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Theorem uzaddcl 9936
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 6066 . . . . 5 |- ( j = 0 -> ( N + j ) = ( N + 0 ) )
21eleq1d 2303 . . . 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 6066 . . . . 5 |- ( j = k -> ( N + j ) = ( N + k ) )
54eleq1d 2303 . . . 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 6066 . . . . 5 |- ( j = ( k + 1 ) -> ( N + j ) = ( N + ( k + 1 ) ) )
87eleq1d 2303 . . . 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 6066 . . . . 5 |- ( j = K -> ( N + j ) = ( N + K ) )
1110eleq1d 2303 . . . 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 9883 . . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. CC )
1413addridd 8438 . . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( N + 0 ) = N )
1514eleq1d 2303 . . . 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 9523 . . . . . . . 8 |- ( k e. NN0 -> k e. CC )
18 ax-1cn 8236 . . . . . . . . 9 |- 1 e. CC
19 addass 8273 . . . . . . . . 9 |- ( ( N e. CC /\ k e. CC /\ 1 e. CC ) -> ( ( N + k ) + 1 ) = ( N + ( k + 1 ) ) )
2018, 19mp3an3 1363 . . . . . . . 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 9933 . . . . . . 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 2312 . . . . 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 9710 . 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 1398   e. wcel 2205   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   NN0cn0 9513   ZZ>=cuz 9871
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259
This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872
This theorem is referenced by:  elfz0add  10476  zpnn0elfzo  10574  ccatass  11321  ccatrn  11322  swrdccat2  11388  pfxccat1  11419  mertenslemi1  12246  eftlub  12401
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