Theorem evennn02n 12064

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 2259	. . . . . . . 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 9077	. . . . . . . . . . . 12 |- 2 e. RR
4	3	a1i 9	. . . . . . . . . . 11 |- ( ( ( 2 x. n ) e. NN0 /\ n e. ZZ ) -> 2 e. RR )
5		zre 9347	. . . . . . . . . . . 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 9098	. . . . . . . . . . . 12 |- 0 < 2
8	7	a1i 9	. . . . . . . . . . 11 |- ( ( ( 2 x. n ) e. NN0 /\ n e. ZZ ) -> 0 < 2 )
9		nn0ge0 9291	. . . . . . . . . . . 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 8898	. . . . . . . . . . 11 |- ( ( ( 2 e. RR /\ n e. RR ) /\ ( 0 < 2 /\ 0 <_ ( 2 x. n ) ) ) -> 0 <_ n )
12	4, 6, 8, 10, 11	syl22anc 1250	. . . . . . . . . 10 |- ( ( ( 2 x. n ) e. NN0 /\ n e. ZZ ) -> 0 <_ n )
13		elnn0z 9356	. . . . . . . . . 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 2494	. 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 9361	. . 3 |- NN0 C_ ZZ
23		rexss 3251	. . 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 12056	. . 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 1364   e. wcel 2167   E.wrex 2476   C_ wss 3157   class class class wbr 4034  (class class class)co 5925   RRcr 7895   0cc0 7896   x. cmul 7901   < clt 8078   <_ cle 8079   2c2 9058   NN0cn0 9266   ZZcz 9343   || cdvds 11969

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012  ax-pre-mulgt0 8013

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-2 9066  df-n0 9267  df-z 9344  df-dvds 11970

This theorem is referenced by: (None)