Theorem evennn02n 12409

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 2292	. . . . . . . 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 9191	. . . . . . . . . . . 12 |- 2 e. RR
4	3	a1i 9	. . . . . . . . . . 11 |- ( ( ( 2 x. n ) e. NN0 /\ n e. ZZ ) -> 2 e. RR )
5		zre 9461	. . . . . . . . . . . 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 9212	. . . . . . . . . . . 12 |- 0 < 2
8	7	a1i 9	. . . . . . . . . . 11 |- ( ( ( 2 x. n ) e. NN0 /\ n e. ZZ ) -> 0 < 2 )
9		nn0ge0 9405	. . . . . . . . . . . 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 9012	. . . . . . . . . . 11 |- ( ( ( 2 e. RR /\ n e. RR ) /\ ( 0 < 2 /\ 0 <_ ( 2 x. n ) ) ) -> 0 <_ n )
12	4, 6, 8, 10, 11	syl22anc 1272	. . . . . . . . . 10 |- ( ( ( 2 x. n ) e. NN0 /\ n e. ZZ ) -> 0 <_ n )
13		elnn0z 9470	. . . . . . . . . 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 2527	. 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 9475	. . 3 |- NN0 C_ ZZ
23		rexss 3291	. . 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 12401	. . 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 1395   e. wcel 2200   E.wrex 2509   C_ wss 3197   class class class wbr 4083  (class class class)co 6007   RRcr 8009   0cc0 8010   x. cmul 8015   < clt 8192   <_ cle 8193   2c2 9172   NN0cn0 9380   ZZcz 9457   || cdvds 12314

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-ltadd 8126  ax-pre-mulgt0 8127

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-inn 9122  df-2 9180  df-n0 9381  df-z 9458  df-dvds 12315

This theorem is referenced by: (None)