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Theorem znval 20954
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 20888 . . . . 5 β„€ring ∈ Ring
32a1i 11 . . . 4 (𝑛 = 𝑁 β†’ β„€ring ∈ Ring)
4 ovexd 7397 . . . . 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 486 . . . . . . . . 9 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ 𝑧 = β„€ring)
76fveq2d 6851 . . . . . . . . . . . 12 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ (RSpanβ€˜π‘§) = (RSpanβ€˜β„€ring))
8 znval.s . . . . . . . . . . . 12 𝑆 = (RSpanβ€˜β„€ring)
97, 8eqtr4di 2795 . . . . . . . . . . 11 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ (RSpanβ€˜π‘§) = 𝑆)
10 simpl 484 . . . . . . . . . . . 12 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ 𝑛 = 𝑁)
1110sneqd 4603 . . . . . . . . . . 11 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ {𝑛} = {𝑁})
129, 11fveq12d 6854 . . . . . . . . . 10 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ ((RSpanβ€˜π‘§)β€˜{𝑛}) = (π‘†β€˜{𝑁}))
136, 12oveq12d 7380 . . . . . . . . 9 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})) = (β„€ring ~QG (π‘†β€˜{𝑁})))
146, 13oveq12d 7380 . . . . . . . 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 2795 . . . . . . 7 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛}))) = π‘ˆ)
175, 16sylan9eqr 2799 . . . . . 6 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ 𝑠 = π‘ˆ)
18 fvex 6860 . . . . . . . . . 10 (β„€RHomβ€˜π‘ ) ∈ V
1918resex 5990 . . . . . . . . 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 6851 . . . . . . . . . . . . . 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 766 . . . . . . . . . . . . . . . . 17 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ 𝑛 = 𝑁)
2423eqeq1d 2739 . . . . . . . . . . . . . . . 16 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ (𝑛 = 0 ↔ 𝑁 = 0))
2523oveq2d 7378 . . . . . . . . . . . . . . . 16 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ (0..^𝑛) = (0..^𝑁))
2624, 25ifbieq2d 4517 . . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . 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 5943 . . . . . . . . . . . . 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 2795 . . . . . . . . . . . 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 2799 . . . . . . . . . . 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 5822 . . . . . . . . . 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 5836 . . . . . . . . . 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 5825 . . . . . . . . 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 2795 . . . . . . . 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 3898 . . . . . . 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 4842 . . . . . 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 7380 . . . . 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 3898 . . . 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 3898 . . 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 20923 . . 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 7395 . . 3 (π‘ˆ sSet ⟨(leβ€˜ndx), ≀ ⟩) ∈ V
4542, 43, 44fvmpt 6953 . 2 (𝑁 ∈ β„•0 β†’ (β„€/nβ„€β€˜π‘) = (π‘ˆ sSet ⟨(leβ€˜ndx), ≀ ⟩))
461, 45eqtrid 2789 1 (𝑁 ∈ β„•0 β†’ π‘Œ = (π‘ˆ sSet ⟨(leβ€˜ndx), ≀ ⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3448  β¦‹csb 3860  ifcif 4491  {csn 4591  βŸ¨cop 4597  β—‘ccnv 5637   β†Ύ cres 5640   ∘ ccom 5642  β€˜cfv 6501  (class class class)co 7362  0cc0 11058   ≀ cle 11197  β„•0cn0 12420  β„€cz 12506  ..^cfzo 13574   sSet csts 17042  ndxcnx 17072  lecple 17147   /s cqus 17394   ~QG cqg 18931  Ringcrg 19971  RSpancrsp 20648  β„€ringczring 20885  β„€RHomczrh 20916  β„€/nβ„€czn 20919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-addf 11137  ax-mulf 11138
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-z 12507  df-dec 12626  df-uz 12771  df-fz 13432  df-struct 17026  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-mulr 17154  df-starv 17155  df-tset 17159  df-ple 17160  df-ds 17162  df-unif 17163  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-grp 18758  df-minusg 18759  df-subg 18932  df-cmn 19571  df-mgp 19904  df-ur 19921  df-ring 19973  df-cring 19974  df-subrg 20236  df-cnfld 20813  df-zring 20886  df-zn 20923
This theorem is referenced by:  znle  20955  znval2  20956
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