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Theorem znval 14192
Description: The value of the ℤ/n structure. It is defined as the quotient ring ℤ / 𝑛, with an "artificial" ordering added. (In other words, ℤ/n 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, ℤ, (0..^𝑁))
znval.l = ((𝐹 ∘ ≤ ) ∘ 𝐹)
Assertion
Ref Expression
znval (𝑁 ∈ ℕ0𝑌 = (𝑈 sSet ⟨(le‘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 𝑌 = (ℤ/nℤ‘𝑁)
2 df-zn 14172 . . 3 ℤ/nℤ = (𝑛 ∈ ℕ0ring / 𝑧(𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) / 𝑠(𝑠 sSet ⟨(le‘ndx), ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓((𝑓 ∘ ≤ ) ∘ 𝑓)⟩))
3 zringring 14149 . . . . 5 ring ∈ Ring
43a1i 9 . . . 4 (𝑛 = 𝑁 → ℤring ∈ Ring)
5 vex 2766 . . . . . . 7 𝑧 ∈ V
6 rspex 14030 . . . . . . . . . 10 (𝑧 ∈ V → (RSpan‘𝑧) ∈ V)
76elv 2767 . . . . . . . . 9 (RSpan‘𝑧) ∈ V
8 vex 2766 . . . . . . . . . 10 𝑛 ∈ V
98snex 4218 . . . . . . . . 9 {𝑛} ∈ V
107, 9fvex 5578 . . . . . . . 8 ((RSpan‘𝑧)‘{𝑛}) ∈ V
11 eqgex 13351 . . . . . . . 8 ((𝑧 ∈ V ∧ ((RSpan‘𝑧)‘{𝑛}) ∈ V) → (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) ∈ V)
125, 10, 11mp2an 426 . . . . . . 7 (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) ∈ V
13 qusex 12968 . . . . . . 7 ((𝑧 ∈ V ∧ (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) ∈ V) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) ∈ V)
145, 12, 13mp2an 426 . . . . . 6 (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) ∈ V
1514a1i 9 . . . . 5 ((𝑛 = 𝑁𝑧 = ℤring) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) ∈ 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 = 𝑁𝑧 = ℤring) → 𝑧 = ℤring)
1817fveq2d 5562 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑧 = ℤring) → (RSpan‘𝑧) = (RSpan‘ℤring))
19 znval.s . . . . . . . . . . . 12 𝑆 = (RSpan‘ℤring)
2018, 19eqtr4di 2247 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑧 = ℤring) → (RSpan‘𝑧) = 𝑆)
21 simpl 109 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑧 = ℤring) → 𝑛 = 𝑁)
2221sneqd 3635 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑧 = ℤring) → {𝑛} = {𝑁})
2320, 22fveq12d 5565 . . . . . . . . . 10 ((𝑛 = 𝑁𝑧 = ℤring) → ((RSpan‘𝑧)‘{𝑛}) = (𝑆‘{𝑁}))
2417, 23oveq12d 5940 . . . . . . . . 9 ((𝑛 = 𝑁𝑧 = ℤring) → (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) = (ℤring ~QG (𝑆‘{𝑁})))
2517, 24oveq12d 5940 . . . . . . . 8 ((𝑛 = 𝑁𝑧 = 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})))
2725, 26eqtr4di 2247 . . . . . . 7 ((𝑛 = 𝑁𝑧 = ℤring) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) = 𝑈)
2816, 27sylan9eqr 2251 . . . . . 6 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → 𝑠 = 𝑈)
29 eqid 2196 . . . . . . . . . . . 12 (ℤRHom‘𝑠) = (ℤRHom‘𝑠)
3029zrhex 14177 . . . . . . . . . . 11 (𝑠 ∈ V → (ℤRHom‘𝑠) ∈ V)
3130elv 2767 . . . . . . . . . 10 (ℤRHom‘𝑠) ∈ V
3231resex 4987 . . . . . . . . 9 ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) ∈ V
3332a1i 9 . . . . . . . 8 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) ∈ V)
34 id 19 . . . . . . . . . . . 12 (𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) → 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))))
3528fveq2d 5562 . . . . . . . . . . . . . 14 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (ℤRHom‘𝑠) = (ℤRHom‘𝑈))
36 simpll 527 . . . . . . . . . . . . . . . . 17 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → 𝑛 = 𝑁)
3736eqeq1d 2205 . . . . . . . . . . . . . . . 16 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (𝑛 = 0 ↔ 𝑁 = 0))
3836oveq2d 5938 . . . . . . . . . . . . . . . 16 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (0..^𝑛) = (0..^𝑁))
3937, 38ifbieq2d 3585 . . . . . . . . . . . . . . 15 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → if(𝑛 = 0, ℤ, (0..^𝑛)) = if(𝑁 = 0, ℤ, (0..^𝑁)))
40 znval.w . . . . . . . . . . . . . . 15 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁))
4139, 40eqtr4di 2247 . . . . . . . . . . . . . 14 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → if(𝑛 = 0, ℤ, (0..^𝑛)) = 𝑊)
4235, 41reseq12d 4947 . . . . . . . . . . . . 13 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) = ((ℤRHom‘𝑈) ↾ 𝑊))
43 znval.f . . . . . . . . . . . . 13 𝐹 = ((ℤRHom‘𝑈) ↾ 𝑊)
4442, 43eqtr4di 2247 . . . . . . . . . . . 12 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) = 𝐹)
4534, 44sylan9eqr 2251 . . . . . . . . . . 11 ((((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → 𝑓 = 𝐹)
4645coeq1d 4827 . . . . . . . . . 10 ((((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → (𝑓 ∘ ≤ ) = (𝐹 ∘ ≤ ))
4745cnveqd 4842 . . . . . . . . . 10 ((((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → 𝑓 = 𝐹)
4846, 47coeq12d 4830 . . . . . . . . 9 ((((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → ((𝑓 ∘ ≤ ) ∘ 𝑓) = ((𝐹 ∘ ≤ ) ∘ 𝐹))
49 znval.l . . . . . . . . 9 = ((𝐹 ∘ ≤ ) ∘ 𝐹)
5048, 49eqtr4di 2247 . . . . . . . 8 ((((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → ((𝑓 ∘ ≤ ) ∘ 𝑓) = )
5133, 50csbied 3131 . . . . . . 7 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓((𝑓 ∘ ≤ ) ∘ 𝑓) = )
5251opeq2d 3815 . . . . . 6 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ⟨(le‘ndx), ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓((𝑓 ∘ ≤ ) ∘ 𝑓)⟩ = ⟨(le‘ndx), ⟩)
5328, 52oveq12d 5940 . . . . 5 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (𝑠 sSet ⟨(le‘ndx), ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓((𝑓 ∘ ≤ ) ∘ 𝑓)⟩) = (𝑈 sSet ⟨(le‘ndx), ⟩))
5415, 53csbied 3131 . . . 4 ((𝑛 = 𝑁𝑧 = ℤring) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) / 𝑠(𝑠 sSet ⟨(le‘ndx), ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓((𝑓 ∘ ≤ ) ∘ 𝑓)⟩) = (𝑈 sSet ⟨(le‘ndx), ⟩))
554, 54csbied 3131 . . 3 (𝑛 = 𝑁ring / 𝑧(𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) / 𝑠(𝑠 sSet ⟨(le‘ndx), ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓((𝑓 ∘ ≤ ) ∘ 𝑓)⟩) = (𝑈 sSet ⟨(le‘ndx), ⟩))
56 id 19 . . 3 (𝑁 ∈ ℕ0𝑁 ∈ ℕ0)
57 rspex 14030 . . . . . . . . . 10 (ℤring ∈ Ring → (RSpan‘ℤring) ∈ V)
583, 57ax-mp 5 . . . . . . . . 9 (RSpan‘ℤring) ∈ V
5919, 58eqeltri 2269 . . . . . . . 8 𝑆 ∈ V
60 snexg 4217 . . . . . . . 8 (𝑁 ∈ ℕ0 → {𝑁} ∈ V)
61 fvexg 5577 . . . . . . . 8 ((𝑆 ∈ V ∧ {𝑁} ∈ V) → (𝑆‘{𝑁}) ∈ V)
6259, 60, 61sylancr 414 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝑆‘{𝑁}) ∈ V)
63 eqgex 13351 . . . . . . 7 ((ℤring ∈ Ring ∧ (𝑆‘{𝑁}) ∈ V) → (ℤring ~QG (𝑆‘{𝑁})) ∈ V)
643, 62, 63sylancr 414 . . . . . 6 (𝑁 ∈ ℕ0 → (ℤring ~QG (𝑆‘{𝑁})) ∈ V)
65 qusex 12968 . . . . . 6 ((ℤring ∈ Ring ∧ (ℤring ~QG (𝑆‘{𝑁})) ∈ V) → (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) ∈ V)
663, 64, 65sylancr 414 . . . . 5 (𝑁 ∈ ℕ0 → (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) ∈ V)
6726, 66eqeltrid 2283 . . . 4 (𝑁 ∈ ℕ0𝑈 ∈ V)
68 plendxnn 12880 . . . . 5 (le‘ndx) ∈ ℕ
6968a1i 9 . . . 4 (𝑁 ∈ ℕ0 → (le‘ndx) ∈ ℕ)
70 eqid 2196 . . . . . . . . . . 11 (ℤRHom‘𝑈) = (ℤRHom‘𝑈)
7170zrhex 14177 . . . . . . . . . 10 (𝑈 ∈ V → (ℤRHom‘𝑈) ∈ V)
7267, 71syl 14 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (ℤRHom‘𝑈) ∈ V)
73 resexg 4986 . . . . . . . . 9 ((ℤRHom‘𝑈) ∈ V → ((ℤRHom‘𝑈) ↾ 𝑊) ∈ V)
7472, 73syl 14 . . . . . . . 8 (𝑁 ∈ ℕ0 → ((ℤRHom‘𝑈) ↾ 𝑊) ∈ V)
7543, 74eqeltrid 2283 . . . . . . 7 (𝑁 ∈ ℕ0𝐹 ∈ V)
76 xrex 9931 . . . . . . . . 9 * ∈ V
7776, 76xpex 4778 . . . . . . . 8 (ℝ* × ℝ*) ∈ V
78 lerelxr 8089 . . . . . . . 8 ≤ ⊆ (ℝ* × ℝ*)
7977, 78ssexi 4171 . . . . . . 7 ≤ ∈ V
80 coexg 5214 . . . . . . 7 ((𝐹 ∈ V ∧ ≤ ∈ V) → (𝐹 ∘ ≤ ) ∈ V)
8175, 79, 80sylancl 413 . . . . . 6 (𝑁 ∈ ℕ0 → (𝐹 ∘ ≤ ) ∈ V)
82 cnvexg 5207 . . . . . . 7 (𝐹 ∈ V → 𝐹 ∈ V)
8375, 82syl 14 . . . . . 6 (𝑁 ∈ ℕ0𝐹 ∈ V)
84 coexg 5214 . . . . . 6 (((𝐹 ∘ ≤ ) ∈ V ∧ 𝐹 ∈ V) → ((𝐹 ∘ ≤ ) ∘ 𝐹) ∈ V)
8581, 83, 84syl2anc 411 . . . . 5 (𝑁 ∈ ℕ0 → ((𝐹 ∘ ≤ ) ∘ 𝐹) ∈ V)
8649, 85eqeltrid 2283 . . . 4 (𝑁 ∈ ℕ0 ∈ V)
87 setsex 12710 . . . 4 ((𝑈 ∈ V ∧ (le‘ndx) ∈ ℕ ∧ ∈ V) → (𝑈 sSet ⟨(le‘ndx), ⟩) ∈ V)
8867, 69, 86, 87syl3anc 1249 . . 3 (𝑁 ∈ ℕ0 → (𝑈 sSet ⟨(le‘ndx), ⟩) ∈ V)
892, 55, 56, 88fvmptd3 5655 . 2 (𝑁 ∈ ℕ0 → (ℤ/nℤ‘𝑁) = (𝑈 sSet ⟨(le‘ndx), ⟩))
901, 89eqtrid 2241 1 (𝑁 ∈ ℕ0𝑌 = (𝑈 sSet ⟨(le‘ndx), ⟩))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2167  Vcvv 2763  csb 3084  ifcif 3561  {csn 3622  cop 3625   × cxp 4661  ccnv 4662  cres 4665  ccom 4667  cfv 5258  (class class class)co 5922  0cc0 7879  *cxr 8060  cle 8062  cn 8990  0cn0 9249  cz 9326  ..^cfzo 10217  ndxcnx 12675   sSet csts 12676  lecple 12762   /s cqus 12943   ~QG cqg 13299  Ringcrg 13552  RSpancrsp 14024  ringczring 14146  ℤRHomczrh 14167  ℤ/nczn 14169
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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-addf 8001  ax-mulf 8002
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-tp 3630  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-ec 6594  df-map 6709  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-9 9056  df-n0 9250  df-z 9327  df-dec 9458  df-uz 9602  df-rp 9729  df-fz 10084  df-cj 11007  df-abs 11164  df-struct 12680  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-starv 12770  df-sca 12771  df-vsca 12772  df-ip 12773  df-tset 12774  df-ple 12775  df-ds 12777  df-unif 12778  df-0g 12929  df-topgen 12931  df-iimas 12945  df-qus 12946  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-subg 13300  df-eqg 13302  df-cmn 13416  df-mgp 13477  df-ur 13516  df-ring 13554  df-cring 13555  df-rhm 13708  df-subrg 13775  df-lsp 13943  df-sra 13991  df-rgmod 13992  df-rsp 14026  df-bl 14102  df-mopn 14103  df-fg 14105  df-metu 14106  df-cnfld 14113  df-zring 14147  df-zrh 14170  df-zn 14172
This theorem is referenced by:  znle  14193  znval2  14194  znbaslemnn  14195
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