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Theorem uzaddcl 11215
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 eluzelcn 11170 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  CC )
2 nn0cn 10879 . . . . . . . 8  |-  ( k  e.  NN0  ->  k  e.  CC )
3 ax-1cn 9596 . . . . . . . . 9  |-  1  e.  CC
4 addass 9625 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  (
( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
53, 4mp3an3 1349 . . . . . . . 8  |-  ( ( N  e.  CC  /\  k  e.  CC )  ->  ( ( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
61, 2, 5syl2anr 480 . . . . . . 7  |-  ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  -> 
( ( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
76adantr 466 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  /\  ( N  +  k
)  e.  ( ZZ>= `  M ) )  -> 
( ( N  +  k )  +  1 )  =  ( N  +  ( k  +  1 ) ) )
8 peano2uz 11212 . . . . . . 7  |-  ( ( N  +  k )  e.  ( ZZ>= `  M
)  ->  ( ( N  +  k )  +  1 )  e.  ( ZZ>= `  M )
)
98adantl 467 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  /\  ( N  +  k
)  e.  ( ZZ>= `  M ) )  -> 
( ( N  +  k )  +  1 )  e.  ( ZZ>= `  M ) )
107, 9eqeltrrd 2518 . . . . 5  |-  ( ( ( k  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  /\  ( N  +  k
)  e.  ( ZZ>= `  M ) )  -> 
( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M ) )
1110exp31 607 . . . 4  |-  ( k  e.  NN0  ->  ( N  e.  ( ZZ>= `  M
)  ->  ( ( N  +  k )  e.  ( ZZ>= `  M )  ->  ( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M ) ) ) )
1211a2d 29 . . 3  |-  ( k  e.  NN0  ->  ( ( N  e.  ( ZZ>= `  M )  ->  ( N  +  k )  e.  ( ZZ>= `  M )
)  ->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M ) ) ) )
131addid1d 9832 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  0 )  =  N )
1413eleq1d 2498 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( N  +  0 )  e.  ( ZZ>= `  M
)  <->  N  e.  ( ZZ>=
`  M ) ) )
1514ibir 245 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  0 )  e.  ( ZZ>= `  M )
)
16 oveq2 6313 . . . . 5  |-  ( j  =  0  ->  ( N  +  j )  =  ( N  + 
0 ) )
1716eleq1d 2498 . . . 4  |-  ( j  =  0  ->  (
( N  +  j )  e.  ( ZZ>= `  M )  <->  ( N  +  0 )  e.  ( ZZ>= `  M )
) )
1817imbi2d 317 . . 3  |-  ( j  =  0  ->  (
( N  e.  (
ZZ>= `  M )  -> 
( N  +  j )  e.  ( ZZ>= `  M ) )  <->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  0 )  e.  ( ZZ>= `  M ) ) ) )
19 oveq2 6313 . . . . 5  |-  ( j  =  k  ->  ( N  +  j )  =  ( N  +  k ) )
2019eleq1d 2498 . . . 4  |-  ( j  =  k  ->  (
( N  +  j )  e.  ( ZZ>= `  M )  <->  ( N  +  k )  e.  ( ZZ>= `  M )
) )
2120imbi2d 317 . . 3  |-  ( j  =  k  ->  (
( N  e.  (
ZZ>= `  M )  -> 
( N  +  j )  e.  ( ZZ>= `  M ) )  <->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  k )  e.  ( ZZ>= `  M ) ) ) )
22 oveq2 6313 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( N  +  j )  =  ( N  +  ( k  +  1 ) ) )
2322eleq1d 2498 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( N  +  j )  e.  ( ZZ>= `  M )  <->  ( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M )
) )
2423imbi2d 317 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( N  e.  (
ZZ>= `  M )  -> 
( N  +  j )  e.  ( ZZ>= `  M ) )  <->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  ( k  +  1 ) )  e.  ( ZZ>= `  M ) ) ) )
25 oveq2 6313 . . . . 5  |-  ( j  =  K  ->  ( N  +  j )  =  ( N  +  K ) )
2625eleq1d 2498 . . . 4  |-  ( j  =  K  ->  (
( N  +  j )  e.  ( ZZ>= `  M )  <->  ( N  +  K )  e.  (
ZZ>= `  M ) ) )
2726imbi2d 317 . . 3  |-  ( j  =  K  ->  (
( N  e.  (
ZZ>= `  M )  -> 
( N  +  j )  e.  ( ZZ>= `  M ) )  <->  ( N  e.  ( ZZ>= `  M )  ->  ( N  +  K
)  e.  ( ZZ>= `  M ) ) ) )
2812, 15, 18, 21, 24, 27nn0indALT 11031 . 2  |-  ( K  e.  NN0  ->  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  K )  e.  (
ZZ>= `  M ) ) )
2928impcom 431 1  |-  ( ( N  e.  ( ZZ>= `  M )  /\  K  e.  NN0 )  ->  ( N  +  K )  e.  ( ZZ>= `  M )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   ` cfv 5601  (class class class)co 6305   CCcc 9536   0cc0 9538   1c1 9539    + caddc 9541   NN0cn0 10869   ZZ>=cuz 11159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160
This theorem is referenced by:  elfz0add  11889  zpnn0elfzo  11983  ccatass  12719  ccatrn  12720  swrdccat1  12798  swrdccat2  12799  splfv1  12847  splval2  12849  revccat  12856  relexpaddg  13095  isercoll2  13710  iseraltlem2  13727  iseraltlem3  13728  mertenslem1  13918  eftlub  14141  vdwlem6  14899  gsumccat  16576  efginvrel2  17312  efgredleme  17328  efgcpbllemb  17340  geolim3  23160  jm2.27c  35568  iunrelexpuztr  35950  pfxccat1  38341
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