Theorem znval 14135

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 14115	. . 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 14092	. . . . 5 |- ℤring e. Ring
4	3	a1i 9	. . . 4 |- ( n = N -> ℤring e. Ring )
5		vex 2763	. . . . . . 7 |- z e. _V
6		rspex 13973	. . . . . . . . . 10 |- ( z e. _V -> (RSpan ` z ) e. _V )
7	6	elv 2764	. . . . . . . . 9 |- (RSpan ` z ) e. _V
8		vex 2763	. . . . . . . . . 10 |- n e. _V
9	8	snex 4215	. . . . . . . . 9 |- { n } e. _V
10	7, 9	fvex 5575	. . . . . . . 8 |- ( (RSpan ` z ) ` { n } ) e. _V
11		eqgex 13294	. . . . . . . 8 |- ( ( z e. _V /\ ( (RSpan ` z ) ` { n } ) e. _V ) -> ( z ~QG ( (RSpan ` z ) ` { n } ) ) e. _V )
12	5, 10, 11	mp2an 426	. . . . . . 7 |- ( z ~QG ( (RSpan ` z ) ` { n } ) ) e. _V
13		qusex 12911	. . . . . . 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 426	. . . . . 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 5559	. . . . . . . . . . . 12 |- ( ( n = N /\ z = ℤring ) -> (RSpan ` z ) = (RSpan ` ℤring ) )
19		znval.s	. . . . . . . . . . . 12 |- S = (RSpan ` ℤring )
20	18, 19	eqtr4di 2244	. . . . . . . . . . 11 |- ( ( n = N /\ z = ℤring ) -> (RSpan ` z ) = S )
21		simpl 109	. . . . . . . . . . . 12 |- ( ( n = N /\ z = ℤring ) -> n = N )
22	21	sneqd 3632	. . . . . . . . . . 11 |- ( ( n = N /\ z = ℤring ) -> { n } = { N } )
23	20, 22	fveq12d 5562	. . . . . . . . . 10 |- ( ( n = N /\ z = ℤring ) -> ( (RSpan ` z ) ` { n } ) = ( S ` { N } ) )
24	17, 23	oveq12d 5937	. . . . . . . . 9 |- ( ( n = N /\ z = ℤring ) -> ( z ~QG ( (RSpan ` z ) ` { n } ) ) = (ℤring ~QG ( S ` { N } ) ) )
25	17, 24	oveq12d 5937	. . . . . . . 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 2244	. . . . . . 7 |- ( ( n = N /\ z = ℤring ) -> ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) = U )
28	16, 27	sylan9eqr 2248	. . . . . 6 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> s = U )
29		eqid 2193	. . . . . . . . . . . 12 |- ( ZRHom ` s ) = ( ZRHom ` s )
30	29	zrhex 14120	. . . . . . . . . . 11 |- ( s e. _V -> ( ZRHom ` s ) e. _V )
31	30	elv 2764	. . . . . . . . . 10 |- ( ZRHom ` s ) e. _V
32	31	resex 4984	. . . . . . . . 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 5559	. . . . . . . . . . . . . 14 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> ( ZRHom ` s ) = ( ZRHom ` U ) )
36		simpll 527	. . . . . . . . . . . . . . . . 17 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> n = N )
37	36	eqeq1d 2202	. . . . . . . . . . . . . . . 16 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> ( n = 0 <-> N = 0 ) )
38	36	oveq2d 5935	. . . . . . . . . . . . . . . 16 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> ( 0..^ n ) = ( 0..^ N ) )
39	37, 38	ifbieq2d 3582	. . . . . . . . . . . . . . 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 2244	. . . . . . . . . . . . . 14 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> if ( n = 0 , ZZ , ( 0..^ n ) ) = W )
42	35, 41	reseq12d 4944	. . . . . . . . . . . . 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 2244	. . . . . . . . . . . 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 2248	. . . . . . . . . . 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 4824	. . . . . . . . . 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 4839	. . . . . . . . . 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 4827	. . . . . . . . 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 2244	. . . . . . . 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 3128	. . . . . . 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 3812	. . . . . 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 5937	. . . . 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 3128	. . . 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 3128	. . 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 13973	. . . . . . . . . 10 |- (ℤring e. Ring -> (RSpan ` ℤring ) e. _V )
58	3, 57	ax-mp 5	. . . . . . . . 9 |- (RSpan ` ℤring ) e. _V
59	19, 58	eqeltri 2266	. . . . . . . 8 |- S e. _V
60		snexg 4214	. . . . . . . 8 |- ( N e. NN0 -> { N } e. _V )
61		fvexg 5574	. . . . . . . 8 |- ( ( S e. _V /\ { N } e. _V ) -> ( S ` { N } ) e. _V )
62	59, 60, 61	sylancr 414	. . . . . . 7 |- ( N e. NN0 -> ( S ` { N } ) e. _V )
63		eqgex 13294	. . . . . . 7 |- ( (ℤring e. Ring /\ ( S ` { N } ) e. _V ) -> (ℤring ~QG ( S ` { N } ) ) e. _V )
64	3, 62, 63	sylancr 414	. . . . . 6 |- ( N e. NN0 -> (ℤring ~QG ( S ` { N } ) ) e. _V )
65		qusex 12911	. . . . . 6 |- ( (ℤring e. Ring /\ (ℤring ~QG ( S ` { N } ) ) e. _V ) -> (ℤring /.s (ℤring ~QG ( S ` { N } ) ) ) e. _V )
66	3, 64, 65	sylancr 414	. . . . 5 |- ( N e. NN0 -> (ℤring /.s (ℤring ~QG ( S ` { N } ) ) ) e. _V )
67	26, 66	eqeltrid 2280	. . . 4 |- ( N e. NN0 -> U e. _V )
68		plendxnn 12823	. . . . 5 |- ( le ` ndx ) e. NN
69	68	a1i 9	. . . 4 |- ( N e. NN0 -> ( le ` ndx ) e. NN )
70		eqid 2193	. . . . . . . . . . 11 |- ( ZRHom ` U ) = ( ZRHom ` U )
71	70	zrhex 14120	. . . . . . . . . 10 |- ( U e. _V -> ( ZRHom ` U ) e. _V )
72	67, 71	syl 14	. . . . . . . . 9 |- ( N e. NN0 -> ( ZRHom ` U ) e. _V )
73		resexg 4983	. . . . . . . . 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 2280	. . . . . . 7 |- ( N e. NN0 -> F e. _V )
76		xrex 9925	. . . . . . . . 9 |- RR* e. _V
77	76, 76	xpex 4775	. . . . . . . 8 |- ( RR* X. RR* ) e. _V
78		lerelxr 8084	. . . . . . . 8 |- <_ C_ ( RR* X. RR* )
79	77, 78	ssexi 4168	. . . . . . 7 |- <_ e. _V
80		coexg 5211	. . . . . . 7 |- ( ( F e. _V /\ <_ e. _V ) -> ( F o. <_ ) e. _V )
81	75, 79, 80	sylancl 413	. . . . . 6 |- ( N e. NN0 -> ( F o. <_ ) e. _V )
82		cnvexg 5204	. . . . . . 7 |- ( F e. _V -> `' F e. _V )
83	75, 82	syl 14	. . . . . 6 |- ( N e. NN0 -> `' F e. _V )
84		coexg 5211	. . . . . 6 |- ( ( ( F o. <_ ) e. _V /\ `' F e. _V ) -> ( ( F o. <_ ) o. `' F ) e. _V )
85	81, 83, 84	syl2anc 411	. . . . 5 |- ( N e. NN0 -> ( ( F o. <_ ) o. `' F ) e. _V )
86	49, 85	eqeltrid 2280	. . . 4 |- ( N e. NN0 -> .<_ e. _V )
87		setsex 12653	. . . 4 |- ( ( U e. _V /\ ( le ` ndx ) e. NN /\ .<_ e. _V ) -> ( U sSet <. ( le ` ndx ) , .<_ >. ) e. _V )
88	67, 69, 86, 87	syl3anc 1249	. . 3 |- ( N e. NN0 -> ( U sSet <. ( le ` ndx ) , .<_ >. ) e. _V )
89	2, 55, 56, 88	fvmptd3 5652	. 2 |- ( N e. NN0 -> (ℤ/nℤ ` N ) = ( U sSet <. ( le ` ndx ) , .<_ >. ) )
90	1, 89	eqtrid 2238	1 |- ( N e. NN0 -> Y = ( U sSet <. ( le ` ndx ) , .<_ >. ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   [_csb 3081   ifcif 3558   {csn 3619   <.cop 3622   X. cxp 4658   `'ccnv 4659   |` cres 4662   o. ccom 4664   ` cfv 5255  (class class class)co 5919   0cc0 7874   RR*cxr 8055   <_ cle 8057   NNcn 8984   NN0cn0 9243   ZZcz 9320  ..^cfzo 10211   ndxcnx 12618   sSet csts 12619   lecple 12705   /._s cqus 12886   ~_QG cqg 13242   Ringcrg 13495  RSpancrsp 13967  ℤ_ringczring 14089   ZRHomczrh 14110  ℤ/nℤczn 14112

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-addf 7996  ax-mulf 7997

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-tp 3627  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-ec 6591  df-map 6706  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047  df-7 9048  df-8 9049  df-9 9050  df-n0 9244  df-z 9321  df-dec 9452  df-uz 9596  df-rp 9723  df-fz 10078  df-cj 10989  df-abs 11146  df-struct 12623  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-mulr 12712  df-starv 12713  df-sca 12714  df-vsca 12715  df-ip 12716  df-tset 12717  df-ple 12718  df-ds 12720  df-unif 12721  df-0g 12872  df-topgen 12874  df-iimas 12888  df-qus 12889  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079  df-subg 13243  df-eqg 13245  df-cmn 13359  df-mgp 13420  df-ur 13459  df-ring 13497  df-cring 13498  df-rhm 13651  df-subrg 13718  df-lsp 13886  df-sra 13934  df-rgmod 13935  df-rsp 13969  df-bl 14045  df-mopn 14046  df-fg 14048  df-metu 14049  df-cnfld 14056  df-zring 14090  df-zrh 14113  df-zn 14115

This theorem is referenced by:  znle  14136  znval2  14137  znbaslemnn  14138