Theorem znval 14732

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 14712	. . 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 14689	. . . . 5 |- ℤring e. Ring
4	3	a1i 9	. . . 4 |- ( n = N -> ℤring e. Ring )
5		vex 2806	. . . . . . 7 |- z e. _V
6		rspex 14570	. . . . . . . . . 10 |- ( z e. _V -> (RSpan ` z ) e. _V )
7	6	elv 2807	. . . . . . . . 9 |- (RSpan ` z ) e. _V
8		vex 2806	. . . . . . . . . 10 |- n e. _V
9	8	snex 4281	. . . . . . . . 9 |- { n } e. _V
10	7, 9	fvex 5668	. . . . . . . 8 |- ( (RSpan ` z ) ` { n } ) e. _V
11		eqgex 13888	. . . . . . . 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 13488	. . . . . . 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 5652	. . . . . . . . . . . 12 |- ( ( n = N /\ z = ℤring ) -> (RSpan ` z ) = (RSpan ` ℤring ) )
19		znval.s	. . . . . . . . . . . 12 |- S = (RSpan ` ℤring )
20	18, 19	eqtr4di 2282	. . . . . . . . . . 11 |- ( ( n = N /\ z = ℤring ) -> (RSpan ` z ) = S )
21		simpl 109	. . . . . . . . . . . 12 |- ( ( n = N /\ z = ℤring ) -> n = N )
22	21	sneqd 3686	. . . . . . . . . . 11 |- ( ( n = N /\ z = ℤring ) -> { n } = { N } )
23	20, 22	fveq12d 5655	. . . . . . . . . 10 |- ( ( n = N /\ z = ℤring ) -> ( (RSpan ` z ) ` { n } ) = ( S ` { N } ) )
24	17, 23	oveq12d 6046	. . . . . . . . 9 |- ( ( n = N /\ z = ℤring ) -> ( z ~QG ( (RSpan ` z ) ` { n } ) ) = (ℤring ~QG ( S ` { N } ) ) )
25	17, 24	oveq12d 6046	. . . . . . . 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 2282	. . . . . . 7 |- ( ( n = N /\ z = ℤring ) -> ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) = U )
28	16, 27	sylan9eqr 2286	. . . . . 6 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> s = U )
29		eqid 2231	. . . . . . . . . . . 12 |- ( ZRHom ` s ) = ( ZRHom ` s )
30	29	zrhex 14717	. . . . . . . . . . 11 |- ( s e. _V -> ( ZRHom ` s ) e. _V )
31	30	elv 2807	. . . . . . . . . 10 |- ( ZRHom ` s ) e. _V
32	31	resex 5060	. . . . . . . . 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 5652	. . . . . . . . . . . . . 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 2240	. . . . . . . . . . . . . . . 16 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> ( n = 0 <-> N = 0 ) )
38	36	oveq2d 6044	. . . . . . . . . . . . . . . 16 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> ( 0..^ n ) = ( 0..^ N ) )
39	37, 38	ifbieq2d 3634	. . . . . . . . . . . . . . 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 2282	. . . . . . . . . . . . . 14 |- ( ( ( n = N /\ z = ℤring ) /\ s = ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) ) -> if ( n = 0 , ZZ , ( 0..^ n ) ) = W )
42	35, 41	reseq12d 5020	. . . . . . . . . . . . 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 2282	. . . . . . . . . . . 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 2286	. . . . . . . . . . 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 4897	. . . . . . . . . 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 4912	. . . . . . . . . 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 4900	. . . . . . . . 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 2282	. . . . . . . 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 3175	. . . . . . 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 3874	. . . . . 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 6046	. . . . 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 3175	. . . 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 3175	. . 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 14570	. . . . . . . . . 10 |- (ℤring e. Ring -> (RSpan ` ℤring ) e. _V )
58	3, 57	ax-mp 5	. . . . . . . . 9 |- (RSpan ` ℤring ) e. _V
59	19, 58	eqeltri 2304	. . . . . . . 8 |- S e. _V
60		snexg 4280	. . . . . . . 8 |- ( N e. NN0 -> { N } e. _V )
61		fvexg 5667	. . . . . . . 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 13888	. . . . . . 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 13488	. . . . . 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 2318	. . . 4 |- ( N e. NN0 -> U e. _V )
68		plendxnn 13366	. . . . 5 |- ( le ` ndx ) e. NN
69	68	a1i 9	. . . 4 |- ( N e. NN0 -> ( le ` ndx ) e. NN )
70		eqid 2231	. . . . . . . . . . 11 |- ( ZRHom ` U ) = ( ZRHom ` U )
71	70	zrhex 14717	. . . . . . . . . 10 |- ( U e. _V -> ( ZRHom ` U ) e. _V )
72	67, 71	syl 14	. . . . . . . . 9 |- ( N e. NN0 -> ( ZRHom ` U ) e. _V )
73		resexg 5059	. . . . . . . . 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 2318	. . . . . . 7 |- ( N e. NN0 -> F e. _V )
76		xrex 10152	. . . . . . . . 9 |- RR* e. _V
77	76, 76	xpex 4848	. . . . . . . 8 |- ( RR* X. RR* ) e. _V
78		lerelxr 8301	. . . . . . . 8 |- <_ C_ ( RR* X. RR* )
79	77, 78	ssexi 4232	. . . . . . 7 |- <_ e. _V
80		coexg 5288	. . . . . . 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 5281	. . . . . . 7 |- ( F e. _V -> `' F e. _V )
83	75, 82	syl 14	. . . . . 6 |- ( N e. NN0 -> `' F e. _V )
84		coexg 5288	. . . . . 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 2318	. . . 4 |- ( N e. NN0 -> .<_ e. _V )
87		setsex 13194	. . . 4 |- ( ( U e. _V /\ ( le ` ndx ) e. NN /\ .<_ e. _V ) -> ( U sSet <. ( le ` ndx ) , .<_ >. ) e. _V )
88	67, 69, 86, 87	syl3anc 1274	. . 3 |- ( N e. NN0 -> ( U sSet <. ( le ` ndx ) , .<_ >. ) e. _V )
89	2, 55, 56, 88	fvmptd3 5749	. 2 |- ( N e. NN0 -> (ℤ/nℤ ` N ) = ( U sSet <. ( le ` ndx ) , .<_ >. ) )
90	1, 89	eqtrid 2276	1 |- ( N e. NN0 -> Y = ( U sSet <. ( le ` ndx ) , .<_ >. ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   [_csb 3128   ifcif 3607   {csn 3673   <.cop 3676   X. cxp 4729   `'ccnv 4730   |` cres 4733   o. ccom 4735   ` cfv 5333  (class class class)co 6028   0cc0 8092   RR*cxr 8272   <_ cle 8274   NNcn 9202   NN0cn0 9461   ZZcz 9540  ..^cfzo 10439   ndxcnx 13159   sSet csts 13160   lecple 13247   /._s cqus 13463   ~_QG cqg 13836   Ringcrg 14090  RSpancrsp 14564  ℤ_ringczring 14686   ZRHomczrh 14707  ℤ/nℤczn 14709

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-addf 8214  ax-mulf 8215

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-ec 6747  df-map 6862  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-5 9264  df-6 9265  df-7 9266  df-8 9267  df-9 9268  df-n0 9462  df-z 9541  df-dec 9673  df-uz 9817  df-rp 9950  df-fz 10306  df-cj 11482  df-abs 11639  df-struct 13164  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-mulr 13254  df-starv 13255  df-sca 13256  df-vsca 13257  df-ip 13258  df-tset 13259  df-ple 13260  df-ds 13262  df-unif 13263  df-0g 13421  df-topgen 13423  df-iimas 13465  df-qus 13466  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-subg 13837  df-eqg 13839  df-cmn 13953  df-mgp 14015  df-ur 14054  df-ring 14092  df-cring 14093  df-rhm 14247  df-subrg 14314  df-lsp 14483  df-sra 14531  df-rgmod 14532  df-rsp 14566  df-bl 14642  df-mopn 14643  df-fg 14645  df-metu 14646  df-cnfld 14653  df-zring 14687  df-zrh 14710  df-zn 14712

This theorem is referenced by:  znle  14733  znval2  14734  znbaslemnn  14735