Theorem evennn02n 12568

Description: A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.)

Assertion

Ref	Expression
evennn02n	|- ( N e. NN0 -> ( 2 || N <-> E. n e. NN0 ( 2 x. n ) = N ) )

Distinct variable group:   n, N

Proof of Theorem evennn02n

Step	Hyp	Ref	Expression
1		eleq1 2295	. . . . . . . 8 |- ( ( 2 x. n ) = N -> ( ( 2 x. n ) e. NN0 <-> N e. NN0 ) )
2		simpr 110	. . . . . . . . . 10 |- ( ( ( 2 x. n ) e. NN0 /\ n e. ZZ ) -> n e. ZZ )
3		2re 9307	. . . . . . . . . . . 12 |- 2 e. RR
4	3	a1i 9	. . . . . . . . . . 11 |- ( ( ( 2 x. n ) e. NN0 /\ n e. ZZ ) -> 2 e. RR )
5		zre 9581	. . . . . . . . . . . 12 |- ( n e. ZZ -> n e. RR )
6	5	adantl 277	. . . . . . . . . . 11 |- ( ( ( 2 x. n ) e. NN0 /\ n e. ZZ ) -> n e. RR )
7		2pos 9328	. . . . . . . . . . . 12 |- 0 < 2
8	7	a1i 9	. . . . . . . . . . 11 |- ( ( ( 2 x. n ) e. NN0 /\ n e. ZZ ) -> 0 < 2 )
9		nn0ge0 9521	. . . . . . . . . . . 12 |- ( ( 2 x. n ) e. NN0 -> 0 <_ ( 2 x. n ) )
10	9	adantr 276	. . . . . . . . . . 11 |- ( ( ( 2 x. n ) e. NN0 /\ n e. ZZ ) -> 0 <_ ( 2 x. n ) )
11		prodge0 9128	. . . . . . . . . . 11 |- ( ( ( 2 e. RR /\ n e. RR ) /\ ( 0 < 2 /\ 0 <_ ( 2 x. n ) ) ) -> 0 <_ n )
12	4, 6, 8, 10, 11	syl22anc 1275	. . . . . . . . . 10 |- ( ( ( 2 x. n ) e. NN0 /\ n e. ZZ ) -> 0 <_ n )
13		elnn0z 9590	. . . . . . . . . 10 |- ( n e. NN0 <-> ( n e. ZZ /\ 0 <_ n ) )
14	2, 12, 13	sylanbrc 417	. . . . . . . . 9 |- ( ( ( 2 x. n ) e. NN0 /\ n e. ZZ ) -> n e. NN0 )
15	14	ex 115	. . . . . . . 8 |- ( ( 2 x. n ) e. NN0 -> ( n e. ZZ -> n e. NN0 ) )
16	1, 15	biimtrrdi 164	. . . . . . 7 |- ( ( 2 x. n ) = N -> ( N e. NN0 -> ( n e. ZZ -> n e. NN0 ) ) )
17	16	com13 80	. . . . . 6 |- ( n e. ZZ -> ( N e. NN0 -> ( ( 2 x. n ) = N -> n e. NN0 ) ) )
18	17	impcom 125	. . . . 5 |- ( ( N e. NN0 /\ n e. ZZ ) -> ( ( 2 x. n ) = N -> n e. NN0 ) )
19	18	pm4.71rd 394	. . . 4 |- ( ( N e. NN0 /\ n e. ZZ ) -> ( ( 2 x. n ) = N <-> ( n e. NN0 /\ ( 2 x. n ) = N ) ) )
20	19	bicomd 141	. . 3 |- ( ( N e. NN0 /\ n e. ZZ ) -> ( ( n e. NN0 /\ ( 2 x. n ) = N ) <-> ( 2 x. n ) = N ) )
21	20	rexbidva 2539	. 2 |- ( N e. NN0 -> ( E. n e. ZZ ( n e. NN0 /\ ( 2 x. n ) = N ) <-> E. n e. ZZ ( 2 x. n ) = N ) )
22		nn0ssz 9595	. . 3 |- NN0 C_ ZZ
23		rexss 3305	. . 3 |- ( NN0 C_ ZZ -> ( E. n e. NN0 ( 2 x. n ) = N <-> E. n e. ZZ ( n e. NN0 /\ ( 2 x. n ) = N ) ) )
24	22, 23	mp1i 10	. 2 |- ( N e. NN0 -> ( E. n e. NN0 ( 2 x. n ) = N <-> E. n e. ZZ ( n e. NN0 /\ ( 2 x. n ) = N ) ) )
25		even2n 12560	. . 3 |- ( 2 || N <-> E. n e. ZZ ( 2 x. n ) = N )
26	25	a1i 9	. 2 |- ( N e. NN0 -> ( 2 || N <-> E. n e. ZZ ( 2 x. n ) = N ) )
27	21, 24, 26	3bitr4rd 221	1 |- ( N e. NN0 -> ( 2 || N <-> E. n e. NN0 ( 2 x. n ) = N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   E.wrex 2521   C_ wss 3211   class class class wbr 4109  (class class class)co 6050   RRcr 8126   0cc0 8127   x. cmul 8132   < clt 8308   <_ cle 8309   2c2 9288   NN0cn0 9496   ZZcz 9577   || cdvds 12473

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243  ax-pre-mulgt0 8244

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 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-2 9296  df-n0 9497  df-z 9578  df-dvds 12474

This theorem is referenced by: (None)