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Theorem cply1coe0bi 20403
Description: A polynomial is constant (i.e. a "lifted scalar") iff all but the first coefficient are zero. (Contributed by AV, 16-Nov-2019.)
Hypotheses
Ref Expression
cply1coe0.k 𝐾 = (Base‘𝑅)
cply1coe0.0 0 = (0g𝑅)
cply1coe0.p 𝑃 = (Poly1𝑅)
cply1coe0.b 𝐵 = (Base‘𝑃)
cply1coe0.a 𝐴 = (algSc‘𝑃)
Assertion
Ref Expression
cply1coe0bi ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (∃𝑠𝐾 𝑀 = (𝐴𝑠) ↔ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ))
Distinct variable groups:   𝑛,𝐾   𝑅,𝑛   𝐴,𝑛,𝑠   𝐵,𝑛,𝑠   𝐾,𝑠   𝑛,𝑀,𝑠   𝑅,𝑠   0 ,𝑠
Allowed substitution hints:   𝑃(𝑛,𝑠)   0 (𝑛)

Proof of Theorem cply1coe0bi
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 cply1coe0.k . . . . . 6 𝐾 = (Base‘𝑅)
2 cply1coe0.0 . . . . . 6 0 = (0g𝑅)
3 cply1coe0.p . . . . . 6 𝑃 = (Poly1𝑅)
4 cply1coe0.b . . . . . 6 𝐵 = (Base‘𝑃)
5 cply1coe0.a . . . . . 6 𝐴 = (algSc‘𝑃)
61, 2, 3, 4, 5cply1coe0 20402 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑠𝐾) → ∀𝑛 ∈ ℕ ((coe1‘(𝐴𝑠))‘𝑛) = 0 )
76ad4ant13 747 . . . 4 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑠𝐾) ∧ 𝑀 = (𝐴𝑠)) → ∀𝑛 ∈ ℕ ((coe1‘(𝐴𝑠))‘𝑛) = 0 )
8 fveq2 6669 . . . . . . . 8 (𝑀 = (𝐴𝑠) → (coe1𝑀) = (coe1‘(𝐴𝑠)))
98fveq1d 6671 . . . . . . 7 (𝑀 = (𝐴𝑠) → ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴𝑠))‘𝑛))
109eqeq1d 2828 . . . . . 6 (𝑀 = (𝐴𝑠) → (((coe1𝑀)‘𝑛) = 0 ↔ ((coe1‘(𝐴𝑠))‘𝑛) = 0 ))
1110ralbidv 3202 . . . . 5 (𝑀 = (𝐴𝑠) → (∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝐴𝑠))‘𝑛) = 0 ))
1211adantl 482 . . . 4 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑠𝐾) ∧ 𝑀 = (𝐴𝑠)) → (∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝐴𝑠))‘𝑛) = 0 ))
137, 12mpbird 258 . . 3 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑠𝐾) ∧ 𝑀 = (𝐴𝑠)) → ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 )
1413rexlimdva2 3292 . 2 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (∃𝑠𝐾 𝑀 = (𝐴𝑠) → ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ))
15 simpr 485 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑀𝐵)
16 0nn0 11906 . . . . . 6 0 ∈ ℕ0
17 eqid 2826 . . . . . . 7 (coe1𝑀) = (coe1𝑀)
1817, 4, 3, 1coe1fvalcl 20315 . . . . . 6 ((𝑀𝐵 ∧ 0 ∈ ℕ0) → ((coe1𝑀)‘0) ∈ 𝐾)
1915, 16, 18sylancl 586 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ((coe1𝑀)‘0) ∈ 𝐾)
2019adantr 481 . . . 4 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) → ((coe1𝑀)‘0) ∈ 𝐾)
21 fveq2 6669 . . . . . 6 (𝑠 = ((coe1𝑀)‘0) → (𝐴𝑠) = (𝐴‘((coe1𝑀)‘0)))
2221eqeq2d 2837 . . . . 5 (𝑠 = ((coe1𝑀)‘0) → (𝑀 = (𝐴𝑠) ↔ 𝑀 = (𝐴‘((coe1𝑀)‘0))))
2322adantl 482 . . . 4 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) ∧ 𝑠 = ((coe1𝑀)‘0)) → (𝑀 = (𝐴𝑠) ↔ 𝑀 = (𝐴‘((coe1𝑀)‘0))))
24 simpl 483 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
25 eqid 2826 . . . . . . . . . 10 (Scalar‘𝑃) = (Scalar‘𝑃)
263ply1ring 20351 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
273ply1lmod 20355 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
28 eqid 2826 . . . . . . . . . 10 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
295, 25, 26, 27, 28, 4asclf 20046 . . . . . . . . 9 (𝑅 ∈ Ring → 𝐴:(Base‘(Scalar‘𝑃))⟶𝐵)
3029adantr 481 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝐴:(Base‘(Scalar‘𝑃))⟶𝐵)
31 eqid 2826 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
3217, 4, 3, 31coe1fvalcl 20315 . . . . . . . . . 10 ((𝑀𝐵 ∧ 0 ∈ ℕ0) → ((coe1𝑀)‘0) ∈ (Base‘𝑅))
3315, 16, 32sylancl 586 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ((coe1𝑀)‘0) ∈ (Base‘𝑅))
343ply1sca 20356 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
3534eqcomd 2832 . . . . . . . . . . 11 (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
3635fveq2d 6673 . . . . . . . . . 10 (𝑅 ∈ Ring → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
3736adantr 481 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
3833, 37eleqtrrd 2921 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ((coe1𝑀)‘0) ∈ (Base‘(Scalar‘𝑃)))
3930, 38ffvelrnd 6850 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝐴‘((coe1𝑀)‘0)) ∈ 𝐵)
4024, 15, 393jca 1122 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑅 ∈ Ring ∧ 𝑀𝐵 ∧ (𝐴‘((coe1𝑀)‘0)) ∈ 𝐵))
4140adantr 481 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) → (𝑅 ∈ Ring ∧ 𝑀𝐵 ∧ (𝐴‘((coe1𝑀)‘0)) ∈ 𝐵))
42 simpr 485 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) ∧ ((coe1𝑀)‘𝑛) = 0 ) → ((coe1𝑀)‘𝑛) = 0 )
433, 5, 1, 2coe1scl 20390 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ ((coe1𝑀)‘0) ∈ 𝐾) → (coe1‘(𝐴‘((coe1𝑀)‘0))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, ((coe1𝑀)‘0), 0 )))
4419, 43syldan 591 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (coe1‘(𝐴‘((coe1𝑀)‘0))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, ((coe1𝑀)‘0), 0 )))
4544adantr 481 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) → (coe1‘(𝐴‘((coe1𝑀)‘0))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, ((coe1𝑀)‘0), 0 )))
46 nnne0 11665 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → 𝑛 ≠ 0)
4746neneqd 3026 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
4847adantl 482 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
4948adantr 481 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → ¬ 𝑛 = 0)
50 eqeq1 2830 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑘 = 0 ↔ 𝑛 = 0))
5150notbid 319 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (¬ 𝑘 = 0 ↔ ¬ 𝑛 = 0))
5251adantl 482 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (¬ 𝑘 = 0 ↔ ¬ 𝑛 = 0))
5349, 52mpbird 258 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → ¬ 𝑘 = 0)
5453iffalsed 4481 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → if(𝑘 = 0, ((coe1𝑀)‘0), 0 ) = 0 )
55 nnnn0 11898 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
5655adantl 482 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
572fvexi 6683 . . . . . . . . . . . . . 14 0 ∈ V
5857a1i 11 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) → 0 ∈ V)
5945, 54, 56, 58fvmptd 6773 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) → ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛) = 0 )
6059eqcomd 2832 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) → 0 = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛))
6160adantr 481 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) ∧ ((coe1𝑀)‘𝑛) = 0 ) → 0 = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛))
6242, 61eqtrd 2861 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) ∧ ((coe1𝑀)‘𝑛) = 0 ) → ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛))
6362ex 413 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) → (((coe1𝑀)‘𝑛) = 0 → ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛)))
6463ralimdva 3182 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 → ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛)))
6564imp 407 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) → ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛))
663, 5, 1ply1sclid 20391 . . . . . . . 8 ((𝑅 ∈ Ring ∧ ((coe1𝑀)‘0) ∈ 𝐾) → ((coe1𝑀)‘0) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘0))
6719, 66syldan 591 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ((coe1𝑀)‘0) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘0))
6867adantr 481 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) → ((coe1𝑀)‘0) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘0))
69 df-n0 11892 . . . . . . . 8 0 = (ℕ ∪ {0})
7069raleqi 3419 . . . . . . 7 (∀𝑛 ∈ ℕ0 ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛) ↔ ∀𝑛 ∈ (ℕ ∪ {0})((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛))
71 c0ex 10629 . . . . . . . 8 0 ∈ V
72 fveq2 6669 . . . . . . . . . 10 (𝑛 = 0 → ((coe1𝑀)‘𝑛) = ((coe1𝑀)‘0))
73 fveq2 6669 . . . . . . . . . 10 (𝑛 = 0 → ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘0))
7472, 73eqeq12d 2842 . . . . . . . . 9 (𝑛 = 0 → (((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛) ↔ ((coe1𝑀)‘0) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘0)))
7574ralunsn 4823 . . . . . . . 8 (0 ∈ V → (∀𝑛 ∈ (ℕ ∪ {0})((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛) ↔ (∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛) ∧ ((coe1𝑀)‘0) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘0))))
7671, 75mp1i 13 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) → (∀𝑛 ∈ (ℕ ∪ {0})((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛) ↔ (∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛) ∧ ((coe1𝑀)‘0) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘0))))
7770, 76syl5bb 284 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) → (∀𝑛 ∈ ℕ0 ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛) ↔ (∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛) ∧ ((coe1𝑀)‘0) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘0))))
7865, 68, 77mpbir2and 709 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) → ∀𝑛 ∈ ℕ0 ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛))
79 eqid 2826 . . . . . 6 (coe1‘(𝐴‘((coe1𝑀)‘0))) = (coe1‘(𝐴‘((coe1𝑀)‘0)))
803, 4, 17, 79eqcoe1ply1eq 20400 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑀𝐵 ∧ (𝐴‘((coe1𝑀)‘0)) ∈ 𝐵) → (∀𝑛 ∈ ℕ0 ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛) → 𝑀 = (𝐴‘((coe1𝑀)‘0))))
8141, 78, 80sylc 65 . . . 4 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) → 𝑀 = (𝐴‘((coe1𝑀)‘0)))
8220, 23, 81rspcedvd 3630 . . 3 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) → ∃𝑠𝐾 𝑀 = (𝐴𝑠))
8382ex 413 . 2 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 → ∃𝑠𝐾 𝑀 = (𝐴𝑠)))
8414, 83impbid 213 1 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (∃𝑠𝐾 𝑀 = (𝐴𝑠) ↔ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3143  wrex 3144  Vcvv 3500  cun 3938  ifcif 4470  {csn 4564  cmpt 5143  wf 6350  cfv 6354  0cc0 10531  cn 11632  0cn0 11891  Basecbs 16478  Scalarcsca 16563  0gc0g 16708  Ringcrg 19233  algSccascl 20019  Poly1cpl1 20280  coe1cco1 20281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-ofr 7404  df-om 7574  df-1st 7685  df-2nd 7686  df-supp 7827  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8284  df-map 8403  df-pm 8404  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fsupp 8828  df-oi 8968  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-fz 12888  df-fzo 13029  df-seq 13365  df-hash 13686  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-sca 16576  df-vsca 16577  df-tset 16579  df-ple 16580  df-0g 16710  df-gsum 16711  df-mre 16852  df-mrc 16853  df-acs 16855  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-mhm 17951  df-submnd 17952  df-grp 18051  df-minusg 18052  df-sbg 18053  df-mulg 18170  df-subg 18221  df-ghm 18301  df-cntz 18392  df-cmn 18844  df-abl 18845  df-mgp 19176  df-ur 19188  df-srg 19192  df-ring 19235  df-subrg 19469  df-lmod 19572  df-lss 19640  df-ascl 20022  df-psr 20071  df-mvr 20072  df-mpl 20073  df-opsr 20075  df-psr1 20283  df-vr1 20284  df-ply1 20285  df-coe1 20286
This theorem is referenced by:  cpmatel2  21256
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