Theorem divconjdvds 12599

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 12542	. . 3 |- ( M || N -> ( M e. ZZ /\ N e. ZZ ) )
2		simpll 531	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) -> M e. ZZ )
3		oveq1 6086	. . . . . . . . . 10 |- ( m = M -> ( m x. ( N / M ) ) = ( M x. ( N / M ) ) )
4	3	eqeq1d 2247	. . . . . . . . 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 9632	. . . . . . . . . . 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 9632	. . . . . . . . . . 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 9638	. . . . . . . . . . . 12 |- 0 e. ZZ
13		zapne 9702	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
14	12, 13	mpan2 429	. . . . . . . . . . 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 9116	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) -> ( M x. ( N / M ) ) = N )
18	2, 5, 17	rspcedvd 2935	. . . . . . 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 1208	. . . . . . . . . 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 12540	. . . . . . . . 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 12539	. . . . . . 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 415	. . . . . 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 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   E.wrex 2529   class class class wbr 4128  (class class class)co 6079   CCcc 8171   0cc0 8173   x. cmul 8178   # cap 8903   / cdiv 8996   ZZcz 9627   || cdvds 12537

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-id 4436  df-po 4439  df-iso 4440  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-n0 9547  df-z 9628  df-dvds 12538

This theorem is referenced by:  dvdsdivcl  12600  isprm5lem  12902