Theorem znval 14971

Description: The value of the ℤ/nℤ structure. It is defined as the quotient ring ZZ / n ZZ, with an "artificial" ordering added. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.) (Revised by AV, 13-Jun-2019.)

Hypotheses

Ref	Expression
znval.s	|- S = (RSpan ` ℤring )
znval.u	|- U = (ℤring /.s (ℤring ~QG ( S ` { N } ) ) )
znval.y	|- Y = (ℤ/nℤ ` N )
znval.f	|- F = ( ( ZRHom ` U ) |` W )
znval.w	|- W = if ( N = 0 , ZZ , ( 0..^ N ) )
znval.l	|- .<_ = ( ( F o. <_ ) o. `' F )

Assertion

Ref	Expression
znval	|- ( N e. NN0 -> Y = ( U sSet <. ( le ` ndx ) , .<_ >. ) )

Proof of Theorem znval

Dummy variables f n s z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		znval.y	. 2 |- Y = (ℤ/nℤ ` N )
2		df-zn 14951	. . 3 |- ℤ/nℤ = ( n e. NN0 |-> [_ℤring / z ]_ [_ ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) / s ]_ ( s sSet <. ( le ` ndx ) , [_ ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) / f ]_ ( ( f o. <_ ) o. `' f ) >. ) )
3		zringring 14928	. . . . 5 |- ℤring e. Ring
4	3	a1i 9	. . . 4 |- ( n = N -> ℤring e. Ring )
5		vex 2824	. . . . . . 7 |- z e. _V
6		rspex 14811	. . . . . . . . . 10 |- ( z e. _V -> (RSpan ` z ) e. _V )
7	6	elv 2825	. . . . . . . . 9 |- (RSpan ` z ) e. _V
8		vex 2824	. . . . . . . . . 10 |- n e. _V
9	8	snex 4322	. . . . . . . . 9 |- { n } e. _V
10	7, 9	fvex 5715	. . . . . . . 8 |- ( (RSpan ` z ) ` { n } ) e. _V
11		eqgex 14024	. . . . . . . 8 |- ( ( z e. _V /\ ( (RSpan ` z ) ` { n } ) e. _V ) -> ( z ~QG ( (RSpan ` z ) ` { n } ) ) e. _V )
12	5, 10, 11	mp2an 430	. . . . . . 7 |- ( z ~QG ( (RSpan ` z ) ` { n } ) ) e. _V
13		qusex 13646	. . . . . . 7 |- ( ( z e. _V /\ ( z ~QG ( (RSpan ` z ) ` { n } ) ) e. _V ) -> ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) e. _V )
14	5, 12, 13	mp2an 430	. . . . . 6 |- ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) e. _V
15	14	a1i 9	. . . . 5 |- ( ( n = N /\ z = ℤring ) -> ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) e. _V )
16		id 19	. . . . . . 7 |- ( s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) -> s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) )
17		simpr 110	. . . . . . . . 9 |- ( ( n = N /\ z = ℤring ) -> z = ℤring )
18	17	fveq2d 5699	. . . . . . . . . . . 12 |- ( ( n = N /\ z = ℤring ) -> (RSpan ` z ) = (RSpan ` ℤring ) )
19		znval.s	. . . . . . . . . . . 12 |- S = (RSpan ` ℤring )
20	18, 19	eqtr4di 2289	. . . . . . . . . . 11 |- ( ( n = N /\ z = ℤring ) -> (RSpan ` z ) = S )
21		simpl 109	. . . . . . . . . . . 12 |- ( ( n = N /\ z = ℤring ) -> n = N )
22	21	sneqd 3722	. . . . . . . . . . 11 |- ( ( n = N /\ z = ℤring ) -> { n } = { N } )
23	20, 22	fveq12d 5702	. . . . . . . . . 10 |- ( ( n = N /\ z = ℤring ) -> ( (RSpan ` z ) ` { n } ) = ( S ` { N } ) )
24	17, 23	oveq12d 6103	. . . . . . . . 9 |- ( ( n = N /\ z = ℤring ) -> ( z ~QG ( (RSpan ` z ) ` { n } ) ) = (ℤring ~QG ( S ` { N } ) ) )
25	17, 24	oveq12d 6103	. . . . . . . 8 |- ( ( n = N /\ z = ℤring ) -> ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) = (ℤring /.s (ℤring ~QG ( S ` { N } ) ) ) )
26		znval.u	. . . . . . . 8 |- U = (ℤring /.s (ℤring ~QG ( S ` { N } ) ) )
27	25, 26	eqtr4di 2289	. . . . . . 7 |- ( ( n = N /\ z = ℤring ) -> ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) = U )
28	16, 27	sylan9eqr 2293	. . . . . 6 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> s = U )
29		eqid 2238	. . . . . . . . . . . 12 |- ( ZRHom ` s ) = ( ZRHom ` s )
30	29	zrhex 14956	. . . . . . . . . . 11 |- ( s e. _V -> ( ZRHom ` s ) e. _V )
31	30	elv 2825	. . . . . . . . . 10 |- ( ZRHom ` s ) e. _V
32	31	resex 5104	. . . . . . . . 9 |- ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) e. _V
33	32	a1i 9	. . . . . . . 8 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) e. _V )
34		id 19	. . . . . . . . . . . 12 |- ( f = ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) -> f = ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) )
35	28	fveq2d 5699	. . . . . . . . . . . . . 14 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> ( ZRHom ` s ) = ( ZRHom ` U ) )
36		simpll 531	. . . . . . . . . . . . . . . . 17 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> n = N )
37	36	eqeq1d 2247	. . . . . . . . . . . . . . . 16 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> ( n = 0 <-> N = 0 ) )
38	36	oveq2d 6101	. . . . . . . . . . . . . . . 16 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> ( 0..^ n ) = ( 0..^ N ) )
39	37, 38	ifbieq2d 3665	. . . . . . . . . . . . . . 15 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> if ( n = 0 , ZZ , ( 0..^ n ) ) = if ( N = 0 , ZZ , ( 0..^ N ) ) )
40		znval.w	. . . . . . . . . . . . . . 15 |- W = if ( N = 0 , ZZ , ( 0..^ N ) )
41	39, 40	eqtr4di 2289	. . . . . . . . . . . . . 14 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> if ( n = 0 , ZZ , ( 0..^ n ) ) = W )
42	35, 41	reseq12d 5064	. . . . . . . . . . . . 13 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) = ( ( ZRHom ` U ) |` W ) )
43		znval.f	. . . . . . . . . . . . 13 |- F = ( ( ZRHom ` U ) |` W )
44	42, 43	eqtr4di 2289	. . . . . . . . . . . 12 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) = F )
45	34, 44	sylan9eqr 2293	. . . . . . . . . . 11 |- ( ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) /\ f = ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) ) -> f = F )
46	45	coeq1d 4941	. . . . . . . . . 10 |- ( ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) /\ f = ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) ) -> ( f o. <_ ) = ( F o. <_ ) )
47	45	cnveqd 4956	. . . . . . . . . 10 |- ( ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) /\ f = ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) ) -> `' f = `' F )
48	46, 47	coeq12d 4944	. . . . . . . . 9 |- ( ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) /\ f = ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) ) -> ( ( f o. <_ ) o. `' f ) = ( ( F o. <_ ) o. `' F ) )
49		znval.l	. . . . . . . . 9 |- .<_ = ( ( F o. <_ ) o. `' F )
50	48, 49	eqtr4di 2289	. . . . . . . 8 |- ( ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) /\ f = ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) ) -> ( ( f o. <_ ) o. `' f ) = .<_ )
51	33, 50	csbied 3194	. . . . . . 7 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> [_ ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) / f ]_ ( ( f o. <_ ) o. `' f ) = .<_ )
52	51	opeq2d 3911	. . . . . 6 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> <. ( le ` ndx ) , [_ ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) / f ]_ ( ( f o. <_ ) o. `' f ) >. = <. ( le ` ndx ) , .<_ >. )
53	28, 52	oveq12d 6103	. . . . 5 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> ( s sSet <. ( le ` ndx ) , [_ ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) / f ]_ ( ( f o. <_ ) o. `' f ) >. ) = ( U sSet <. ( le ` ndx ) , .<_ >. ) )
54	15, 53	csbied 3194	. . . 4 |- ( ( n = N /\ z = ℤring ) -> [_ ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) / s ]_ ( s sSet <. ( le ` ndx ) , [_ ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) / f ]_ ( ( f o. <_ ) o. `' f ) >. ) = ( U sSet <. ( le ` ndx ) , .<_ >. ) )
55	4, 54	csbied 3194	. . 3 |- ( n = N -> [_ℤring / z ]_ [_ ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) / s ]_ ( s sSet <. ( le ` ndx ) , [_ ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) / f ]_ ( ( f o. <_ ) o. `' f ) >. ) = ( U sSet <. ( le ` ndx ) , .<_ >. ) )
56		id 19	. . 3 |- ( N e. NN0 -> N e. NN0 )
57		rspex 14811	. . . . . . . . . 10 |- (ℤring e. Ring -> (RSpan ` ℤring ) e. _V )
58	3, 57	ax-mp 5	. . . . . . . . 9 |- (RSpan ` ℤring ) e. _V
59	19, 58	eqeltri 2311	. . . . . . . 8 |- S e. _V
60		snexg 4321	. . . . . . . 8 |- ( N e. NN0 -> { N } e. _V )
61		fvexg 5714	. . . . . . . 8 |- ( ( S e. _V /\ { N } e. _V ) -> ( S ` { N } ) e. _V )
62	59, 60, 61	sylancr 418	. . . . . . 7 |- ( N e. NN0 -> ( S ` { N } ) e. _V )
63		eqgex 14024	. . . . . . 7 |- ( (ℤring e. Ring /\ ( S ` { N } ) e. _V ) -> (ℤring ~QG ( S ` { N } ) ) e. _V )
64	3, 62, 63	sylancr 418	. . . . . 6 |- ( N e. NN0 -> (ℤring ~QG ( S ` { N } ) ) e. _V )
65		qusex 13646	. . . . . 6 |- ( (ℤring e. Ring /\ (ℤring ~QG ( S ` { N } ) ) e. _V ) -> (ℤring /.s (ℤring ~QG ( S ` { N } ) ) ) e. _V )
66	3, 64, 65	sylancr 418	. . . . 5 |- ( N e. NN0 -> (ℤring /.s (ℤring ~QG ( S ` { N } ) ) ) e. _V )
67	26, 66	eqeltrid 2325	. . . 4 |- ( N e. NN0 -> U e. _V )
68		plendxnn 13557	. . . . 5 |- ( le ` ndx ) e. NN
69	68	a1i 9	. . . 4 |- ( N e. NN0 -> ( le ` ndx ) e. NN )
70		eqid 2238	. . . . . . . . . . 11 |- ( ZRHom ` U ) = ( ZRHom ` U )
71	70	zrhex 14956	. . . . . . . . . 10 |- ( U e. _V -> ( ZRHom ` U ) e. _V )
72	67, 71	syl 14	. . . . . . . . 9 |- ( N e. NN0 -> ( ZRHom ` U ) e. _V )
73		resexg 5103	. . . . . . . . 9 |- ( ( ZRHom ` U ) e. _V -> ( ( ZRHom ` U ) |` W ) e. _V )
74	72, 73	syl 14	. . . . . . . 8 |- ( N e. NN0 -> ( ( ZRHom ` U ) |` W ) e. _V )
75	43, 74	eqeltrid 2325	. . . . . . 7 |- ( N e. NN0 -> F e. _V )
76		xrex 10258	. . . . . . . . 9 |- RR* e. _V
77	76, 76	xpex 4891	. . . . . . . 8 |- ( RR* X. RR* ) e. _V
78		lerelxr 8388	. . . . . . . 8 |- <_ C_ ( RR* X. RR* )
79	77, 78	ssexi 4271	. . . . . . 7 |- <_ e. _V
80		coexg 5332	. . . . . . 7 |- ( ( F e. _V /\ <_ e. _V ) -> ( F o. <_ ) e. _V )
81	75, 79, 80	sylancl 417	. . . . . 6 |- ( N e. NN0 -> ( F o. <_ ) e. _V )
82		cnvexg 5325	. . . . . . 7 |- ( F e. _V -> `' F e. _V )
83	75, 82	syl 14	. . . . . 6 |- ( N e. NN0 -> `' F e. _V )
84		coexg 5332	. . . . . 6 |- ( ( ( F o. <_ ) e. _V /\ `' F e. _V ) -> ( ( F o. <_ ) o. `' F ) e. _V )
85	81, 83, 84	syl2anc 415	. . . . 5 |- ( N e. NN0 -> ( ( F o. <_ ) o. `' F ) e. _V )
86	49, 85	eqeltrid 2325	. . . 4 |- ( N e. NN0 -> .<_ e. _V )
87		setsex 13384	. . . 4 |- ( ( U e. _V /\ ( le ` ndx ) e. NN /\ .<_ e. _V ) -> ( U sSet <. ( le ` ndx ) , .<_ >. ) e. _V )
88	67, 69, 86, 87	syl3anc 1278	. . 3 |- ( N e. NN0 -> ( U sSet <. ( le ` ndx ) , .<_ >. ) e. _V )
89	2, 55, 56, 88	fvmptd3 5799	. 2 |- ( N e. NN0 -> (ℤ/nℤ ` N ) = ( U sSet <. ( le ` ndx ) , .<_ >. ) )
90	1, 89	eqtrid 2283	1 |- ( N e. NN0 -> Y = ( U sSet <. ( le ` ndx ) , .<_ >. ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   [_csb 3147   ifcif 3638   {csn 3709   <.cop 3712   X. cxp 4772   `'ccnv 4773   |` cres 4776   o. ccom 4778   ` cfv 5377  (class class class)co 6085   0cc0 8179   RR*cxr 8359   <_ cle 8361   NNcn 9304   NN0cn0 9563   ZZcz 9644  ..^cfzo 10549   ndxcnx 13349   sSet csts 13350   lecple 13438   /._s cqus 13623   ~_QG cqg 13972   Ringcrg 14300  RSpancrsp 14805  ℤ_ringczring 14925   ZRHomczrh 14946  ℤ/nℤczn 14948

This proof depends on 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-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-addf 8301  ax-mulf 8302

This proof 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-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-ec 6809  df-map 6924  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-dec 9778  df-uz 9922  df-rp 10055  df-fz 10412  df-cj 11607  df-abs 11765  df-struct 13354  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-mulr 13445  df-starv 13446  df-sca 13447  df-vsca 13448  df-ip 13449  df-tset 13450  df-ple 13451  df-ds 13453  df-unif 13454  df-0g 13612  df-topgen 13614  df-iimas 13624  df-qus 13625  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-subg 13973  df-eqg 13975  df-cmn 14089  df-mgp 14218  df-ur 14263  df-ring 14302  df-cring 14303  df-rhm 14459  df-subrg 14527  df-lsp 14724  df-sra 14772  df-rgmod 14773  df-rsp 14807  df-bl 14883  df-mopn 14884  df-fg 14886  df-metu 14887  df-cnfld 14894  df-zring 14926  df-zrh 14949  df-zn 14951

This theorem is used by:  znle  14972  znval2  14973  znbaslemnn  14974