Theorem divconjdvds 12031

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 11974	. . 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 5932	. . . . . . . . . 10 |- ( m = M -> ( m x. ( N / M ) ) = ( M x. ( N / M ) ) )
4	3	eqeq1d 2205	. . . . . . . . 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 9348	. . . . . . . . . . 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 9348	. . . . . . . . . . 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 9354	. . . . . . . . . . . 12 |- 0 e. ZZ
13		zapne 9417	. . . . . . . . . . . 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 8836	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) -> ( M x. ( N / M ) ) = N )
18	2, 5, 17	rspcedvd 2874	. . . . . . 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 1179	. . . . . . . . . 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 11972	. . . . . . . . 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 11971	. . . . . . 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 980   = wceq 1364   e. wcel 2167   =/= wne 2367   E.wrex 2476   class class class wbr 4034  (class class class)co 5925   CCcc 7894   0cc0 7896   x. cmul 7901   # cap 8625   / cdiv 8716   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-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

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-rmo 2483  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-po 4332  df-iso 4333  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-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-n0 9267  df-z 9344  df-dvds 11970

This theorem is referenced by:  dvdsdivcl  12032  isprm5lem  12334