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Theorem uzaddcl 9660
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 5930 . . . . 5 |- ( j = 0 -> ( N + j ) = ( N + 0 ) )
21eleq1d 2265 . . . 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 5930 . . . . 5 |- ( j = k -> ( N + j ) = ( N + k ) )
54eleq1d 2265 . . . 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 5930 . . . . 5 |- ( j = ( k + 1 ) -> ( N + j ) = ( N + ( k + 1 ) ) )
87eleq1d 2265 . . . 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 5930 . . . . 5 |- ( j = K -> ( N + j ) = ( N + K ) )
1110eleq1d 2265 . . . 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 9612 . . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. CC )
1413addridd 8175 . . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( N + 0 ) = N )
1514eleq1d 2265 . . . 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 9259 . . . . . . . 8 |- ( k e. NN0 -> k e. CC )
18 ax-1cn 7972 . . . . . . . . 9 |- 1 e. CC
19 addass 8009 . . . . . . . . 9 |- ( ( N e. CC /\ k e. CC /\ 1 e. CC ) -> ( ( N + k ) + 1 ) = ( N + ( k + 1 ) ) )
2018, 19mp3an3 1337 . . . . . . . 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 9657 . . . . . . 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 2274 . . . . 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 9440 . 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 1364   e. wcel 2167   ` cfv 5258  (class class class)co 5922   CCcc 7877   0cc0 7879   1c1 7880   + caddc 7882   NN0cn0 9249   ZZ>=cuz 9601
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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602
This theorem is referenced by:  elfz0add  10195  zpnn0elfzo  10283  mertenslemi1  11700  eftlub  11855
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