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Theorem nzss 36579
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 14259 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  M  ||  M )
4 breq2 4370 . . . . . . . . . 10  |-  ( x  =  M  ->  ( M  ||  x  <->  M  ||  M
) )
54elabg 3161 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  ( M  e.  { x  |  M  ||  x }  <->  M 
||  M ) )
63, 5mpbird 235 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  { x  |  M  ||  x } )
7 reldvds 36577 . . . . . . . . 9  |-  Rel  ||
8 relimasn 5153 . . . . . . . . 9  |-  ( Rel  ||  ->  (  ||  " { M } )  =  {
x  |  M  ||  x } )
97, 8ax-mp 5 . . . . . . . 8  |-  (  ||  " { M } )  =  { x  |  M  ||  x }
106, 9syl6eleqr 2517 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  (  ||  " { M } ) )
11 ssel 3401 . . . . . . 7  |-  ( ( 
||  " { M }
)  C_  (  ||  " { N } )  ->  ( M  e.  (  ||  " { M } )  ->  M  e.  (  ||  " { N } ) ) )
1210, 11syl5 33 . . . . . 6  |-  ( ( 
||  " { M }
)  C_  (  ||  " { N } )  ->  ( M  e.  ZZ  ->  M  e.  (  ||  " { N } ) ) )
13 breq2 4370 . . . . . . 7  |-  ( x  =  M  ->  ( N  ||  x  <->  N  ||  M
) )
14 relimasn 5153 . . . . . . . 8  |-  ( Rel  ||  ->  (  ||  " { N } )  =  {
x  |  N  ||  x } )
157, 14ax-mp 5 . . . . . . 7  |-  (  ||  " { N } )  =  { x  |  N  ||  x }
1613, 15elab2g 3162 . . . . . 6  |-  ( M  e.  ZZ  ->  ( M  e.  (  ||  " { N } )  <-> 
N  ||  M )
)
1712, 16mpbidi 219 . . . . 5  |-  ( ( 
||  " { M }
)  C_  (  ||  " { N } )  ->  ( M  e.  ZZ  ->  N  ||  M
) )
1817com12 32 . . . 4  |-  ( M  e.  ZZ  ->  (
(  ||  " { M } )  C_  (  ||  " { N }
)  ->  N  ||  M
) )
1918adantr 466 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  V )  ->  ( (  ||  " { M } )  C_  (  ||  " { N }
)  ->  N  ||  M
) )
20 ssid 3426 . . . . . . 7  |-  { 0 }  C_  { 0 }
21 simpl 458 . . . . . . . . . . . . 13  |-  ( ( N  ||  M  /\  N  =  0 )  ->  N  ||  M
)
22 breq1 4369 . . . . . . . . . . . . . 14  |-  ( N  =  0  ->  ( N  ||  M  <->  0  ||  M ) )
23 dvdszrcl 14253 . . . . . . . . . . . . . . . 16  |-  ( N 
||  M  ->  ( N  e.  ZZ  /\  M  e.  ZZ ) )
2423simprd 464 . . . . . . . . . . . . . . 15  |-  ( N 
||  M  ->  M  e.  ZZ )
25 0dvds 14266 . . . . . . . . . . . . . . 15  |-  ( M  e.  ZZ  ->  (
0  ||  M  <->  M  = 
0 ) )
2624, 25syl 17 . . . . . . . . . . . . . 14  |-  ( N 
||  M  ->  (
0  ||  M  <->  M  = 
0 ) )
2722, 26sylan9bbr 705 . . . . . . . . . . . . 13  |-  ( ( N  ||  M  /\  N  =  0 )  ->  ( N  ||  M 
<->  M  =  0 ) )
2821, 27mpbid 213 . . . . . . . . . . . 12  |-  ( ( N  ||  M  /\  N  =  0 )  ->  M  =  0 )
2928breq1d 4376 . . . . . . . . . . 11  |-  ( ( N  ||  M  /\  N  =  0 )  ->  ( M  ||  x 
<->  0  ||  x ) )
30 0dvds 14266 . . . . . . . . . . 11  |-  ( x  e.  ZZ  ->  (
0  ||  x  <->  x  = 
0 ) )
3129, 30sylan9bb 704 . . . . . . . . . 10  |-  ( ( ( N  ||  M  /\  N  =  0
)  /\  x  e.  ZZ )  ->  ( M 
||  x  <->  x  = 
0 ) )
3231rabbidva 3012 . . . . . . . . 9  |-  ( ( N  ||  M  /\  N  =  0 )  ->  { x  e.  ZZ  |  M  ||  x }  =  {
x  e.  ZZ  |  x  =  0 }
)
33 0z 10899 . . . . . . . . . 10  |-  0  e.  ZZ
34 rabsn 4010 . . . . . . . . . 10  |-  ( 0  e.  ZZ  ->  { x  e.  ZZ  |  x  =  0 }  =  {
0 } )
3533, 34ax-mp 5 . . . . . . . . 9  |-  { x  e.  ZZ  |  x  =  0 }  =  {
0 }
3632, 35syl6eq 2478 . . . . . . . 8  |-  ( ( N  ||  M  /\  N  =  0 )  ->  { x  e.  ZZ  |  M  ||  x }  =  {
0 } )
37 breq1 4369 . . . . . . . . . . 11  |-  ( N  =  0  ->  ( N  ||  x  <->  0  ||  x ) )
3837rabbidv 3013 . . . . . . . . . 10  |-  ( N  =  0  ->  { x  e.  ZZ  |  N  ||  x }  =  {
x  e.  ZZ  | 
0  ||  x }
)
3930rabbiia 3010 . . . . . . . . . . 11  |-  { x  e.  ZZ  |  0  ||  x }  =  {
x  e.  ZZ  |  x  =  0 }
4039, 35eqtri 2450 . . . . . . . . . 10  |-  { x  e.  ZZ  |  0  ||  x }  =  {
0 }
4138, 40syl6eq 2478 . . . . . . . . 9  |-  ( N  =  0  ->  { x  e.  ZZ  |  N  ||  x }  =  {
0 } )
4241adantl 467 . . . . . . . 8  |-  ( ( N  ||  M  /\  N  =  0 )  ->  { x  e.  ZZ  |  N  ||  x }  =  {
0 } )
4336, 42sseq12d 3436 . . . . . . 7  |-  ( ( N  ||  M  /\  N  =  0 )  ->  ( { x  e.  ZZ  |  M  ||  x }  C_  { x  e.  ZZ  |  N  ||  x }  <->  { 0 }  C_  { 0 } ) )
4420, 43mpbiri 236 . . . . . 6  |-  ( ( N  ||  M  /\  N  =  0 )  ->  { x  e.  ZZ  |  M  ||  x }  C_  { x  e.  ZZ  |  N  ||  x } )
4524zcnd 10992 . . . . . . . . . . . 12  |-  ( N 
||  M  ->  M  e.  CC )
4645ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( N  ||  M  /\  N  =/=  0
)  /\  n  e.  ZZ )  ->  M  e.  CC )
4723simpld 460 . . . . . . . . . . . . 13  |-  ( N 
||  M  ->  N  e.  ZZ )
4847zcnd 10992 . . . . . . . . . . . 12  |-  ( N 
||  M  ->  N  e.  CC )
4948ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( N  ||  M  /\  N  =/=  0
)  /\  n  e.  ZZ )  ->  N  e.  CC )
50 simplr 760 . . . . . . . . . . 11  |-  ( ( ( N  ||  M  /\  N  =/=  0
)  /\  n  e.  ZZ )  ->  N  =/=  0 )
5146, 49, 50divcan2d 10336 . . . . . . . . . 10  |-  ( ( ( N  ||  M  /\  N  =/=  0
)  /\  n  e.  ZZ )  ->  ( N  x.  ( M  /  N ) )  =  M )
5251breq1d 4376 . . . . . . . . 9  |-  ( ( ( N  ||  M  /\  N  =/=  0
)  /\  n  e.  ZZ )  ->  ( ( N  x.  ( M  /  N ) ) 
||  n  <->  M  ||  n
) )
5347adantr 466 . . . . . . . . . . 11  |-  ( ( N  ||  M  /\  N  =/=  0 )  ->  N  e.  ZZ )
54 dvdsval2 14251 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  ZZ  /\  N  =/=  0  /\  M  e.  ZZ )  ->  ( N  ||  M  <->  ( M  /  N )  e.  ZZ ) )
5554biimpd 210 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  ZZ  /\  N  =/=  0  /\  M  e.  ZZ )  ->  ( N  ||  M  ->  ( M  /  N )  e.  ZZ ) )
56553com23 1211 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  N  =/=  0 )  ->  ( N  ||  M  ->  ( M  /  N )  e.  ZZ ) )
57563expa 1205 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ )  /\  N  =/=  0
)  ->  ( N  ||  M  ->  ( M  /  N )  e.  ZZ ) )
5823, 57sylan 473 . . . . . . . . . . . . 13  |-  ( ( N  ||  M  /\  N  =/=  0 )  -> 
( N  ||  M  ->  ( M  /  N
)  e.  ZZ ) )
5958imp 430 . . . . . . . . . . . 12  |-  ( ( ( N  ||  M  /\  N  =/=  0
)  /\  N  ||  M
)  ->  ( M  /  N )  e.  ZZ )
6059anabss1 821 . . . . . . . . . . 11  |-  ( ( N  ||  M  /\  N  =/=  0 )  -> 
( M  /  N
)  e.  ZZ )
6153, 60jca 534 . . . . . . . . . 10  |-  ( ( N  ||  M  /\  N  =/=  0 )  -> 
( N  e.  ZZ  /\  ( M  /  N
)  e.  ZZ ) )
62 muldvds1 14270 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  ( M  /  N
)  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( N  x.  ( M  /  N
) )  ||  n  ->  N  ||  n ) )
63623expa 1205 . . . . . . . . . 10  |-  ( ( ( N  e.  ZZ  /\  ( M  /  N
)  e.  ZZ )  /\  n  e.  ZZ )  ->  ( ( N  x.  ( M  /  N ) )  ||  n  ->  N  ||  n
) )
6461, 63sylan 473 . . . . . . . . 9  |-  ( ( ( N  ||  M  /\  N  =/=  0
)  /\  n  e.  ZZ )  ->  ( ( N  x.  ( M  /  N ) ) 
||  n  ->  N  ||  n ) )
6552, 64sylbird 238 . . . . . . . 8  |-  ( ( ( N  ||  M  /\  N  =/=  0
)  /\  n  e.  ZZ )  ->  ( M 
||  n  ->  N  ||  n ) )
6665ss2rabdv 3485 . . . . . . 7  |-  ( ( N  ||  M  /\  N  =/=  0 )  ->  { n  e.  ZZ  |  M  ||  n }  C_ 
{ n  e.  ZZ  |  N  ||  n }
)
67 breq2 4370 . . . . . . . 8  |-  ( n  =  x  ->  ( M  ||  n  <->  M  ||  x
) )
6867cbvrabv 3021 . . . . . . 7  |-  { n  e.  ZZ  |  M  ||  n }  =  {
x  e.  ZZ  |  M  ||  x }
69 breq2 4370 . . . . . . . 8  |-  ( n  =  x  ->  ( N  ||  n  <->  N  ||  x
) )
7069cbvrabv 3021 . . . . . . 7  |-  { n  e.  ZZ  |  N  ||  n }  =  {
x  e.  ZZ  |  N  ||  x }
7166, 68, 703sstr3g 3447 . . . . . 6  |-  ( ( N  ||  M  /\  N  =/=  0 )  ->  { x  e.  ZZ  |  M  ||  x }  C_ 
{ x  e.  ZZ  |  N  ||  x }
)
7244, 71pm2.61dane 2688 . . . . 5  |-  ( N 
||  M  ->  { x  e.  ZZ  |  M  ||  x }  C_  { x  e.  ZZ  |  N  ||  x } )
73 breq1 4369 . . . . . . . . . 10  |-  ( n  =  M  ->  (
n  ||  x  <->  M  ||  x
) )
7473rabbidv 3013 . . . . . . . . 9  |-  ( n  =  M  ->  { x  e.  ZZ  |  n  ||  x }  =  {
x  e.  ZZ  |  M  ||  x } )
7573abbidv 2546 . . . . . . . . 9  |-  ( n  =  M  ->  { x  |  n  ||  x }  =  { x  |  M  ||  x } )
7674, 75eqeq12d 2443 . . . . . . . 8  |-  ( n  =  M  ->  ( { x  e.  ZZ  |  n  ||  x }  =  { x  |  n 
||  x }  <->  { x  e.  ZZ  |  M  ||  x }  =  {
x  |  M  ||  x } ) )
77 simpr 462 . . . . . . . . . . 11  |-  ( ( y  e.  ZZ  /\  n  ||  y )  ->  n  ||  y )
78 dvdszrcl 14253 . . . . . . . . . . . . 13  |-  ( n 
||  y  ->  (
n  e.  ZZ  /\  y  e.  ZZ )
)
7978simprd 464 . . . . . . . . . . . 12  |-  ( n 
||  y  ->  y  e.  ZZ )
8079ancri 554 . . . . . . . . . . 11  |-  ( n 
||  y  ->  (
y  e.  ZZ  /\  n  ||  y ) )
8177, 80impbii 190 . . . . . . . . . 10  |-  ( ( y  e.  ZZ  /\  n  ||  y )  <->  n  ||  y
)
82 breq2 4370 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
n  ||  x  <->  n  ||  y
) )
8382elrab 3171 . . . . . . . . . 10  |-  ( y  e.  { x  e.  ZZ  |  n  ||  x }  <->  ( y  e.  ZZ  /\  n  ||  y ) )
84 vex 3025 . . . . . . . . . . 11  |-  y  e. 
_V
8584, 82elab 3160 . . . . . . . . . 10  |-  ( y  e.  { x  |  n  ||  x }  <->  n 
||  y )
8681, 83, 853bitr4i 280 . . . . . . . . 9  |-  ( y  e.  { x  e.  ZZ  |  n  ||  x }  <->  y  e.  {
x  |  n  ||  x } )
8786eqriv 2425 . . . . . . . 8  |-  { x  e.  ZZ  |  n  ||  x }  =  {
x  |  n  ||  x }
8876, 87vtoclg 3082 . . . . . . 7  |-  ( M  e.  ZZ  ->  { x  e.  ZZ  |  M  ||  x }  =  {
x  |  M  ||  x } )
8988adantr 466 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  V )  ->  { x  e.  ZZ  |  M  ||  x }  =  { x  |  M  ||  x } )
90 breq1 4369 . . . . . . . . . 10  |-  ( n  =  N  ->  (
n  ||  x  <->  N  ||  x
) )
9190rabbidv 3013 . . . . . . . . 9  |-  ( n  =  N  ->  { x  e.  ZZ  |  n  ||  x }  =  {
x  e.  ZZ  |  N  ||  x } )
9290abbidv 2546 . . . . . . . . 9  |-  ( n  =  N  ->  { x  |  n  ||  x }  =  { x  |  N  ||  x } )
9391, 92eqeq12d 2443 . . . . . . . 8  |-  ( n  =  N  ->  ( { x  e.  ZZ  |  n  ||  x }  =  { x  |  n 
||  x }  <->  { x  e.  ZZ  |  N  ||  x }  =  {
x  |  N  ||  x } ) )
9493, 87vtoclg 3082 . . . . . . 7  |-  ( N  e.  V  ->  { x  e.  ZZ  |  N  ||  x }  =  {
x  |  N  ||  x } )
9594adantl 467 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  V )  ->  { x  e.  ZZ  |  N  ||  x }  =  { x  |  N  ||  x } )
9689, 95sseq12d 3436 . . . . 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 222 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  V )  ->  ( N  ||  M  ->  { x  |  M  ||  x }  C_  { x  |  N  ||  x }
) )
989, 15sseq12i 3433 . . . 4  |-  ( ( 
||  " { M }
)  C_  (  ||  " { N } )  <->  { x  |  M  ||  x }  C_  { x  |  N  ||  x }
)
9997, 98syl6ibr 230 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  V )  ->  ( N  ||  M  ->  (  ||  " { M } )  C_  (  ||  " { N }
) ) )
10019, 99impbid 193 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  V )  ->  ( (  ||  " { M } )  C_  (  ||  " { N }
)  <->  N  ||  M ) )
1011, 2, 100syl2anc 665 1  |-  ( ph  ->  ( (  ||  " { M } )  C_  (  ||  " { N }
)  <->  N  ||  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   {cab 2414    =/= wne 2599   {crab 2718    C_ wss 3379   {csn 3941   class class class wbr 4366   "cima 4799   Rel wrel 4801  (class class class)co 6249   CCcc 9488   0cc0 9490    x. cmul 9495    / cdiv 10220   ZZcz 10888    || cdvds 14248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-n0 10821  df-z 10889  df-dvds 14249
This theorem is referenced by:  nzin  36580
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