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Theorem uzaddcl 9781
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 6009 . . . . 5 |- ( j = 0 -> ( N + j ) = ( N + 0 ) )
21eleq1d 2298 . . . 4 |- ( j = 0 -> ( ( N + j ) e. ( ZZ>= ` M ) <-> ( N + 0 ) e. ( ZZ>= ` M ) ) )
32imbi2d 230 . . 3 |- ( j = 0 -> ( ( N e. ( ZZ>= ` M ) -> ( N + j ) e. ( ZZ>= ` M ) ) <-> ( N e. ( ZZ>= ` M ) -> ( N + 0 ) e. ( ZZ>= ` M ) ) ) )
4 oveq2 6009 . . . . 5 |- ( j = k -> ( N + j ) = ( N + k ) )
54eleq1d 2298 . . . 4 |- ( j = k -> ( ( N + j ) e. ( ZZ>= ` M ) <-> ( N + k ) e. ( ZZ>= ` M ) ) )
65imbi2d 230 . . 3 |- ( j = k -> ( ( N e. ( ZZ>= ` M ) -> ( N + j ) e. ( ZZ>= ` M ) ) <-> ( N e. ( ZZ>= ` M ) -> ( N + k ) e. ( ZZ>= ` M ) ) ) )
7 oveq2 6009 . . . . 5 |- ( j = ( k + 1 ) -> ( N + j ) = ( N + ( k + 1 ) ) )
87eleq1d 2298 . . . 4 |- ( j = ( k + 1 ) -> ( ( N + j ) e. ( ZZ>= ` M ) <-> ( N + ( k + 1 ) ) e. ( ZZ>= ` M ) ) )
98imbi2d 230 . . 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 6009 . . . . 5 |- ( j = K -> ( N + j ) = ( N + K ) )
1110eleq1d 2298 . . . 4 |- ( j = K -> ( ( N + j ) e. ( ZZ>= ` M ) <-> ( N + K ) e. ( ZZ>= ` M ) ) )
1211imbi2d 230 . . 3 |- ( j = K -> ( ( N e. ( ZZ>= ` M ) -> ( N + j ) e. ( ZZ>= ` M ) ) <-> ( N e. ( ZZ>= ` M ) -> ( N + K ) e. ( ZZ>= ` M ) ) ) )
13 eluzelcn 9733 . . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. CC )
1413addridd 8295 . . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( N + 0 ) = N )
1514eleq1d 2298 . . . 4 |- ( N e. ( ZZ>= ` M ) -> ( ( N + 0 ) e. ( ZZ>= ` M ) <-> N e. ( ZZ>= ` M ) ) )
1615ibir 177 . . 3 |- ( N e. ( ZZ>= ` M ) -> ( N + 0 ) e. ( ZZ>= ` M ) )
17 nn0cn 9379 . . . . . . . 8 |- ( k e. NN0 -> k e. CC )
18 ax-1cn 8092 . . . . . . . . 9 |- 1 e. CC
19 addass 8129 . . . . . . . . 9 |- ( ( N e. CC /\ k e. CC /\ 1 e. CC ) -> ( ( N + k ) + 1 ) = ( N + ( k + 1 ) ) )
2018, 19mp3an3 1360 . . . . . . . 8 |- ( ( N e. CC /\ k e. CC ) -> ( ( N + k ) + 1 ) = ( N + ( k + 1 ) ) )
2113, 17, 20syl2anr 290 . . . . . . 7 |- ( ( k e. NN0 /\ N e. ( ZZ>= ` M ) ) -> ( ( N + k ) + 1 ) = ( N + ( k + 1 ) ) )
2221adantr 276 . . . . . 6 |- ( ( ( k e. NN0 /\ N e. ( ZZ>= ` M ) ) /\ ( N + k ) e. ( ZZ>= ` M ) ) -> ( ( N + k ) + 1 ) = ( N + ( k + 1 ) ) )
23 peano2uz 9778 . . . . . . 7 |- ( ( N + k ) e. ( ZZ>= ` M ) -> ( ( N + k ) + 1 ) e. ( ZZ>= ` M ) )
2423adantl 277 . . . . . 6 |- ( ( ( k e. NN0 /\ N e. ( ZZ>= ` M ) ) /\ ( N + k ) e. ( ZZ>= ` M ) ) -> ( ( N + k ) + 1 ) e. ( ZZ>= ` M ) )
2522, 24eqeltrrd 2307 . . . . 5 |- ( ( ( k e. NN0 /\ N e. ( ZZ>= ` M ) ) /\ ( N + k ) e. ( ZZ>= ` M ) ) -> ( N + ( k + 1 ) ) e. ( ZZ>= ` M ) )
2625exp31 364 . . . 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 9561 . 2 |- ( K e. NN0 -> ( N e. ( ZZ>= ` M ) -> ( N + K ) e. ( ZZ>= ` M ) ) )
2928impcom 125 1 |- ( ( N e. ( ZZ>= ` M ) /\ K e. NN0 ) -> ( N + K ) e. ( ZZ>= ` M ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6001   CCcc 7997   0cc0 7999   1c1 8000   + caddc 8002   NN0cn0 9369   ZZ>=cuz 9722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115
This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723
This theorem is referenced by:  elfz0add  10316  zpnn0elfzo  10413  ccatass  11143  ccatrn  11144  swrdccat2  11203  pfxccat1  11234  mertenslemi1  12046  eftlub  12201
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