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Theorem uzaddcl 10002
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 6096 . . . . 5 |- ( j = 0 -> ( N + j ) = ( N + 0 ) )
21eleq1d 2307 . . . 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 6096 . . . . 5 |- ( j = k -> ( N + j ) = ( N + k ) )
54eleq1d 2307 . . . 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 6096 . . . . 5 |- ( j = ( k + 1 ) -> ( N + j ) = ( N + ( k + 1 ) ) )
87eleq1d 2307 . . . 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 6096 . . . . 5 |- ( j = K -> ( N + j ) = ( N + K ) )
1110eleq1d 2307 . . . 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 9949 . . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. CC )
1413addridd 8482 . . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( N + 0 ) = N )
1514eleq1d 2307 . . . 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 9584 . . . . . . . 8 |- ( k e. NN0 -> k e. CC )
18 ax-1cn 8278 . . . . . . . . 9 |- 1 e. CC
19 addass 8315 . . . . . . . . 9 |- ( ( N e. CC /\ k e. CC /\ 1 e. CC ) -> ( ( N + k ) + 1 ) = ( N + ( k + 1 ) ) )
2018, 19mp3an3 1367 . . . . . . . 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 9999 . . . . . . 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 2316 . . . . 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 9771 . 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
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   ` cfv 5380  (class class class)co 6088   CCcc 8183   0cc0 8185   1c1 8186   + caddc 8188   NN0cn0 9574   ZZ>=cuz 9937
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-ltadd 8301
This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938
This theorem is used by:  elfz0add  10544  zpnn0elfzo  10642  ccatass  11398  ccatrn  11399  swrdccat2  11465  pfxccat1  11496  mertenslemi1  12327  eftlub  12482
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