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Theorem znval 16847
 Description: The value of the ℤ/nℤ structure. It is defined as the quotient ring , with an "artificial" ordering added to make it a Toset. (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.)
Hypotheses
Ref Expression
znval.z flds
znval.s RSpan
znval.u s ~QG
znval.y ℤ/n
znval.f RHom
znval.w ..^
znval.l
Assertion
Ref Expression
znval sSet

Proof of Theorem znval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 znval.y . 2 ℤ/n
2 ovex 6135 . . . . 5 flds
32a1i 11 . . . 4 flds
4 ovex 6135 . . . . . 6 s ~QG RSpan
54a1i 11 . . . . 5 flds s ~QG RSpan
6 id 21 . . . . . . 7 s ~QG RSpan s ~QG RSpan
7 simpr 449 . . . . . . . . . 10 flds flds
8 znval.z . . . . . . . . . 10 flds
97, 8syl6eqr 2492 . . . . . . . . 9 flds
109fveq2d 5761 . . . . . . . . . . . 12 flds RSpan RSpan
11 znval.s . . . . . . . . . . . 12 RSpan
1210, 11syl6eqr 2492 . . . . . . . . . . 11 flds RSpan
13 simpl 445 . . . . . . . . . . . 12 flds
1413sneqd 3851 . . . . . . . . . . 11 flds
1512, 14fveq12d 5763 . . . . . . . . . 10 flds RSpan
169, 15oveq12d 6128 . . . . . . . . 9 flds ~QG RSpan ~QG
179, 16oveq12d 6128 . . . . . . . 8 flds s ~QG RSpan s ~QG
18 znval.u . . . . . . . 8 s ~QG
1917, 18syl6eqr 2492 . . . . . . 7 flds s ~QG RSpan
206, 19sylan9eqr 2496 . . . . . 6 flds s ~QG RSpan
21 fvex 5771 . . . . . . . . . 10 RHom
2221resex 5215 . . . . . . . . 9 RHom ..^
2322a1i 11 . . . . . . . 8 flds s ~QG RSpan RHom ..^
24 id 21 . . . . . . . . . . . 12 RHom ..^ RHom ..^
2520fveq2d 5761 . . . . . . . . . . . . . 14 flds s ~QG RSpan RHom RHom
26 simpll 732 . . . . . . . . . . . . . . . . 17 flds s ~QG RSpan
2726eqeq1d 2450 . . . . . . . . . . . . . . . 16 flds s ~QG RSpan
2826oveq2d 6126 . . . . . . . . . . . . . . . 16 flds s ~QG RSpan ..^ ..^
2927, 28ifbieq2d 3783 . . . . . . . . . . . . . . 15 flds s ~QG RSpan ..^ ..^
30 znval.w . . . . . . . . . . . . . . 15 ..^
3129, 30syl6eqr 2492 . . . . . . . . . . . . . 14 flds s ~QG RSpan ..^
3225, 31reseq12d 5176 . . . . . . . . . . . . 13 flds s ~QG RSpan RHom ..^ RHom
33 znval.f . . . . . . . . . . . . 13 RHom
3432, 33syl6eqr 2492 . . . . . . . . . . . 12 flds s ~QG RSpan RHom ..^
3524, 34sylan9eqr 2496 . . . . . . . . . . 11 flds s ~QG RSpan RHom ..^
3635coeq1d 5063 . . . . . . . . . 10 flds s ~QG RSpan RHom ..^
3735cnveqd 5077 . . . . . . . . . 10 flds s ~QG RSpan RHom ..^
3836, 37coeq12d 5066 . . . . . . . . 9 flds s ~QG RSpan RHom ..^
39 znval.l . . . . . . . . 9
4038, 39syl6eqr 2492 . . . . . . . 8 flds s ~QG RSpan RHom ..^
4123, 40csbied 3292 . . . . . . 7 flds s ~QG RSpan RHom ..^
4241opeq2d 4015 . . . . . 6 flds s ~QG RSpan RHom ..^
4320, 42oveq12d 6128 . . . . 5 flds s ~QG RSpan sSet RHom ..^ sSet
445, 43csbied 3292 . . . 4 flds s ~QG RSpan sSet RHom ..^ sSet
453, 44csbied 3292 . . 3 flds s ~QG RSpan sSet RHom ..^ sSet
46 df-zn 16816 . . 3 ℤ/n flds s ~QG RSpan sSet RHom ..^
47 ovex 6135 . . 3 sSet
4845, 46, 47fvmpt 5835 . 2 ℤ/n sSet
491, 48syl5eq 2486 1 sSet
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1727  cvv 2962  csb 3267  cif 3763  csn 3838  cop 3841  ccnv 4906   cres 4909   ccom 4911  cfv 5483  (class class class)co 6110  cc0 9021   cle 9152  cn0 10252  cz 10313  ..^cfzo 11166  cnx 13497   sSet csts 13498   ↾s cress 13501  cple 13567   s cqus 13762   ~QG cqg 14971  RSpancrsp 16274  ℂfldccnfld 16734  RHomczrh 16809  ℤ/nℤczn 16812 This theorem is referenced by:  znle  16848  znval2  16849 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-res 4919  df-iota 5447  df-fun 5485  df-fv 5491  df-ov 6113  df-zn 16816
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