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Theorem mulsucdiv2z 11892
Description: An integer multiplied with its successor divided by 2 yields an integer, i.e. an integer multiplied with its successor is even. (Contributed by AV, 19-Jul-2021.)
Assertion
Ref Expression
mulsucdiv2z |- ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ )
Proof of Theorem mulsucdiv2z
StepHypRef Expression
1 zeo 9360 . 2 |- ( N e. ZZ -> ( ( N / 2 ) e. ZZ \/ ( ( N + 1 ) / 2 ) e. ZZ ) )
2 peano2z 9291 . . . . . 6 |- ( N e. ZZ -> ( N + 1 ) e. ZZ )
3 zmulcl 9308 . . . . . 6 |- ( ( ( N / 2 ) e. ZZ /\ ( N + 1 ) e. ZZ ) -> ( ( N / 2 ) x. ( N + 1 ) ) e. ZZ )
42, 3sylan2 286 . . . . 5 |- ( ( ( N / 2 ) e. ZZ /\ N e. ZZ ) -> ( ( N / 2 ) x. ( N + 1 ) ) e. ZZ )
5 zcn 9260 . . . . . . . 8 |- ( N e. ZZ -> N e. CC )
62zcnd 9378 . . . . . . . 8 |- ( N e. ZZ -> ( N + 1 ) e. CC )
7 2cnd 8994 . . . . . . . 8 |- ( N e. ZZ -> 2 e. CC )
8 2ap0 9014 . . . . . . . . 9 |- 2 # 0
98a1i 9 . . . . . . . 8 |- ( N e. ZZ -> 2 # 0 )
105, 6, 7, 9div23apd 8787 . . . . . . 7 |- ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) = ( ( N / 2 ) x. ( N + 1 ) ) )
1110eleq1d 2246 . . . . . 6 |- ( N e. ZZ -> ( ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ <-> ( ( N / 2 ) x. ( N + 1 ) ) e. ZZ ) )
1211adantl 277 . . . . 5 |- ( ( ( N / 2 ) e. ZZ /\ N e. ZZ ) -> ( ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ <-> ( ( N / 2 ) x. ( N + 1 ) ) e. ZZ ) )
134, 12mpbird 167 . . . 4 |- ( ( ( N / 2 ) e. ZZ /\ N e. ZZ ) -> ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ )
1413ex 115 . . 3 |- ( ( N / 2 ) e. ZZ -> ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ ) )
15 zmulcl 9308 . . . . . 6 |- ( ( N e. ZZ /\ ( ( N + 1 ) / 2 ) e. ZZ ) -> ( N x. ( ( N + 1 ) / 2 ) ) e. ZZ )
1615ancoms 268 . . . . 5 |- ( ( ( ( N + 1 ) / 2 ) e. ZZ /\ N e. ZZ ) -> ( N x. ( ( N + 1 ) / 2 ) ) e. ZZ )
175, 6, 7, 9divassapd 8785 . . . . . . 7 |- ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) = ( N x. ( ( N + 1 ) / 2 ) ) )
1817eleq1d 2246 . . . . . 6 |- ( N e. ZZ -> ( ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ <-> ( N x. ( ( N + 1 ) / 2 ) ) e. ZZ ) )
1918adantl 277 . . . . 5 |- ( ( ( ( N + 1 ) / 2 ) e. ZZ /\ N e. ZZ ) -> ( ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ <-> ( N x. ( ( N + 1 ) / 2 ) ) e. ZZ ) )
2016, 19mpbird 167 . . . 4 |- ( ( ( ( N + 1 ) / 2 ) e. ZZ /\ N e. ZZ ) -> ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ )
2120ex 115 . . 3 |- ( ( ( N + 1 ) / 2 ) e. ZZ -> ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ ) )
2214, 21jaoi 716 . 2 |- ( ( ( N / 2 ) e. ZZ \/ ( ( N + 1 ) / 2 ) e. ZZ ) -> ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ ) )
231, 22mpcom 36 1 |- ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   e. wcel 2148   class class class wbr 4005  (class class class)co 5877   0cc0 7813   1c1 7814   + caddc 7816   x. cmul 7818   # cap 8540   / cdiv 8631   2c2 8972   ZZcz 9255
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-n0 9179  df-z 9256
This theorem is referenced by:  sqoddm1div8z  11893
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