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Theorem zaddcllemneg 8284
Description: Lemma for zaddcl 8285. Special case in which -u N is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
Assertion
Ref Expression
zaddcllemneg |- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) -> ( M + N ) e. ZZ )
Proof of Theorem zaddcllemneg
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 905 . . . . 5 |- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) -> N e. RR )
21recnd 7054 . . . 4 |- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) -> N e. CC )
32negnegd 7313 . . 3 |- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) -> -u -u N = N )
43oveq2d 5528 . 2 |- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) -> ( M + -u -u N ) = ( M + N ) )
5 negeq 7204 . . . . . . . 8 |- ( x = 1 -> -u x = -u 1 )
65oveq2d 5528 . . . . . . 7 |- ( x = 1 -> ( M + -u x ) = ( M + -u 1 ) )
76eleq1d 2106 . . . . . 6 |- ( x = 1 -> ( ( M + -u x ) e. ZZ <-> ( M + -u 1 ) e. ZZ ) )
87imbi2d 219 . . . . 5 |- ( x = 1 -> ( ( ( M e. ZZ /\ N e. RR ) -> ( M + -u x ) e. ZZ ) <-> ( ( M e. ZZ /\ N e. RR ) -> ( M + -u 1 ) e. ZZ ) ) )
9 negeq 7204 . . . . . . . 8 |- ( x = y -> -u x = -u y )
109oveq2d 5528 . . . . . . 7 |- ( x = y -> ( M + -u x ) = ( M + -u y ) )
1110eleq1d 2106 . . . . . 6 |- ( x = y -> ( ( M + -u x ) e. ZZ <-> ( M + -u y ) e. ZZ ) )
1211imbi2d 219 . . . . 5 |- ( x = y -> ( ( ( M e. ZZ /\ N e. RR ) -> ( M + -u x ) e. ZZ ) <-> ( ( M e. ZZ /\ N e. RR ) -> ( M + -u y ) e. ZZ ) ) )
13 negeq 7204 . . . . . . . 8 |- ( x = ( y + 1 ) -> -u x = -u ( y + 1 ) )
1413oveq2d 5528 . . . . . . 7 |- ( x = ( y + 1 ) -> ( M + -u x ) = ( M + -u ( y + 1 ) ) )
1514eleq1d 2106 . . . . . 6 |- ( x = ( y + 1 ) -> ( ( M + -u x ) e. ZZ <-> ( M + -u ( y + 1 ) ) e. ZZ ) )
1615imbi2d 219 . . . . 5 |- ( x = ( y + 1 ) -> ( ( ( M e. ZZ /\ N e. RR ) -> ( M + -u x ) e. ZZ ) <-> ( ( M e. ZZ /\ N e. RR ) -> ( M + -u ( y + 1 ) ) e. ZZ ) ) )
17 negeq 7204 . . . . . . . 8 |- ( x = -u N -> -u x = -u -u N )
1817oveq2d 5528 . . . . . . 7 |- ( x = -u N -> ( M + -u x ) = ( M + -u -u N ) )
1918eleq1d 2106 . . . . . 6 |- ( x = -u N -> ( ( M + -u x ) e. ZZ <-> ( M + -u -u N ) e. ZZ ) )
2019imbi2d 219 . . . . 5 |- ( x = -u N -> ( ( ( M e. ZZ /\ N e. RR ) -> ( M + -u x ) e. ZZ ) <-> ( ( M e. ZZ /\ N e. RR ) -> ( M + -u -u N ) e. ZZ ) ) )
21 zcn 8250 . . . . . . . 8 |- ( M e. ZZ -> M e. CC )
2221adantr 261 . . . . . . 7 |- ( ( M e. ZZ /\ N e. RR ) -> M e. CC )
23 1cnd 7043 . . . . . . 7 |- ( ( M e. ZZ /\ N e. RR ) -> 1 e. CC )
2422, 23negsubd 7328 . . . . . 6 |- ( ( M e. ZZ /\ N e. RR ) -> ( M + -u 1 ) = ( M - 1 ) )
25 peano2zm 8283 . . . . . . 7 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
2625adantr 261 . . . . . 6 |- ( ( M e. ZZ /\ N e. RR ) -> ( M - 1 ) e. ZZ )
2724, 26eqeltrd 2114 . . . . 5 |- ( ( M e. ZZ /\ N e. RR ) -> ( M + -u 1 ) e. ZZ )
28 nncn 7922 . . . . . . . . . . 11 |- ( y e. NN -> y e. CC )
2928ad2antrr 457 . . . . . . . . . 10 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> y e. CC )
30 1cnd 7043 . . . . . . . . . 10 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> 1 e. CC )
3129, 30negdi2d 7336 . . . . . . . . 9 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> -u ( y + 1 ) = ( -u y - 1 ) )
3231oveq2d 5528 . . . . . . . 8 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> ( M + -u ( y + 1 ) ) = ( M + ( -u y - 1 ) ) )
3322ad2antlr 458 . . . . . . . . . 10 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> M e. CC )
3429negcld 7309 . . . . . . . . . 10 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> -u y e. CC )
3533, 34, 30addsubassd 7342 . . . . . . . . 9 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> ( ( M + -u y ) - 1 ) = ( M + ( -u y - 1 ) ) )
36 peano2zm 8283 . . . . . . . . . 10 |- ( ( M + -u y ) e. ZZ -> ( ( M + -u y ) - 1 ) e. ZZ )
3736adantl 262 . . . . . . . . 9 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> ( ( M + -u y ) - 1 ) e. ZZ )
3835, 37eqeltrrd 2115 . . . . . . . 8 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> ( M + ( -u y - 1 ) ) e. ZZ )
3932, 38eqeltrd 2114 . . . . . . 7 |- ( ( ( y e. NN /\ ( M e. ZZ /\ N e. RR ) ) /\ ( M + -u y ) e. ZZ ) -> ( M + -u ( y + 1 ) ) e. ZZ )
4039exp31 346 . . . . . 6 |- ( y e. NN -> ( ( M e. ZZ /\ N e. RR ) -> ( ( M + -u y ) e. ZZ -> ( M + -u ( y + 1 ) ) e. ZZ ) ) )
4140a2d 23 . . . . 5 |- ( y e. NN -> ( ( ( M e. ZZ /\ N e. RR ) -> ( M + -u y ) e. ZZ ) -> ( ( M e. ZZ /\ N e. RR ) -> ( M + -u ( y + 1 ) ) e. ZZ ) ) )
428, 12, 16, 20, 27, 41nnind 7930 . . . 4 |- ( -u N e. NN -> ( ( M e. ZZ /\ N e. RR ) -> ( M + -u -u N ) e. ZZ ) )
4342impcom 116 . . 3 |- ( ( ( M e. ZZ /\ N e. RR ) /\ -u N e. NN ) -> ( M + -u -u N ) e. ZZ )
44433impa 1099 . 2 |- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) -> ( M + -u -u N ) e. ZZ )
454, 44eqeltrrd 2115 1 |- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) -> ( M + N ) e. ZZ )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   = wceq 1243   e. wcel 1393  (class class class)co 5512   CCcc 6887   RRcr 6888   1c1 6890   + caddc 6892   - cmin 7182   -ucneg 7183   NNcn 7914   ZZcz 8245
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-n0 8182  df-z 8246
This theorem is referenced by:  zaddcl  8285
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