Theorem divconjdvds 12415

Description: If a nonzero integer M divides another integer N, the other integer N divided by the nonzero integer M (i.e. the divisor conjugate of N to M) divides the other integer N. Theorem 1.1(k) in [ApostolNT] p. 14. (Contributed by AV, 7-Aug-2021.)

Assertion

Ref	Expression
divconjdvds	|- ( ( M || N /\ M =/= 0 ) -> ( N / M ) || N )

Proof of Theorem divconjdvds

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvdszrcl 12358	. . 3 |- ( M || N -> ( M e. ZZ /\ N e. ZZ ) )
2		simpll 527	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) -> M e. ZZ )
3		oveq1 6025	. . . . . . . . . 10 |- ( m = M -> ( m x. ( N / M ) ) = ( M x. ( N / M ) ) )
4	3	eqeq1d 2240	. . . . . . . . 9 |- ( m = M -> ( ( m x. ( N / M ) ) = N <-> ( M x. ( N / M ) ) = N ) )
5	4	adantl 277	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) /\ m = M ) -> ( ( m x. ( N / M ) ) = N <-> ( M x. ( N / M ) ) = N ) )
6		zcn 9484	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
7	6	adantl 277	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. CC )
8	7	adantr 276	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) -> N e. CC )
9		zcn 9484	. . . . . . . . . . 11 |- ( M e. ZZ -> M e. CC )
10	9	adantr 276	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> M e. CC )
11	10	adantr 276	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) -> M e. CC )
12		0z 9490	. . . . . . . . . . . 12 |- 0 e. ZZ
13		zapne 9554	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
14	12, 13	mpan2 425	. . . . . . . . . . 11 |- ( M e. ZZ -> ( M # 0 <-> M =/= 0 ) )
15	14	adantr 276	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
16	15	biimpar 297	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) -> M # 0 )
17	8, 11, 16	divcanap2d 8972	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) -> ( M x. ( N / M ) ) = N )
18	2, 5, 17	rspcedvd 2916	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) -> E. m e. ZZ ( m x. ( N / M ) ) = N )
19	18	adantr 276	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) /\ M || N ) -> E. m e. ZZ ( m x. ( N / M ) ) = N )
20		simpr 110	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) /\ M || N ) -> M || N )
21		simpr 110	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) -> M =/= 0 )
22		simpr 110	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
23	22	adantr 276	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) -> N e. ZZ )
24	2, 21, 23	3jca 1203	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) -> ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) )
25	24	adantr 276	. . . . . . . . 9 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) /\ M || N ) -> ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) )
26		dvdsval2 12356	. . . . . . . . 9 |- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( M || N <-> ( N / M ) e. ZZ ) )
27	25, 26	syl 14	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) /\ M || N ) -> ( M || N <-> ( N / M ) e. ZZ ) )
28	20, 27	mpbid 147	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) /\ M || N ) -> ( N / M ) e. ZZ )
29	23	adantr 276	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) /\ M || N ) -> N e. ZZ )
30		divides 12355	. . . . . . 7 |- ( ( ( N / M ) e. ZZ /\ N e. ZZ ) -> ( ( N / M ) || N <-> E. m e. ZZ ( m x. ( N / M ) ) = N ) )
31	28, 29, 30	syl2anc 411	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) /\ M || N ) -> ( ( N / M ) || N <-> E. m e. ZZ ( m x. ( N / M ) ) = N ) )
32	19, 31	mpbird 167	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) /\ M || N ) -> ( N / M ) || N )
33	32	exp31 364	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M =/= 0 -> ( M || N -> ( N / M ) || N ) ) )
34	33	com3r 79	. . 3 |- ( M || N -> ( ( M e. ZZ /\ N e. ZZ ) -> ( M =/= 0 -> ( N / M ) || N ) ) )
35	1, 34	mpd 13	. 2 |- ( M || N -> ( M =/= 0 -> ( N / M ) || N ) )
36	35	imp 124	1 |- ( ( M || N /\ M =/= 0 ) -> ( N / M ) || N )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   E.wrex 2511   class class class wbr 4088  (class class class)co 6018   CCcc 8030   0cc0 8032   x. cmul 8037   # cap 8761   / cdiv 8852   ZZcz 9479   || cdvds 12353

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-n0 9403  df-z 9480  df-dvds 12354

This theorem is referenced by:  dvdsdivcl  12416  isprm5lem  12718