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Theorem mulsucdiv2z 12436
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 9575 . 2 |- ( N e. ZZ -> ( ( N / 2 ) e. ZZ \/ ( ( N + 1 ) / 2 ) e. ZZ ) )
2 peano2z 9505 . . . . . 6 |- ( N e. ZZ -> ( N + 1 ) e. ZZ )
3 zmulcl 9523 . . . . . 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 9474 . . . . . . . 8 |- ( N e. ZZ -> N e. CC )
62zcnd 9593 . . . . . . . 8 |- ( N e. ZZ -> ( N + 1 ) e. CC )
7 2cnd 9206 . . . . . . . 8 |- ( N e. ZZ -> 2 e. CC )
8 2ap0 9226 . . . . . . . . 9 |- 2 # 0
98a1i 9 . . . . . . . 8 |- ( N e. ZZ -> 2 # 0 )
105, 6, 7, 9div23apd 8998 . . . . . . 7 |- ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) = ( ( N / 2 ) x. ( N + 1 ) ) )
1110eleq1d 2298 . . . . . 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 9523 . . . . . 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 8996 . . . . . . 7 |- ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) = ( N x. ( ( N + 1 ) / 2 ) ) )
1817eleq1d 2298 . . . . . 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 721 . 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 713   e. wcel 2200   class class class wbr 4086  (class class class)co 6013   0cc0 8022   1c1 8023   + caddc 8025   x. cmul 8027   # cap 8751   / cdiv 8842   2c2 9184   ZZcz 9469
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140
This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-id 4388  df-po 4391  df-iso 4392  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-n0 9393  df-z 9470
This theorem is referenced by:  sqoddm1div8z  12437
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