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Theorem mulsucdiv2z 11844
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 9317 . 2 |- ( N e. ZZ -> ( ( N / 2 ) e. ZZ \/ ( ( N + 1 ) / 2 ) e. ZZ ) )
2 peano2z 9248 . . . . . 6 |- ( N e. ZZ -> ( N + 1 ) e. ZZ )
3 zmulcl 9265 . . . . . 6 |- ( ( ( N / 2 ) e. ZZ /\ ( N + 1 ) e. ZZ ) -> ( ( N / 2 ) x. ( N + 1 ) ) e. ZZ )
42, 3sylan2 284 . . . . 5 |- ( ( ( N / 2 ) e. ZZ /\ N e. ZZ ) -> ( ( N / 2 ) x. ( N + 1 ) ) e. ZZ )
5 zcn 9217 . . . . . . . 8 |- ( N e. ZZ -> N e. CC )
62zcnd 9335 . . . . . . . 8 |- ( N e. ZZ -> ( N + 1 ) e. CC )
7 2cnd 8951 . . . . . . . 8 |- ( N e. ZZ -> 2 e. CC )
8 2ap0 8971 . . . . . . . . 9 |- 2 # 0
98a1i 9 . . . . . . . 8 |- ( N e. ZZ -> 2 # 0 )
105, 6, 7, 9div23apd 8745 . . . . . . 7 |- ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) = ( ( N / 2 ) x. ( N + 1 ) ) )
1110eleq1d 2239 . . . . . 6 |- ( N e. ZZ -> ( ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ <-> ( ( N / 2 ) x. ( N + 1 ) ) e. ZZ ) )
1211adantl 275 . . . . 5 |- ( ( ( N / 2 ) e. ZZ /\ N e. ZZ ) -> ( ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ <-> ( ( N / 2 ) x. ( N + 1 ) ) e. ZZ ) )
134, 12mpbird 166 . . . 4 |- ( ( ( N / 2 ) e. ZZ /\ N e. ZZ ) -> ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ )
1413ex 114 . . 3 |- ( ( N / 2 ) e. ZZ -> ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ ) )
15 zmulcl 9265 . . . . . 6 |- ( ( N e. ZZ /\ ( ( N + 1 ) / 2 ) e. ZZ ) -> ( N x. ( ( N + 1 ) / 2 ) ) e. ZZ )
1615ancoms 266 . . . . 5 |- ( ( ( ( N + 1 ) / 2 ) e. ZZ /\ N e. ZZ ) -> ( N x. ( ( N + 1 ) / 2 ) ) e. ZZ )
175, 6, 7, 9divassapd 8743 . . . . . . 7 |- ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) = ( N x. ( ( N + 1 ) / 2 ) ) )
1817eleq1d 2239 . . . . . 6 |- ( N e. ZZ -> ( ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ <-> ( N x. ( ( N + 1 ) / 2 ) ) e. ZZ ) )
1918adantl 275 . . . . 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 166 . . . 4 |- ( ( ( ( N + 1 ) / 2 ) e. ZZ /\ N e. ZZ ) -> ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ )
2120ex 114 . . 3 |- ( ( ( N + 1 ) / 2 ) e. ZZ -> ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ ) )
2214, 21jaoi 711 . 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 103   <-> wb 104   \/ wo 703   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   # cap 8500   / cdiv 8589   2c2 8929   ZZcz 9212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-n0 9136  df-z 9213
This theorem is referenced by:  sqoddm1div8z  11845
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