( ZZ>= `  M
)  <->  N  e.  ( ZZ>=
`  M ) ) )
1615ibir 242 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  0 )  e.  ( ZZ>= `  M )
)
17 oveq2 6097 . . . . 5  |-  ( j  =  0  ->  ( N  +  j )  =  ( N  + 
0 ) )
1817eleq1d 2507 . . . 4  |-  ( j  =  0  ->  (
( N  +  j )  e.  ( ZZ>= `  M )  <->  ( N  +  0 )  e.  ( ZZ>= `  M )
) )
1918imbi2d 316 . . 3  |-  ( j  =  0  ->  (
( N  e.  (
ZZ>= `  M )  -> 
( N  +  j )  e.  ( ZZ>= `  M ) )  <->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  0 )  e.  ( ZZ>= `  M ) ) ) )
20 oveq2 6097 . . . . 5  |-  ( j  =  k  ->  ( N  +  j )  =  ( N  +  k ) )
2120eleq1d 2507 . . . 4  |-  ( j  =  k  ->  (
( N  +  j )  e.  ( ZZ>= `  M )  <->  ( N  +  k )  e.  ( ZZ>= `  M )
) )
2221imbi2d 316 . . 3  |-  ( j  =  k  ->  (
( N  e.  (
ZZ>= `  M )  -> 
( N  +  j )  e.  ( ZZ>= `  M ) )  <->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  k )  e.  ( ZZ>= `  M ) ) ) )
23 oveq2 6097 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( N  +  j )  =  ( N  +  ( k  +  1 ) ) )
2423eleq1d 2507 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( N  +  j )  e.  ( ZZ>= `  M )  <->  ( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M )
) )
2524imbi2d 316 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( N  e.  (
ZZ>= `  M )  -> 
( N  +  j )  e.  ( ZZ>= `  M ) )  <->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M ) ) ) )
26 oveq2 6097 . . . . 5  |-  ( j  =  K  ->  ( N  +  j )  =  ( N  +  K ) )
2726eleq1d 2507 . . . 4  |-  ( j  =  K  ->  (
( N  +  j )  e.  ( ZZ>= `  M )  <->  ( N  +  K )  e.  (
ZZ>= `  M ) ) )
2827imbi2d 316 . . 3  |-  ( j  =  K  ->  (
( N  e.  (
ZZ>= `  M )  -> 
( N  +  j )  e.  ( ZZ>= `  M ) )  <->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  K
)  e.  ( ZZ>= `  M ) ) ) )
2913, 16, 19, 22, 25, 28nn0indALT 10737 . 2  |-  ( K  e.  NN0  ->  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  K )  e.  (
ZZ>= `  M ) ) )
3029impcom 430 1  |-  ( ( N  e.  ( ZZ>= `  M )  /\  K  e.  NN0 )  ->  ( N  +  K )  e.  ( ZZ>= `  M )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5416  (class class class)co 6089   CCcc 9278   0cc0 9280   1c1 9281    + caddc 9283   NN0cn0 10577   ZZ>=cuz 10859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860
This theorem is referenced by:  ccatass  12284  swrdccat1  12349  swrdccat2  12350  splfv1  12395  splval2  12397  revccat  12404  isercoll2  13144  iseraltlem2  13158  iseraltlem3  13159  mertenslem1  13342  eftlub  13391  vdwlem6  14045  gsumccat  15517  efginvrel2  16222  efgredleme  16238  efgcpbllemb  16250  geolim3  21803  jm2.27c  29353  zpnn0elfzo  30218
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