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Theorem nn0o 12333
Description: An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Proof shortened by AV, 2-Jun-2020.)
Assertion
Ref Expression
nn0o |- ( ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( ( N - 1 ) / 2 ) e. NN0 )
Proof of Theorem nn0o
StepHypRef Expression
1 nn0o1gt2 12331 . 2 |- ( ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( N = 1 \/ 2 < N ) )
2 1m1e0 9140 . . . . . . . 8 |- ( 1 - 1 ) = 0
32oveq1i 5977 . . . . . . 7 |- ( ( 1 - 1 ) / 2 ) = ( 0 / 2 )
4 2cn 9142 . . . . . . . 8 |- 2 e. CC
5 2ap0 9164 . . . . . . . 8 |- 2 # 0
64, 5div0api 8854 . . . . . . 7 |- ( 0 / 2 ) = 0
73, 6eqtri 2228 . . . . . 6 |- ( ( 1 - 1 ) / 2 ) = 0
8 0nn0 9345 . . . . . 6 |- 0 e. NN0
97, 8eqeltri 2280 . . . . 5 |- ( ( 1 - 1 ) / 2 ) e. NN0
10 oveq1 5974 . . . . . . . 8 |- ( N = 1 -> ( N - 1 ) = ( 1 - 1 ) )
1110oveq1d 5982 . . . . . . 7 |- ( N = 1 -> ( ( N - 1 ) / 2 ) = ( ( 1 - 1 ) / 2 ) )
1211eleq1d 2276 . . . . . 6 |- ( N = 1 -> ( ( ( N - 1 ) / 2 ) e. NN0 <-> ( ( 1 - 1 ) / 2 ) e. NN0 ) )
1312adantr 276 . . . . 5 |- ( ( N = 1 /\ ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) ) -> ( ( ( N - 1 ) / 2 ) e. NN0 <-> ( ( 1 - 1 ) / 2 ) e. NN0 ) )
149, 13mpbiri 168 . . . 4 |- ( ( N = 1 /\ ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) ) -> ( ( N - 1 ) / 2 ) e. NN0 )
1514ex 115 . . 3 |- ( N = 1 -> ( ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( ( N - 1 ) / 2 ) e. NN0 ) )
16 2z 9435 . . . . . . . 8 |- 2 e. ZZ
1716a1i 9 . . . . . . 7 |- ( ( 2 < N /\ ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) ) -> 2 e. ZZ )
18 nn0z 9427 . . . . . . . 8 |- ( N e. NN0 -> N e. ZZ )
1918ad2antrl 490 . . . . . . 7 |- ( ( 2 < N /\ ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) ) -> N e. ZZ )
20 2re 9141 . . . . . . . . . 10 |- 2 e. RR
21 nn0re 9339 . . . . . . . . . 10 |- ( N e. NN0 -> N e. RR )
22 ltle 8195 . . . . . . . . . 10 |- ( ( 2 e. RR /\ N e. RR ) -> ( 2 < N -> 2 <_ N ) )
2320, 21, 22sylancr 414 . . . . . . . . 9 |- ( N e. NN0 -> ( 2 < N -> 2 <_ N ) )
2423adantr 276 . . . . . . . 8 |- ( ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( 2 < N -> 2 <_ N ) )
2524impcom 125 . . . . . . 7 |- ( ( 2 < N /\ ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) ) -> 2 <_ N )
26 eluz2 9689 . . . . . . 7 |- ( N e. ( ZZ>= ` 2 ) <-> ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) )
2717, 19, 25, 26syl3anbrc 1184 . . . . . 6 |- ( ( 2 < N /\ ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) ) -> N e. ( ZZ>= ` 2 ) )
28 simprr 531 . . . . . 6 |- ( ( 2 < N /\ ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) ) -> ( ( N + 1 ) / 2 ) e. NN0 )
2927, 28jca 306 . . . . 5 |- ( ( 2 < N /\ ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) ) -> ( N e. ( ZZ>= ` 2 ) /\ ( ( N + 1 ) / 2 ) e. NN0 ) )
30 nno 12332 . . . . 5 |- ( ( N e. ( ZZ>= ` 2 ) /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( ( N - 1 ) / 2 ) e. NN )
31 nnnn0 9337 . . . . 5 |- ( ( ( N - 1 ) / 2 ) e. NN -> ( ( N - 1 ) / 2 ) e. NN0 )
3229, 30, 313syl 17 . . . 4 |- ( ( 2 < N /\ ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) ) -> ( ( N - 1 ) / 2 ) e. NN0 )
3332ex 115 . . 3 |- ( 2 < N -> ( ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( ( N - 1 ) / 2 ) e. NN0 ) )
3415, 33jaoi 718 . 2 |- ( ( N = 1 \/ 2 < N ) -> ( ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( ( N - 1 ) / 2 ) e. NN0 ) )
351, 34mpcom 36 1 |- ( ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( ( N - 1 ) / 2 ) e. NN0 )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   = wceq 1373   e. wcel 2178   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   RRcr 7959   0cc0 7960   1c1 7961   + caddc 7963   < clt 8142   <_ cle 8143   - cmin 8278   / cdiv 8780   NNcn 9071   2c2 9122   NN0cn0 9330   ZZcz 9407   ZZ>=cuz 9683
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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078
This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-po 4361  df-iso 4362  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684
This theorem is referenced by:  nn0ob  12334
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