Theorem nzss 38516

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.

Step	Hyp	Ref	Expression
1		nzss.m	. 2 |- ( ph -> M e. ZZ )
2		nzss.n	. 2 |- ( ph -> N e. V )
3		iddvds 14995	. . . . . . . . 9 |- ( M e. ZZ -> M || M )
4		breq2 4657	. . . . . . . . . 10 |- ( x = M -> ( M || x <-> M || M ) )
5	4	elabg 3351	. . . . . . . . 9 |- ( M e. ZZ -> ( M e. { x | M || x } <-> M || M ) )
6	3, 5	mpbird 247	. . . . . . . 8 |- ( M e. ZZ -> M e. { x | M || x } )
7		reldvds 38514	. . . . . . . . 9 |- Rel ||
8		relimasn 5488	. . . . . . . . 9 |- ( Rel || -> ( || " { M } ) = { x | M || x } )
9	7, 8	ax-mp 5	. . . . . . . 8 |- ( || " { M } ) = { x | M || x }
10	6, 9	syl6eleqr 2712	. . . . . . 7 |- ( M e. ZZ -> M e. ( || " { M } ) )
11		ssel 3597	. . . . . . 7 |- ( ( || " { M } ) C_ ( || " { N } ) -> ( M e. ( || " { M } ) -> M e. ( || " { N } ) ) )
12	10, 11	syl5 34	. . . . . 6 |- ( ( || " { M } ) C_ ( || " { N } ) -> ( M e. ZZ -> M e. ( || " { N } ) ) )
13		breq2 4657	. . . . . . 7 |- ( x = M -> ( N || x <-> N || M ) )
14		relimasn 5488	. . . . . . . 8 |- ( Rel || -> ( || " { N } ) = { x | N || x } )
15	7, 14	ax-mp 5	. . . . . . 7 |- ( || " { N } ) = { x | N || x }
16	13, 15	elab2g 3353	. . . . . 6 |- ( M e. ZZ -> ( M e. ( || " { N } ) <-> N || M ) )
17	12, 16	mpbidi 231	. . . . 5 |- ( ( || " { M } ) C_ ( || " { N } ) -> ( M e. ZZ -> N || M ) )
18	17	com12 32	. . . 4 |- ( M e. ZZ -> ( ( || " { M } ) C_ ( || " { N } ) -> N || M ) )
19	18	adantr 481	. . 3 |- ( ( M e. ZZ /\ N e. V ) -> ( ( || " { M } ) C_ ( || " { N } ) -> N || M ) )
20		ssid 3624	. . . . . . 7 |- { 0 } C_ { 0 }
21		simpl 473	. . . . . . . . . . . . 13 |- ( ( N || M /\ N = 0 ) -> N || M )
22		breq1 4656	. . . . . . . . . . . . . 14 |- ( N = 0 -> ( N || M <-> 0 || M ) )
23		dvdszrcl 14988	. . . . . . . . . . . . . . . 16 |- ( N || M -> ( N e. ZZ /\ M e. ZZ ) )
24	23	simprd 479	. . . . . . . . . . . . . . 15 |- ( N || M -> M e. ZZ )
25		0dvds 15002	. . . . . . . . . . . . . . 15 |- ( M e. ZZ -> ( 0 || M <-> M = 0 ) )
26	24, 25	syl 17	. . . . . . . . . . . . . 14 |- ( N || M -> ( 0 || M <-> M = 0 ) )
27	22, 26	sylan9bbr 737	. . . . . . . . . . . . 13 |- ( ( N || M /\ N = 0 ) -> ( N || M <-> M = 0 ) )
28	21, 27	mpbid 222	. . . . . . . . . . . 12 |- ( ( N || M /\ N = 0 ) -> M = 0 )
29	28	breq1d 4663	. . . . . . . . . . 11 |- ( ( N || M /\ N = 0 ) -> ( M || x <-> 0 || x ) )
30		0dvds 15002	. . . . . . . . . . 11 |- ( x e. ZZ -> ( 0 || x <-> x = 0 ) )
31	29, 30	sylan9bb 736	. . . . . . . . . 10 |- ( ( ( N || M /\ N = 0 ) /\ x e. ZZ ) -> ( M || x <-> x = 0 ) )
32	31	rabbidva 3188	. . . . . . . . 9 |- ( ( N || M /\ N = 0 ) -> { x e. ZZ | M || x } = { x e. ZZ | x = 0 } )
33		0z 11388	. . . . . . . . . 10 |- 0 e. ZZ
34		rabsn 4256	. . . . . . . . . 10 |- ( 0 e. ZZ -> { x e. ZZ | x = 0 } = { 0 } )
35	33, 34	ax-mp 5	. . . . . . . . 9 |- { x e. ZZ | x = 0 } = { 0 }
36	32, 35	syl6eq 2672	. . . . . . . 8 |- ( ( N || M /\ N = 0 ) -> { x e. ZZ | M || x } = { 0 } )
37		breq1 4656	. . . . . . . . . . 11 |- ( N = 0 -> ( N || x <-> 0 || x ) )
38	37	rabbidv 3189	. . . . . . . . . 10 |- ( N = 0 -> { x e. ZZ | N || x } = { x e. ZZ | 0 || x } )
39	30	rabbiia 3185	. . . . . . . . . . 11 |- { x e. ZZ | 0 || x } = { x e. ZZ | x = 0 }
40	39, 35	eqtri 2644	. . . . . . . . . 10 |- { x e. ZZ | 0 || x } = { 0 }
41	38, 40	syl6eq 2672	. . . . . . . . 9 |- ( N = 0 -> { x e. ZZ | N || x } = { 0 } )
42	41	adantl 482	. . . . . . . 8 |- ( ( N || M /\ N = 0 ) -> { x e. ZZ | N || x } = { 0 } )
43	36, 42	sseq12d 3634	. . . . . . 7 |- ( ( N || M /\ N = 0 ) -> ( { x e. ZZ | M || x } C_ { x e. ZZ | N || x } <-> { 0 } C_ { 0 } ) )
44	20, 43	mpbiri 248	. . . . . 6 |- ( ( N || M /\ N = 0 ) -> { x e. ZZ | M || x } C_ { x e. ZZ | N || x } )
45	24	zcnd 11483	. . . . . . . . . . . 12 |- ( N || M -> M e. CC )
46	45	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( N || M /\ N =/= 0 ) /\ n e. ZZ ) -> M e. CC )
47	23	simpld 475	. . . . . . . . . . . . 13 |- ( N || M -> N e. ZZ )
48	47	zcnd 11483	. . . . . . . . . . . 12 |- ( N || M -> N e. CC )
49	48	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( N || M /\ N =/= 0 ) /\ n e. ZZ ) -> N e. CC )
50		simplr 792	. . . . . . . . . . 11 |- ( ( ( N || M /\ N =/= 0 ) /\ n e. ZZ ) -> N =/= 0 )
51	46, 49, 50	divcan2d 10803	. . . . . . . . . 10 |- ( ( ( N || M /\ N =/= 0 ) /\ n e. ZZ ) -> ( N x. ( M / N ) ) = M )
52	51	breq1d 4663	. . . . . . . . 9 |- ( ( ( N || M /\ N =/= 0 ) /\ n e. ZZ ) -> ( ( N x. ( M / N ) ) || n <-> M || n ) )
53	47	adantr 481	. . . . . . . . . . 11 |- ( ( N || M /\ N =/= 0 ) -> N e. ZZ )
54		dvdsval2 14986	. . . . . . . . . . . . . . . . 17 |- ( ( N e. ZZ /\ N =/= 0 /\ M e. ZZ ) -> ( N || M <-> ( M / N ) e. ZZ ) )
55	54	biimpd 219	. . . . . . . . . . . . . . . 16 |- ( ( N e. ZZ /\ N =/= 0 /\ M e. ZZ ) -> ( N || M -> ( M / N ) e. ZZ ) )
56	55	3com23 1271	. . . . . . . . . . . . . . 15 |- ( ( N e. ZZ /\ M e. ZZ /\ N =/= 0 ) -> ( N || M -> ( M / N ) e. ZZ ) )
57	56	3expa 1265	. . . . . . . . . . . . . 14 |- ( ( ( N e. ZZ /\ M e. ZZ ) /\ N =/= 0 ) -> ( N || M -> ( M / N ) e. ZZ ) )
58	23, 57	sylan 488	. . . . . . . . . . . . 13 |- ( ( N || M /\ N =/= 0 ) -> ( N || M -> ( M / N ) e. ZZ ) )
59	58	imp 445	. . . . . . . . . . . 12 |- ( ( ( N || M /\ N =/= 0 ) /\ N || M ) -> ( M / N ) e. ZZ )
60	59	anabss1 855	. . . . . . . . . . 11 |- ( ( N || M /\ N =/= 0 ) -> ( M / N ) e. ZZ )
61	53, 60	jca 554	. . . . . . . . . 10 |- ( ( N || M /\ N =/= 0 ) -> ( N e. ZZ /\ ( M / N ) e. ZZ ) )
62		muldvds1 15006	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ ( M / N ) e. ZZ /\ n e. ZZ ) -> ( ( N x. ( M / N ) ) || n -> N || n ) )
63	62	3expa 1265	. . . . . . . . . 10 |- ( ( ( N e. ZZ /\ ( M / N ) e. ZZ ) /\ n e. ZZ ) -> ( ( N x. ( M / N ) ) || n -> N || n ) )
64	61, 63	sylan 488	. . . . . . . . 9 |- ( ( ( N || M /\ N =/= 0 ) /\ n e. ZZ ) -> ( ( N x. ( M / N ) ) || n -> N || n ) )
65	52, 64	sylbird 250	. . . . . . . 8 |- ( ( ( N || M /\ N =/= 0 ) /\ n e. ZZ ) -> ( M || n -> N || n ) )
66	65	ss2rabdv 3683	. . . . . . 7 |- ( ( N || M /\ N =/= 0 ) -> { n e. ZZ | M || n } C_ { n e. ZZ | N || n } )
67		breq2 4657	. . . . . . . 8 |- ( n = x -> ( M || n <-> M || x ) )
68	67	cbvrabv 3199	. . . . . . 7 |- { n e. ZZ | M || n } = { x e. ZZ | M || x }
69		breq2 4657	. . . . . . . 8 |- ( n = x -> ( N || n <-> N || x ) )
70	69	cbvrabv 3199	. . . . . . 7 |- { n e. ZZ | N || n } = { x e. ZZ | N || x }
71	66, 68, 70	3sstr3g 3645	. . . . . 6 |- ( ( N || M /\ N =/= 0 ) -> { x e. ZZ | M || x } C_ { x e. ZZ | N || x } )
72	44, 71	pm2.61dane 2881	. . . . 5 |- ( N || M -> { x e. ZZ | M || x } C_ { x e. ZZ | N || x } )
73		breq1 4656	. . . . . . . . . 10 |- ( n = M -> ( n || x <-> M || x ) )
74	73	rabbidv 3189	. . . . . . . . 9 |- ( n = M -> { x e. ZZ | n || x } = { x e. ZZ | M || x } )
75	73	abbidv 2741	. . . . . . . . 9 |- ( n = M -> { x | n || x } = { x | M || x } )
76	74, 75	eqeq12d 2637	. . . . . . . 8 |- ( n = M -> ( { x e. ZZ | n || x } = { x | n || x } <-> { x e. ZZ | M || x } = { x | M || x } ) )
77		simpr 477	. . . . . . . . . . 11 |- ( ( y e. ZZ /\ n || y ) -> n || y )
78		dvdszrcl 14988	. . . . . . . . . . . . 13 |- ( n || y -> ( n e. ZZ /\ y e. ZZ ) )
79	78	simprd 479	. . . . . . . . . . . 12 |- ( n || y -> y e. ZZ )
80	79	ancri 575	. . . . . . . . . . 11 |- ( n || y -> ( y e. ZZ /\ n || y ) )
81	77, 80	impbii 199	. . . . . . . . . 10 |- ( ( y e. ZZ /\ n || y ) <-> n || y )
82		breq2 4657	. . . . . . . . . . 11 |- ( x = y -> ( n || x <-> n || y ) )
83	82	elrab 3363	. . . . . . . . . 10 |- ( y e. { x e. ZZ | n || x } <-> ( y e. ZZ /\ n || y ) )
84		vex 3203	. . . . . . . . . . 11 |- y e. _V
85	84, 82	elab 3350	. . . . . . . . . 10 |- ( y e. { x | n || x } <-> n || y )
86	81, 83, 85	3bitr4i 292	. . . . . . . . 9 |- ( y e. { x e. ZZ | n || x } <-> y e. { x | n || x } )
87	86	eqriv 2619	. . . . . . . 8 |- { x e. ZZ | n || x } = { x | n || x }
88	76, 87	vtoclg 3266	. . . . . . 7 |- ( M e. ZZ -> { x e. ZZ | M || x } = { x | M || x } )
89	88	adantr 481	. . . . . 6 |- ( ( M e. ZZ /\ N e. V ) -> { x e. ZZ | M || x } = { x | M || x } )
90		breq1 4656	. . . . . . . . . 10 |- ( n = N -> ( n || x <-> N || x ) )
91	90	rabbidv 3189	. . . . . . . . 9 |- ( n = N -> { x e. ZZ | n || x } = { x e. ZZ | N || x } )
92	90	abbidv 2741	. . . . . . . . 9 |- ( n = N -> { x | n || x } = { x | N || x } )
93	91, 92	eqeq12d 2637	. . . . . . . 8 |- ( n = N -> ( { x e. ZZ | n || x } = { x | n || x } <-> { x e. ZZ | N || x } = { x | N || x } ) )
94	93, 87	vtoclg 3266	. . . . . . 7 |- ( N e. V -> { x e. ZZ | N || x } = { x | N || x } )
95	94	adantl 482	. . . . . 6 |- ( ( M e. ZZ /\ N e. V ) -> { x e. ZZ | N || x } = { x | N || x } )
96	89, 95	sseq12d 3634	. . . . 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 } ) )
97	72, 96	syl5ib 234	. . . 4 |- ( ( M e. ZZ /\ N e. V ) -> ( N || M -> { x | M || x } C_ { x | N || x } ) )
98	9, 15	sseq12i 3631	. . . 4 |- ( ( || " { M } ) C_ ( || " { N } ) <-> { x | M || x } C_ { x | N || x } )
99	97, 98	syl6ibr 242	. . 3 |- ( ( M e. ZZ /\ N e. V ) -> ( N || M -> ( || " { M } ) C_ ( || " { N } ) ) )
100	19, 99	impbid 202	. 2 |- ( ( M e. ZZ /\ N e. V ) -> ( ( || " { M } ) C_ ( || " { N } ) <-> N || M ) )
101	1, 2, 100	syl2anc 693	1 |- ( ph -> ( ( || " { M } ) C_ ( || " { N } ) <-> N || M ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   {crab 2916   C_ wss 3574   {csn 4177   class class class wbr 4653   "cima 5117   Rel wrel 5119  (class class class)co 6650   CCcc 9934   0cc0 9936   x. cmul 9941   / cdiv 10684   ZZcz 11377   || cdvds 14983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-dvds 14984

This theorem is referenced by:  nzin  38517