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Theorem znval 21087
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 21020 . . . . 5 β„€ring ∈ Ring
32a1i 11 . . . 4 (𝑛 = 𝑁 β†’ β„€ring ∈ Ring)
4 ovexd 7444 . . . . 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 6896 . . . . . . . . . . . 12 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ (RSpanβ€˜π‘§) = (RSpanβ€˜β„€ring))
8 znval.s . . . . . . . . . . . 12 𝑆 = (RSpanβ€˜β„€ring)
97, 8eqtr4di 2791 . . . . . . . . . . 11 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ (RSpanβ€˜π‘§) = 𝑆)
10 simpl 484 . . . . . . . . . . . 12 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ 𝑛 = 𝑁)
1110sneqd 4641 . . . . . . . . . . 11 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ {𝑛} = {𝑁})
129, 11fveq12d 6899 . . . . . . . . . 10 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ ((RSpanβ€˜π‘§)β€˜{𝑛}) = (π‘†β€˜{𝑁}))
136, 12oveq12d 7427 . . . . . . . . 9 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})) = (β„€ring ~QG (π‘†β€˜{𝑁})))
146, 13oveq12d 7427 . . . . . . . 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 2791 . . . . . . 7 ((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) β†’ (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛}))) = π‘ˆ)
175, 16sylan9eqr 2795 . . . . . 6 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ 𝑠 = π‘ˆ)
18 fvex 6905 . . . . . . . . . 10 (β„€RHomβ€˜π‘ ) ∈ V
1918resex 6030 . . . . . . . . 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 6896 . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . 16 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ (𝑛 = 0 ↔ 𝑁 = 0))
2523oveq2d 7425 . . . . . . . . . . . . . . . 16 (((𝑛 = 𝑁 ∧ 𝑧 = β„€ring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛})))) β†’ (0..^𝑛) = (0..^𝑁))
2624, 25ifbieq2d 4555 . . . . . . . . . . . . . . 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 2791 . . . . . . . . . . . . . 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 5983 . . . . . . . . . . . . 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 2791 . . . . . . . . . . . 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 2795 . . . . . . . . . . 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 5862 . . . . . . . . . 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 5876 . . . . . . . . . 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 5865 . . . . . . . . 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 2791 . . . . . . . 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 3932 . . . . . . 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 4881 . . . . . 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 7427 . . . . 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 3932 . . . 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 3932 . . 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 21056 . . 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 7442 . . 3 (π‘ˆ sSet ⟨(leβ€˜ndx), ≀ ⟩) ∈ V
4542, 43, 44fvmpt 6999 . 2 (𝑁 ∈ β„•0 β†’ (β„€/nβ„€β€˜π‘) = (π‘ˆ sSet ⟨(leβ€˜ndx), ≀ ⟩))
461, 45eqtrid 2785 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 3475  β¦‹csb 3894  ifcif 4529  {csn 4629  βŸ¨cop 4635  β—‘ccnv 5676   β†Ύ cres 5679   ∘ ccom 5681  β€˜cfv 6544  (class class class)co 7409  0cc0 11110   ≀ cle 11249  β„•0cn0 12472  β„€cz 12558  ..^cfzo 13627   sSet csts 17096  ndxcnx 17126  lecple 17204   /s cqus 17451   ~QG cqg 19002  Ringcrg 20056  RSpancrsp 20784  β„€ringczring 21017  β„€RHomczrh 21049  β„€/nβ„€czn 21052
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 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-addf 11189  ax-mulf 11190
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-fz 13485  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-starv 17212  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-subg 19003  df-cmn 19650  df-mgp 19988  df-ur 20005  df-ring 20058  df-cring 20059  df-subrg 20317  df-cnfld 20945  df-zring 21018  df-zn 21056
This theorem is referenced by:  znle  21088  znval2  21089
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