Theorem divconjdvds 11857

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 11801	. . 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 5884	. . . . . . . . . 10 |- ( m = M -> ( m x. ( N / M ) ) = ( M x. ( N / M ) ) )
4	3	eqeq1d 2186	. . . . . . . . 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 9260	. . . . . . . . . . 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 9260	. . . . . . . . . . 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 9266	. . . . . . . . . . . 12 |- 0 e. ZZ
13		zapne 9329	. . . . . . . . . . . 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 8751	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) -> ( M x. ( N / M ) ) = N )
18	2, 5, 17	rspcedvd 2849	. . . . . . 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 1177	. . . . . . . . . 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 11799	. . . . . . . . 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 11798	. . . . . . 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 978   = wceq 1353   e. wcel 2148   =/= wne 2347   E.wrex 2456   class class class wbr 4005  (class class class)co 5877   CCcc 7811   0cc0 7813   x. cmul 7818   # cap 8540   / cdiv 8631   ZZcz 9255   || cdvds 11796

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-n0 9179  df-z 9256  df-dvds 11797

This theorem is referenced by:  dvdsdivcl  11858  isprm5lem  12143