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Theorem nn0o 12548
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 12546 . 2 |- ( ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( N = 1 \/ 2 < N ) )
2 1m1e0 9271 . . . . . . . 8 |- ( 1 - 1 ) = 0
32oveq1i 6038 . . . . . . 7 |- ( ( 1 - 1 ) / 2 ) = ( 0 / 2 )
4 2cn 9273 . . . . . . . 8 |- 2 e. CC
5 2ap0 9295 . . . . . . . 8 |- 2 # 0
64, 5div0api 8985 . . . . . . 7 |- ( 0 / 2 ) = 0
73, 6eqtri 2252 . . . . . 6 |- ( ( 1 - 1 ) / 2 ) = 0
8 0nn0 9476 . . . . . 6 |- 0 e. NN0
97, 8eqeltri 2304 . . . . 5 |- ( ( 1 - 1 ) / 2 ) e. NN0
10 oveq1 6035 . . . . . . . 8 |- ( N = 1 -> ( N - 1 ) = ( 1 - 1 ) )
1110oveq1d 6043 . . . . . . 7 |- ( N = 1 -> ( ( N - 1 ) / 2 ) = ( ( 1 - 1 ) / 2 ) )
1211eleq1d 2300 . . . . . 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 9568 . . . . . . . 8 |- 2 e. ZZ
1716a1i 9 . . . . . . 7 |- ( ( 2 < N /\ ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) ) -> 2 e. ZZ )
18 nn0z 9560 . . . . . . . 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 9272 . . . . . . . . . 10 |- 2 e. RR
21 nn0re 9470 . . . . . . . . . 10 |- ( N e. NN0 -> N e. RR )
22 ltle 8326 . . . . . . . . . 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 9822 . . . . . . 7 |- ( N e. ( ZZ>= ` 2 ) <-> ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) )
2717, 19, 25, 26syl3anbrc 1208 . . . . . 6 |- ( ( 2 < N /\ ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) ) -> N e. ( ZZ>= ` 2 ) )
28 simprr 533 . . . . . 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 12547 . . . . 5 |- ( ( N e. ( ZZ>= ` 2 ) /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( ( N - 1 ) / 2 ) e. NN )
31 nnnn0 9468 . . . . 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 724 . 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 716   = wceq 1398   e. wcel 2202   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   RRcr 8091   0cc0 8092   1c1 8093   + caddc 8095   < clt 8273   <_ cle 8274   - cmin 8409   / cdiv 8911   NNcn 9202   2c2 9253   NN0cn0 9461   ZZcz 9540   ZZ>=cuz 9816
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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210
This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817
This theorem is referenced by:  nn0ob  12549
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