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Theorem mulsucdiv2z 12445
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 9584 . 2 |- ( N e. ZZ -> ( ( N / 2 ) e. ZZ \/ ( ( N + 1 ) / 2 ) e. ZZ ) )
2 peano2z 9514 . . . . . 6 |- ( N e. ZZ -> ( N + 1 ) e. ZZ )
3 zmulcl 9532 . . . . . 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 9483 . . . . . . . 8 |- ( N e. ZZ -> N e. CC )
62zcnd 9602 . . . . . . . 8 |- ( N e. ZZ -> ( N + 1 ) e. CC )
7 2cnd 9215 . . . . . . . 8 |- ( N e. ZZ -> 2 e. CC )
8 2ap0 9235 . . . . . . . . 9 |- 2 # 0
98a1i 9 . . . . . . . 8 |- ( N e. ZZ -> 2 # 0 )
105, 6, 7, 9div23apd 9007 . . . . . . 7 |- ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) = ( ( N / 2 ) x. ( N + 1 ) ) )
1110eleq1d 2300 . . . . . 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 9532 . . . . . 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 9005 . . . . . . 7 |- ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) = ( N x. ( ( N + 1 ) / 2 ) ) )
1817eleq1d 2300 . . . . . 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 723 . 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 715   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   0cc0 8031   1c1 8032   + caddc 8034   x. cmul 8036   # cap 8760   / cdiv 8851   2c2 9193   ZZcz 9478
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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149
This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-n0 9402  df-z 9479
This theorem is referenced by:  sqoddm1div8z  12446
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