Theorem divconjdvds 12471

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 12414	. . 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 6035	. . . . . . . . . 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 9527	. . . . . . . . . . 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 9527	. . . . . . . . . . 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 9533	. . . . . . . . . . . 12 |- 0 e. ZZ
13		zapne 9597	. . . . . . . . . . . 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 9015	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) -> ( M x. ( N / M ) ) = N )
18	2, 5, 17	rspcedvd 2917	. . . . . . 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 1204	. . . . . . . . . 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 12412	. . . . . . . . 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 12411	. . . . . . 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 1005   = wceq 1398   e. wcel 2202   =/= wne 2403   E.wrex 2512   class class class wbr 4093  (class class class)co 6028   CCcc 8073   0cc0 8075   x. cmul 8080   # cap 8804   / cdiv 8895   ZZcz 9522   || cdvds 12409

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-n0 9446  df-z 9523  df-dvds 12410

This theorem is referenced by:  dvdsdivcl  12472  isprm5lem  12774