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Theorem znval 21078
Description: The value of the β„€/nβ„€ structure. It is defined as the quotient ring β„€ / 𝑛℀, with an "artificial" ordering added to make it a Toset. (In other words, β„€/nβ„€ 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 = (RSpanβ€˜β„€ring)
znval.u π‘ˆ = (β„€ring /s (β„€ring ~QG (π‘†β€˜{𝑁})))
znval.y π‘Œ = (β„€/nβ„€β€˜π‘)
znval.f 𝐹 = ((β„€RHomβ€˜π‘ˆ) β†Ύ π‘Š)
znval.w π‘Š = if(𝑁 = 0, β„€, (0..^𝑁))
znval.l ≀ = ((𝐹 ∘ ≀ ) ∘ ◑𝐹)
Assertion
Ref Expression
znval (𝑁 ∈ β„•0 β†’ π‘Œ = (π‘ˆ sSet ⟨(leβ€˜ndx), ≀ ⟩))
Proof of Theorem znval
Dummy variables 𝑓 𝑛 𝑠 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 znval.y . 2 π‘Œ = (β„€/nβ„€β€˜π‘)
2 zringring 21012 . . . . 5 β„€ring ∈ Ring
32a1i 11 . . . 4 (𝑛 = 𝑁 β†’ β„€ring ∈ Ring)
4 ovexd 7440 . . . . 5 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛}))) ∈ V)
5 id 22 . . . . . . 7 (𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛}))) β†’ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛}))))
6 simpr 485 . . . . . . . . 9 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ 𝑧 = β„€ring)
76fveq2d 6892 . . . . . . . . . . . 12 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ (RSpanβ€˜π‘§) = (RSpanβ€˜β„€ring))
8 znval.s . . . . . . . . . . . 12 𝑆 = (RSpanβ€˜β„€ring)
97, 8eqtr4di 2790 . . . . . . . . . . 11 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ (RSpanβ€˜π‘§) = 𝑆)
10 simpl 483 . . . . . . . . . . . 12 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ 𝑛 = 𝑁)
1110sneqd 4639 . . . . . . . . . . 11 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ {𝑛} = {𝑁})
129, 11fveq12d 6895 . . . . . . . . . 10 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ ((RSpanβ€˜π‘§)β€˜{𝑛}) = (π‘†β€˜{𝑁}))
136, 12oveq12d 7423 . . . . . . . . 9 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})) = (β„€ring ~QG (π‘†β€˜{𝑁})))
146, 13oveq12d 7423 . . . . . . . 8 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛}))) = (β„€ring /s (β„€ring ~QG (π‘†β€˜{𝑁}))))
15 znval.u . . . . . . . 8 π‘ˆ = (β„€ring /s (β„€ring ~QG (π‘†β€˜{𝑁})))
1614, 15eqtr4di 2790 . . . . . . 7 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛}))) = π‘ˆ)
175, 16sylan9eqr 2794 . . . . . 6 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ 𝑠 = π‘ˆ)
18 fvex 6901 . . . . . . . . . 10 (β„€RHomβ€˜π‘ ) ∈ V
1918resex 6027 . . . . . . . . 9 ((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛))) ∈ V
2019a1i 11 . . . . . . . 8 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ ((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛))) ∈ V)
21 id 22 . . . . . . . . . . . 12 (𝑓 = ((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛))) β†’ 𝑓 = ((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛))))
2217fveq2d 6892 . . . . . . . . . . . . . 14 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ (β„€RHomβ€˜π‘ ) = (β„€RHomβ€˜π‘ˆ))
23 simpll 765 . . . . . . . . . . . . . . . . 17 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ 𝑛 = 𝑁)
2423eqeq1d 2734 . . . . . . . . . . . . . . . 16 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ (𝑛 = 0 ↔ 𝑁 = 0))
2523oveq2d 7421 . . . . . . . . . . . . . . . 16 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ (0..^𝑛) = (0..^𝑁))
2624, 25ifbieq2d 4553 . . . . . . . . . . . . . . 15 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ if(𝑛 = 0, β„€, (0..^𝑛)) = if(𝑁 = 0, β„€, (0..^𝑁)))
27 znval.w . . . . . . . . . . . . . . 15 π‘Š = if(𝑁 = 0, β„€, (0..^𝑁))
2826, 27eqtr4di 2790 . . . . . . . . . . . . . 14 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ if(𝑛 = 0, β„€, (0..^𝑛)) = π‘Š)
2922, 28reseq12d 5980 . . . . . . . . . . . . 13 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ ((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛))) = ((β„€RHomβ€˜π‘ˆ) β†Ύ π‘Š))
30 znval.f . . . . . . . . . . . . 13 𝐹 = ((β„€RHomβ€˜π‘ˆ) β†Ύ π‘Š)
3129, 30eqtr4di 2790 . . . . . . . . . . . 12 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ ((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛))) = 𝐹)
3221, 31sylan9eqr 2794 . . . . . . . . . . 11 ((((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) ∧ 𝑓 = ((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛)))) β†’ 𝑓 = 𝐹)
3332coeq1d 5859 . . . . . . . . . 10 ((((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) ∧ 𝑓 = ((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛)))) β†’ (𝑓 ∘ ≀ ) = (𝐹 ∘ ≀ ))
3432cnveqd 5873 . . . . . . . . . 10 ((((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) ∧ 𝑓 = ((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛)))) β†’ ◑𝑓 = ◑𝐹)
3533, 34coeq12d 5862 . . . . . . . . 9 ((((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) ∧ 𝑓 = ((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛)))) β†’ ((𝑓 ∘ ≀ ) ∘ ◑𝑓) = ((𝐹 ∘ ≀ ) ∘ ◑𝐹))
36 znval.l . . . . . . . . 9 ≀ = ((𝐹 ∘ ≀ ) ∘ ◑𝐹)
3735, 36eqtr4di 2790 . . . . . . . 8 ((((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) ∧ 𝑓 = ((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛)))) β†’ ((𝑓 ∘ ≀ ) ∘ ◑𝑓) = ≀ )
3820, 37csbied 3930 . . . . . . 7 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ ⦋((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛))) / π‘“β¦Œ((𝑓 ∘ ≀ ) ∘ ◑𝑓) = ≀ )
3938opeq2d 4879 . . . . . 6 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ ⟨(leβ€˜ndx), ⦋((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛))) / π‘“β¦Œ((𝑓 ∘ ≀ ) ∘ ◑𝑓)⟩ = ⟨(leβ€˜ndx), ≀ ⟩)
4017, 39oveq12d 7423 . . . . 5 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ (𝑠 sSet ⟨(leβ€˜ndx), ⦋((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛))) / π‘“β¦Œ((𝑓 ∘ ≀ ) ∘ ◑𝑓)⟩) = (π‘ˆ sSet ⟨(leβ€˜ndx), ≀ ⟩))
414, 40csbied 3930 . . . 4 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ ⦋(𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛}))) / π‘ β¦Œ(𝑠 sSet ⟨(leβ€˜ndx), ⦋((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛))) / π‘“β¦Œ((𝑓 ∘ ≀ ) ∘ ◑𝑓)⟩) = (π‘ˆ sSet ⟨(leβ€˜ndx), ≀ ⟩))
423, 41csbied 3930 . . 3 (𝑛 = 𝑁 β†’ ⦋℀ring / π‘§β¦Œβ¦‹(𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛}))) / π‘ β¦Œ(𝑠 sSet ⟨(leβ€˜ndx), ⦋((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛))) / π‘“β¦Œ((𝑓 ∘ ≀ ) ∘ ◑𝑓)⟩) = (π‘ˆ sSet ⟨(leβ€˜ndx), ≀ ⟩))
43 df-zn 21047 . . 3 β„€/nβ„€ = (𝑛 ∈ β„•0 ↦ ⦋℀ring / π‘§β¦Œβ¦‹(𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛}))) / π‘ β¦Œ(𝑠 sSet ⟨(leβ€˜ndx), ⦋((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛))) / π‘“β¦Œ((𝑓 ∘ ≀ ) ∘ ◑𝑓)⟩))
44 ovex 7438 . . 3 (π‘ˆ sSet ⟨(leβ€˜ndx), ≀ ⟩) ∈ V
4542, 43, 44fvmpt 6995 . 2 (𝑁 ∈ β„•0 β†’ (β„€/nβ„€β€˜π‘) = (π‘ˆ sSet ⟨(leβ€˜ndx), ≀ ⟩))
461, 45eqtrid 2784 1 (𝑁 ∈ β„•0 β†’ π‘Œ = (π‘ˆ sSet ⟨(leβ€˜ndx), ≀ ⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  β¦‹csb 3892  ifcif 4527  {csn 4627  βŸ¨cop 4633  β—‘ccnv 5674   β†Ύ cres 5677   ∘ ccom 5679  β€˜cfv 6540  (class class class)co 7405  0cc0 11106   ≀ cle 11245  β„•0cn0 12468  β„€cz 12554  ..^cfzo 13623   sSet csts 17092  ndxcnx 17122  lecple 17200   /s cqus 17447   ~QG cqg 18996  Ringcrg 20049  RSpancrsp 20776  β„€ringczring 21009  β„€RHomczrh 21040  β„€/nβ„€czn 21043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-addf 11185  ax-mulf 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-starv 17208  df-tset 17212  df-ple 17213  df-ds 17215  df-unif 17216  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-subg 18997  df-cmn 19644  df-mgp 19982  df-ur 19999  df-ring 20051  df-cring 20052  df-subrg 20353  df-cnfld 20937  df-zring 21010  df-zn 21047
This theorem is referenced by:  znle  21079  znval2  21080
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