Theorem znval 14192

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 14172	. . 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 14149	. . . . 5 |- ℤring e. Ring
4	3	a1i 9	. . . 4 |- ( n = N -> ℤring e. Ring )
5		vex 2766	. . . . . . 7 |- z e. _V
6		rspex 14030	. . . . . . . . . 10 |- ( z e. _V -> (RSpan ` z ) e. _V )
7	6	elv 2767	. . . . . . . . 9 |- (RSpan ` z ) e. _V
8		vex 2766	. . . . . . . . . 10 |- n e. _V
9	8	snex 4218	. . . . . . . . 9 |- { n } e. _V
10	7, 9	fvex 5578	. . . . . . . 8 |- ( (RSpan ` z ) ` { n } ) e. _V
11		eqgex 13351	. . . . . . . 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 12968	. . . . . . 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 5562	. . . . . . . . . . . 12 |- ( ( n = N /\ z = ℤring ) -> (RSpan ` z ) = (RSpan ` ℤring ) )
19		znval.s	. . . . . . . . . . . 12 |- S = (RSpan ` ℤring )
20	18, 19	eqtr4di 2247	. . . . . . . . . . 11 |- ( ( n = N /\ z = ℤring ) -> (RSpan ` z ) = S )
21		simpl 109	. . . . . . . . . . . 12 |- ( ( n = N /\ z = ℤring ) -> n = N )
22	21	sneqd 3635	. . . . . . . . . . 11 |- ( ( n = N /\ z = ℤring ) -> { n } = { N } )
23	20, 22	fveq12d 5565	. . . . . . . . . 10 |- ( ( n = N /\ z = ℤring ) -> ( (RSpan ` z ) ` { n } ) = ( S ` { N } ) )
24	17, 23	oveq12d 5940	. . . . . . . . 9 |- ( ( n = N /\ z = ℤring ) -> ( z ~QG ( (RSpan ` z ) ` { n } ) ) = (ℤring ~QG ( S ` { N } ) ) )
25	17, 24	oveq12d 5940	. . . . . . . 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 2247	. . . . . . 7 |- ( ( n = N /\ z = ℤring ) -> ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) = U )
28	16, 27	sylan9eqr 2251	. . . . . 6 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> s = U )
29		eqid 2196	. . . . . . . . . . . 12 |- ( ZRHom ` s ) = ( ZRHom ` s )
30	29	zrhex 14177	. . . . . . . . . . 11 |- ( s e. _V -> ( ZRHom ` s ) e. _V )
31	30	elv 2767	. . . . . . . . . 10 |- ( ZRHom ` s ) e. _V
32	31	resex 4987	. . . . . . . . 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 5562	. . . . . . . . . . . . . 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 2205	. . . . . . . . . . . . . . . 16 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> ( n = 0 <-> N = 0 ) )
38	36	oveq2d 5938	. . . . . . . . . . . . . . . 16 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> ( 0..^ n ) = ( 0..^ N ) )
39	37, 38	ifbieq2d 3585	. . . . . . . . . . . . . . 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 2247	. . . . . . . . . . . . . 14 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> if ( n = 0 , ZZ , ( 0..^ n ) ) = W )
42	35, 41	reseq12d 4947	. . . . . . . . . . . . 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 2247	. . . . . . . . . . . 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 2251	. . . . . . . . . . 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 4827	. . . . . . . . . 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 4842	. . . . . . . . . 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 4830	. . . . . . . . 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 2247	. . . . . . . 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 3131	. . . . . . 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 3815	. . . . . 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 5940	. . . . 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 3131	. . . 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 3131	. . 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 14030	. . . . . . . . . 10 |- (ℤring e. Ring -> (RSpan ` ℤring ) e. _V )
58	3, 57	ax-mp 5	. . . . . . . . 9 |- (RSpan ` ℤring ) e. _V
59	19, 58	eqeltri 2269	. . . . . . . 8 |- S e. _V
60		snexg 4217	. . . . . . . 8 |- ( N e. NN0 -> { N } e. _V )
61		fvexg 5577	. . . . . . . 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 13351	. . . . . . 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 12968	. . . . . 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 2283	. . . 4 |- ( N e. NN0 -> U e. _V )
68		plendxnn 12880	. . . . 5 |- ( le ` ndx ) e. NN
69	68	a1i 9	. . . 4 |- ( N e. NN0 -> ( le ` ndx ) e. NN )
70		eqid 2196	. . . . . . . . . . 11 |- ( ZRHom ` U ) = ( ZRHom ` U )
71	70	zrhex 14177	. . . . . . . . . 10 |- ( U e. _V -> ( ZRHom ` U ) e. _V )
72	67, 71	syl 14	. . . . . . . . 9 |- ( N e. NN0 -> ( ZRHom ` U ) e. _V )
73		resexg 4986	. . . . . . . . 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 2283	. . . . . . 7 |- ( N e. NN0 -> F e. _V )
76		xrex 9931	. . . . . . . . 9 |- RR* e. _V
77	76, 76	xpex 4778	. . . . . . . 8 |- ( RR* X. RR* ) e. _V
78		lerelxr 8089	. . . . . . . 8 |- <_ C_ ( RR* X. RR* )
79	77, 78	ssexi 4171	. . . . . . 7 |- <_ e. _V
80		coexg 5214	. . . . . . 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 5207	. . . . . . 7 |- ( F e. _V -> `' F e. _V )
83	75, 82	syl 14	. . . . . 6 |- ( N e. NN0 -> `' F e. _V )
84		coexg 5214	. . . . . 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 2283	. . . 4 |- ( N e. NN0 -> .<_ e. _V )
87		setsex 12710	. . . 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 5655	. 2 |- ( N e. NN0 -> (ℤ/nℤ ` N ) = ( U sSet <. ( le ` ndx ) , .<_ >. ) )
90	1, 89	eqtrid 2241	1 |- ( N e. NN0 -> Y = ( U sSet <. ( le ` ndx ) , .<_ >. ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   _Vcvv 2763   [_csb 3084   ifcif 3561   {csn 3622   <.cop 3625   X. cxp 4661   `'ccnv 4662   |` cres 4665   o. ccom 4667   ` cfv 5258  (class class class)co 5922   0cc0 7879   RR*cxr 8060   <_ cle 8062   NNcn 8990   NN0cn0 9249   ZZcz 9326  ..^cfzo 10217   ndxcnx 12675   sSet csts 12676   lecple 12762   /._s cqus 12943   ~_QG cqg 13299   Ringcrg 13552  RSpancrsp 14024  ℤ_ringczring 14146   ZRHomczrh 14167  ℤ/nℤczn 14169

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-addf 8001  ax-mulf 8002

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-tp 3630  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-ec 6594  df-map 6709  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-9 9056  df-n0 9250  df-z 9327  df-dec 9458  df-uz 9602  df-rp 9729  df-fz 10084  df-cj 11007  df-abs 11164  df-struct 12680  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-starv 12770  df-sca 12771  df-vsca 12772  df-ip 12773  df-tset 12774  df-ple 12775  df-ds 12777  df-unif 12778  df-0g 12929  df-topgen 12931  df-iimas 12945  df-qus 12946  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-subg 13300  df-eqg 13302  df-cmn 13416  df-mgp 13477  df-ur 13516  df-ring 13554  df-cring 13555  df-rhm 13708  df-subrg 13775  df-lsp 13943  df-sra 13991  df-rgmod 13992  df-rsp 14026  df-bl 14102  df-mopn 14103  df-fg 14105  df-metu 14106  df-cnfld 14113  df-zring 14147  df-zrh 14170  df-zn 14172

This theorem is referenced by:  znle  14193  znval2  14194  znbaslemnn  14195