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Theorem nzss 31426
Description: The set of multiples of m, mℤ, is a subset of those of n, nℤ, iff n divides m. Lemma 2.1(a) of https://www.mscs.dal.ca/~selinger/3343/handouts/ideals.pdf p. 5, with mℤ and nℤ as images of the divides relation under m and n. (Contributed by Steve Rodriguez, 20-Jan-2020.)
Hypotheses
Ref Expression
nzss.m  |-  ( ph  ->  M  e.  ZZ )
nzss.n  |-  ( ph  ->  N  e.  V )
Assertion
Ref Expression
nzss  |-  ( ph  ->  ( (  ||  " { M } )  C_  (  ||  " { N }
)  <->  N  ||  M ) )

Proof of Theorem nzss
Dummy variables  x  y  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nzss.m . 2  |-  ( ph  ->  M  e.  ZZ )
2 nzss.n . 2  |-  ( ph  ->  N  e.  V )
3 iddvds 14009 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  M  ||  M )
4 breq2 4460 . . . . . . . . . 10  |-  ( x  =  M  ->  ( M  ||  x  <->  M  ||  M
) )
54elabg 3247 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  ( M  e.  { x  |  M  ||  x }  <->  M 
||  M ) )
63, 5mpbird 232 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  { x  |  M  ||  x } )
7 reldvds 31424 . . . . . . . . 9  |-  Rel  ||
8 relimasn 5370 . . . . . . . . 9  |-  ( Rel  ||  ->  (  ||  " { M } )  =  {
x  |  M  ||  x } )
97, 8ax-mp 5 . . . . . . . 8  |-  (  ||  " { M } )  =  { x  |  M  ||  x }
106, 9syl6eleqr 2556 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  (  ||  " { M } ) )
11 ssel 3493 . . . . . . 7  |-  ( ( 
||  " { M }
)  C_  (  ||  " { N } )  ->  ( M  e.  (  ||  " { M } )  ->  M  e.  (  ||  " { N } ) ) )
1210, 11syl5 32 . . . . . 6  |-  ( ( 
||  " { M }
)  C_  (  ||  " { N } )  ->  ( M  e.  ZZ  ->  M  e.  (  ||  " { N } ) ) )
13 breq2 4460 . . . . . . 7  |-  ( x  =  M  ->  ( N  ||  x  <->  N  ||  M
) )
14 relimasn 5370 . . . . . . . 8  |-  ( Rel  ||  ->  (  ||  " { N } )  =  {
x  |  N  ||  x } )
157, 14ax-mp 5 . . . . . . 7  |-  (  ||  " { N } )  =  { x  |  N  ||  x }
1613, 15elab2g 3248 . . . . . 6  |-  ( M  e.  ZZ  ->  ( M  e.  (  ||  " { N } )  <-> 
N  ||  M )
)
1712, 16mpbidi 216 . . . . 5  |-  ( ( 
||  " { M }
)  C_  (  ||  " { N } )  ->  ( M  e.  ZZ  ->  N  ||  M
) )
1817com12 31 . . . 4  |-  ( M  e.  ZZ  ->  (
(  ||  " { M } )  C_  (  ||  " { N }
)  ->  N  ||  M
) )
1918adantr 465 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  V )  ->  ( (  ||  " { M } )  C_  (  ||  " { N }
)  ->  N  ||  M
) )
20 ssid 3518 . . . . . . 7  |-  { 0 }  C_  { 0 }
21 simpl 457 . . . . . . . . . . . . 13  |-  ( ( N  ||  M  /\  N  =  0 )  ->  N  ||  M
)
22 breq1 4459 . . . . . . . . . . . . . 14  |-  ( N  =  0  ->  ( N  ||  M  <->  0  ||  M ) )
23 dvdszrcl 14003 . . . . . . . . . . . . . . . 16  |-  ( N 
||  M  ->  ( N  e.  ZZ  /\  M  e.  ZZ ) )
2423simprd 463 . . . . . . . . . . . . . . 15  |-  ( N 
||  M  ->  M  e.  ZZ )
25 0dvds 14016 . . . . . . . . . . . . . . 15  |-  ( M  e.  ZZ  ->  (
0  ||  M  <->  M  = 
0 ) )
2624, 25syl 16 . . . . . . . . . . . . . 14  |-  ( N 
||  M  ->  (
0  ||  M  <->  M  = 
0 ) )
2722, 26sylan9bbr 700 . . . . . . . . . . . . 13  |-  ( ( N  ||  M  /\  N  =  0 )  ->  ( N  ||  M 
<->  M  =  0 ) )
2821, 27mpbid 210 . . . . . . . . . . . 12  |-  ( ( N  ||  M  /\  N  =  0 )  ->  M  =  0 )
2928breq1d 4466 . . . . . . . . . . 11  |-  ( ( N  ||  M  /\  N  =  0 )  ->  ( M  ||  x 
<->  0  ||  x ) )
30 0dvds 14016 . . . . . . . . . . 11  |-  ( x  e.  ZZ  ->  (
0  ||  x  <->  x  = 
0 ) )
3129, 30sylan9bb 699 . . . . . . . . . 10  |-  ( ( ( N  ||  M  /\  N  =  0
)  /\  x  e.  ZZ )  ->  ( M 
||  x  <->  x  = 
0 ) )
3231rabbidva 3100 . . . . . . . . 9  |-  ( ( N  ||  M  /\  N  =  0 )  ->  { x  e.  ZZ  |  M  ||  x }  =  {
x  e.  ZZ  |  x  =  0 }
)
33 0z 10896 . . . . . . . . . 10  |-  0  e.  ZZ
34 rabsn 4099 . . . . . . . . . 10  |-  ( 0  e.  ZZ  ->  { x  e.  ZZ  |  x  =  0 }  =  {
0 } )
3533, 34ax-mp 5 . . . . . . . . 9  |-  { x  e.  ZZ  |  x  =  0 }  =  {
0 }
3632, 35syl6eq 2514 . . . . . . . 8  |-  ( ( N  ||  M  /\  N  =  0 )  ->  { x  e.  ZZ  |  M  ||  x }  =  {
0 } )
37 breq1 4459 . . . . . . . . . . 11  |-  ( N  =  0  ->  ( N  ||  x  <->  0  ||  x ) )
3837rabbidv 3101 . . . . . . . . . 10  |-  ( N  =  0  ->  { x  e.  ZZ  |  N  ||  x }  =  {
x  e.  ZZ  | 
0  ||  x }
)
3930rabbiia 3098 . . . . . . . . . . 11  |-  { x  e.  ZZ  |  0  ||  x }  =  {
x  e.  ZZ  |  x  =  0 }
4039, 35eqtri 2486 . . . . . . . . . 10  |-  { x  e.  ZZ  |  0  ||  x }  =  {
0 }
4138, 40syl6eq 2514 . . . . . . . . 9  |-  ( N  =  0  ->  { x  e.  ZZ  |  N  ||  x }  =  {
0 } )
4241adantl 466 . . . . . . . 8  |-  ( ( N  ||  M  /\  N  =  0 )  ->  { x  e.  ZZ  |  N  ||  x }  =  {
0 } )
4336, 42sseq12d 3528 . . . . . . 7  |-  ( ( N  ||  M  /\  N  =  0 )  ->  ( { x  e.  ZZ  |  M  ||  x }  C_  { x  e.  ZZ  |  N  ||  x }  <->  { 0 }  C_  { 0 } ) )
4420, 43mpbiri 233 . . . . . 6  |-  ( ( N  ||  M  /\  N  =  0 )  ->  { x  e.  ZZ  |  M  ||  x }  C_  { x  e.  ZZ  |  N  ||  x } )
4524zcnd 10991 . . . . . . . . . . . 12  |-  ( N 
||  M  ->  M  e.  CC )
4645ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( N  ||  M  /\  N  =/=  0
)  /\  n  e.  ZZ )  ->  M  e.  CC )
4723simpld 459 . . . . . . . . . . . . 13  |-  ( N 
||  M  ->  N  e.  ZZ )
4847zcnd 10991 . . . . . . . . . . . 12  |-  ( N 
||  M  ->  N  e.  CC )
4948ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( N  ||  M  /\  N  =/=  0
)  /\  n  e.  ZZ )  ->  N  e.  CC )
50 simplr 755 . . . . . . . . . . 11  |-  ( ( ( N  ||  M  /\  N  =/=  0
)  /\  n  e.  ZZ )  ->  N  =/=  0 )
5146, 49, 50divcan2d 10343 . . . . . . . . . 10  |-  ( ( ( N  ||  M  /\  N  =/=  0
)  /\  n  e.  ZZ )  ->  ( N  x.  ( M  /  N ) )  =  M )
5251breq1d 4466 . . . . . . . . 9  |-  ( ( ( N  ||  M  /\  N  =/=  0
)  /\  n  e.  ZZ )  ->  ( ( N  x.  ( M  /  N ) ) 
||  n  <->  M  ||  n
) )
5347adantr 465 . . . . . . . . . . 11  |-  ( ( N  ||  M  /\  N  =/=  0 )  ->  N  e.  ZZ )
54 dvdsval2 14001 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  ZZ  /\  N  =/=  0  /\  M  e.  ZZ )  ->  ( N  ||  M  <->  ( M  /  N )  e.  ZZ ) )
5554biimpd 207 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  ZZ  /\  N  =/=  0  /\  M  e.  ZZ )  ->  ( N  ||  M  ->  ( M  /  N )  e.  ZZ ) )
56553com23 1202 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  N  =/=  0 )  ->  ( N  ||  M  ->  ( M  /  N )  e.  ZZ ) )
57563expa 1196 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ )  /\  N  =/=  0
)  ->  ( N  ||  M  ->  ( M  /  N )  e.  ZZ ) )
5823, 57sylan 471 . . . . . . . . . . . . 13  |-  ( ( N  ||  M  /\  N  =/=  0 )  -> 
( N  ||  M  ->  ( M  /  N
)  e.  ZZ ) )
5958imp 429 . . . . . . . . . . . 12  |-  ( ( ( N  ||  M  /\  N  =/=  0
)  /\  N  ||  M
)  ->  ( M  /  N )  e.  ZZ )
6059anabss1 814 . . . . . . . . . . 11  |-  ( ( N  ||  M  /\  N  =/=  0 )  -> 
( M  /  N
)  e.  ZZ )
6153, 60jca 532 . . . . . . . . . 10  |-  ( ( N  ||  M  /\  N  =/=  0 )  -> 
( N  e.  ZZ  /\  ( M  /  N
)  e.  ZZ ) )
62 muldvds1 14020 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  ( M  /  N
)  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( N  x.  ( M  /  N
) )  ||  n  ->  N  ||  n ) )
63623expa 1196 . . . . . . . . . 10  |-  ( ( ( N  e.  ZZ  /\  ( M  /  N
)  e.  ZZ )  /\  n  e.  ZZ )  ->  ( ( N  x.  ( M  /  N ) )  ||  n  ->  N  ||  n
) )
6461, 63sylan 471 . . . . . . . . 9  |-  ( ( ( N  ||  M  /\  N  =/=  0
)  /\  n  e.  ZZ )  ->  ( ( N  x.  ( M  /  N ) ) 
||  n  ->  N  ||  n ) )
6552, 64sylbird 235 . . . . . . . 8  |-  ( ( ( N  ||  M  /\  N  =/=  0
)  /\  n  e.  ZZ )  ->  ( M 
||  n  ->  N  ||  n ) )
6665ss2rabdv 3577 . . . . . . 7  |-  ( ( N  ||  M  /\  N  =/=  0 )  ->  { n  e.  ZZ  |  M  ||  n }  C_ 
{ n  e.  ZZ  |  N  ||  n }
)
67 breq2 4460 . . . . . . . 8  |-  ( n  =  x  ->  ( M  ||  n  <->  M  ||  x
) )
6867cbvrabv 3108 . . . . . . 7  |-  { n  e.  ZZ  |  M  ||  n }  =  {
x  e.  ZZ  |  M  ||  x }
69 breq2 4460 . . . . . . . 8  |-  ( n  =  x  ->  ( N  ||  n  <->  N  ||  x
) )
7069cbvrabv 3108 . . . . . . 7  |-  { n  e.  ZZ  |  N  ||  n }  =  {
x  e.  ZZ  |  N  ||  x }
7166, 68, 703sstr3g 3539 . . . . . 6  |-  ( ( N  ||  M  /\  N  =/=  0 )  ->  { x  e.  ZZ  |  M  ||  x }  C_ 
{ x  e.  ZZ  |  N  ||  x }
)
7244, 71pm2.61dane 2775 . . . . 5  |-  ( N 
||  M  ->  { x  e.  ZZ  |  M  ||  x }  C_  { x  e.  ZZ  |  N  ||  x } )
73 breq1 4459 . . . . . . . . . 10  |-  ( n  =  M  ->  (
n  ||  x  <->  M  ||  x
) )
7473rabbidv 3101 . . . . . . . . 9  |-  ( n  =  M  ->  { x  e.  ZZ  |  n  ||  x }  =  {
x  e.  ZZ  |  M  ||  x } )
7573abbidv 2593 . . . . . . . . 9  |-  ( n  =  M  ->  { x  |  n  ||  x }  =  { x  |  M  ||  x } )
7674, 75eqeq12d 2479 . . . . . . . 8  |-  ( n  =  M  ->  ( { x  e.  ZZ  |  n  ||  x }  =  { x  |  n 
||  x }  <->  { x  e.  ZZ  |  M  ||  x }  =  {
x  |  M  ||  x } ) )
77 simpr 461 . . . . . . . . . . 11  |-  ( ( y  e.  ZZ  /\  n  ||  y )  ->  n  ||  y )
78 dvdszrcl 14003 . . . . . . . . . . . . 13  |-  ( n 
||  y  ->  (
n  e.  ZZ  /\  y  e.  ZZ )
)
7978simprd 463 . . . . . . . . . . . 12  |-  ( n 
||  y  ->  y  e.  ZZ )
8079ancri 552 . . . . . . . . . . 11  |-  ( n 
||  y  ->  (
y  e.  ZZ  /\  n  ||  y ) )
8177, 80impbii 188 . . . . . . . . . 10  |-  ( ( y  e.  ZZ  /\  n  ||  y )  <->  n  ||  y
)
82 breq2 4460 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
n  ||  x  <->  n  ||  y
) )
8382elrab 3257 . . . . . . . . . 10  |-  ( y  e.  { x  e.  ZZ  |  n  ||  x }  <->  ( y  e.  ZZ  /\  n  ||  y ) )
84 vex 3112 . . . . . . . . . . 11  |-  y  e. 
_V
8584, 82elab 3246 . . . . . . . . . 10  |-  ( y  e.  { x  |  n  ||  x }  <->  n 
||  y )
8681, 83, 853bitr4i 277 . . . . . . . . 9  |-  ( y  e.  { x  e.  ZZ  |  n  ||  x }  <->  y  e.  {
x  |  n  ||  x } )
8786eqriv 2453 . . . . . . . 8  |-  { x  e.  ZZ  |  n  ||  x }  =  {
x  |  n  ||  x }
8876, 87vtoclg 3167 . . . . . . 7  |-  ( M  e.  ZZ  ->  { x  e.  ZZ  |  M  ||  x }  =  {
x  |  M  ||  x } )
8988adantr 465 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  V )  ->  { x  e.  ZZ  |  M  ||  x }  =  { x  |  M  ||  x } )
90 breq1 4459 . . . . . . . . . 10  |-  ( n  =  N  ->  (
n  ||  x  <->  N  ||  x
) )
9190rabbidv 3101 . . . . . . . . 9  |-  ( n  =  N  ->  { x  e.  ZZ  |  n  ||  x }  =  {
x  e.  ZZ  |  N  ||  x } )
9290abbidv 2593 . . . . . . . . 9  |-  ( n  =  N  ->  { x  |  n  ||  x }  =  { x  |  N  ||  x } )
9391, 92eqeq12d 2479 . . . . . . . 8  |-  ( n  =  N  ->  ( { x  e.  ZZ  |  n  ||  x }  =  { x  |  n 
||  x }  <->  { x  e.  ZZ  |  N  ||  x }  =  {
x  |  N  ||  x } ) )
9493, 87vtoclg 3167 . . . . . . 7  |-  ( N  e.  V  ->  { x  e.  ZZ  |  N  ||  x }  =  {
x  |  N  ||  x } )
9594adantl 466 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  V )  ->  { x  e.  ZZ  |  N  ||  x }  =  { x  |  N  ||  x } )
9689, 95sseq12d 3528 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  V )  ->  ( { x  e.  ZZ  |  M  ||  x }  C_  { x  e.  ZZ  |  N  ||  x }  <->  { x  |  M  ||  x }  C_  { x  |  N  ||  x }
) )
9772, 96syl5ib 219 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  V )  ->  ( N  ||  M  ->  { x  |  M  ||  x }  C_  { x  |  N  ||  x }
) )
989, 15sseq12i 3525 . . . 4  |-  ( ( 
||  " { M }
)  C_  (  ||  " { N } )  <->  { x  |  M  ||  x }  C_  { x  |  N  ||  x }
)
9997, 98syl6ibr 227 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  V )  ->  ( N  ||  M  ->  (  ||  " { M } )  C_  (  ||  " { N }
) ) )
10019, 99impbid 191 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  V )  ->  ( (  ||  " { M } )  C_  (  ||  " { N }
)  <->  N  ||  M ) )
1011, 2, 100syl2anc 661 1  |-  ( ph  ->  ( (  ||  " { M } )  C_  (  ||  " { N }
)  <->  N  ||  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   {cab 2442    =/= wne 2652   {crab 2811    C_ wss 3471   {csn 4032   class class class wbr 4456   "cima 5011   Rel wrel 5013  (class class class)co 6296   CCcc 9507   0cc0 9509    x. cmul 9514    / cdiv 10227   ZZcz 10885    || cdvds 13998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-n0 10817  df-z 10886  df-dvds 13999
This theorem is referenced by:  nzin  31427
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