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Theorem cply1coe0bi 19491
 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 simpl 472 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
21anim1i 590 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑠𝐾) → (𝑅 ∈ Ring ∧ 𝑠𝐾))
32adantr 480 . . . . . 6 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑠𝐾) ∧ 𝑀 = (𝐴𝑠)) → (𝑅 ∈ Ring ∧ 𝑠𝐾))
4 cply1coe0.k . . . . . . 7 𝐾 = (Base‘𝑅)
5 cply1coe0.0 . . . . . . 7 0 = (0g𝑅)
6 cply1coe0.p . . . . . . 7 𝑃 = (Poly1𝑅)
7 cply1coe0.b . . . . . . 7 𝐵 = (Base‘𝑃)
8 cply1coe0.a . . . . . . 7 𝐴 = (algSc‘𝑃)
94, 5, 6, 7, 8cply1coe0 19490 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑠𝐾) → ∀𝑛 ∈ ℕ ((coe1‘(𝐴𝑠))‘𝑛) = 0 )
103, 9syl 17 . . . . 5 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑠𝐾) ∧ 𝑀 = (𝐴𝑠)) → ∀𝑛 ∈ ℕ ((coe1‘(𝐴𝑠))‘𝑛) = 0 )
11 fveq2 6103 . . . . . . . . 9 (𝑀 = (𝐴𝑠) → (coe1𝑀) = (coe1‘(𝐴𝑠)))
1211fveq1d 6105 . . . . . . . 8 (𝑀 = (𝐴𝑠) → ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴𝑠))‘𝑛))
1312eqeq1d 2612 . . . . . . 7 (𝑀 = (𝐴𝑠) → (((coe1𝑀)‘𝑛) = 0 ↔ ((coe1‘(𝐴𝑠))‘𝑛) = 0 ))
1413ralbidv 2969 . . . . . 6 (𝑀 = (𝐴𝑠) → (∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝐴𝑠))‘𝑛) = 0 ))
1514adantl 481 . . . . 5 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑠𝐾) ∧ 𝑀 = (𝐴𝑠)) → (∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝐴𝑠))‘𝑛) = 0 ))
1610, 15mpbird 246 . . . 4 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑠𝐾) ∧ 𝑀 = (𝐴𝑠)) → ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 )
1716ex 449 . . 3 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑠𝐾) → (𝑀 = (𝐴𝑠) → ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ))
1817rexlimdva 3013 . 2 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (∃𝑠𝐾 𝑀 = (𝐴𝑠) → ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ))
19 simpr 476 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑀𝐵)
20 0nn0 11184 . . . . . 6 0 ∈ ℕ0
21 eqid 2610 . . . . . . 7 (coe1𝑀) = (coe1𝑀)
2221, 7, 6, 4coe1fvalcl 19403 . . . . . 6 ((𝑀𝐵 ∧ 0 ∈ ℕ0) → ((coe1𝑀)‘0) ∈ 𝐾)
2319, 20, 22sylancl 693 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ((coe1𝑀)‘0) ∈ 𝐾)
2423adantr 480 . . . 4 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) → ((coe1𝑀)‘0) ∈ 𝐾)
25 fveq2 6103 . . . . . 6 (𝑠 = ((coe1𝑀)‘0) → (𝐴𝑠) = (𝐴‘((coe1𝑀)‘0)))
2625eqeq2d 2620 . . . . 5 (𝑠 = ((coe1𝑀)‘0) → (𝑀 = (𝐴𝑠) ↔ 𝑀 = (𝐴‘((coe1𝑀)‘0))))
2726adantl 481 . . . 4 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) ∧ 𝑠 = ((coe1𝑀)‘0)) → (𝑀 = (𝐴𝑠) ↔ 𝑀 = (𝐴‘((coe1𝑀)‘0))))
28 eqid 2610 . . . . . . . . . 10 (Scalar‘𝑃) = (Scalar‘𝑃)
296ply1ring 19439 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
306ply1lmod 19443 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
31 eqid 2610 . . . . . . . . . 10 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
328, 28, 29, 30, 31, 7asclf 19158 . . . . . . . . 9 (𝑅 ∈ Ring → 𝐴:(Base‘(Scalar‘𝑃))⟶𝐵)
3332adantr 480 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝐴:(Base‘(Scalar‘𝑃))⟶𝐵)
34 eqid 2610 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
3521, 7, 6, 34coe1fvalcl 19403 . . . . . . . . . 10 ((𝑀𝐵 ∧ 0 ∈ ℕ0) → ((coe1𝑀)‘0) ∈ (Base‘𝑅))
3619, 20, 35sylancl 693 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ((coe1𝑀)‘0) ∈ (Base‘𝑅))
376ply1sca 19444 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
3837eqcomd 2616 . . . . . . . . . . 11 (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
3938fveq2d 6107 . . . . . . . . . 10 (𝑅 ∈ Ring → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
4039adantr 480 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
4136, 40eleqtrrd 2691 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ((coe1𝑀)‘0) ∈ (Base‘(Scalar‘𝑃)))
4233, 41ffvelrnd 6268 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝐴‘((coe1𝑀)‘0)) ∈ 𝐵)
431, 19, 423jca 1235 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑅 ∈ Ring ∧ 𝑀𝐵 ∧ (𝐴‘((coe1𝑀)‘0)) ∈ 𝐵))
4443adantr 480 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) → (𝑅 ∈ Ring ∧ 𝑀𝐵 ∧ (𝐴‘((coe1𝑀)‘0)) ∈ 𝐵))
45 simpr 476 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) ∧ ((coe1𝑀)‘𝑛) = 0 ) → ((coe1𝑀)‘𝑛) = 0 )
466, 8, 4, 5coe1scl 19478 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ ((coe1𝑀)‘0) ∈ 𝐾) → (coe1‘(𝐴‘((coe1𝑀)‘0))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, ((coe1𝑀)‘0), 0 )))
4723, 46syldan 486 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (coe1‘(𝐴‘((coe1𝑀)‘0))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, ((coe1𝑀)‘0), 0 )))
4847adantr 480 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) → (coe1‘(𝐴‘((coe1𝑀)‘0))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, ((coe1𝑀)‘0), 0 )))
49 nnne0 10930 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → 𝑛 ≠ 0)
5049neneqd 2787 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
5150adantl 481 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
5251adantr 480 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → ¬ 𝑛 = 0)
53 eqeq1 2614 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑘 = 0 ↔ 𝑛 = 0))
5453notbid 307 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (¬ 𝑘 = 0 ↔ ¬ 𝑛 = 0))
5554adantl 481 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (¬ 𝑘 = 0 ↔ ¬ 𝑛 = 0))
5652, 55mpbird 246 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → ¬ 𝑘 = 0)
5756iffalsed 4047 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → if(𝑘 = 0, ((coe1𝑀)‘0), 0 ) = 0 )
58 nnnn0 11176 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
5958adantl 481 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
60 fvex 6113 . . . . . . . . . . . . . . 15 (0g𝑅) ∈ V
615, 60eqeltri 2684 . . . . . . . . . . . . . 14 0 ∈ V
6261a1i 11 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) → 0 ∈ V)
6348, 57, 59, 62fvmptd 6197 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) → ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛) = 0 )
6463eqcomd 2616 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) → 0 = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛))
6564adantr 480 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) ∧ ((coe1𝑀)‘𝑛) = 0 ) → 0 = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛))
6645, 65eqtrd 2644 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) ∧ ((coe1𝑀)‘𝑛) = 0 ) → ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛))
6766ex 449 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ) → (((coe1𝑀)‘𝑛) = 0 → ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛)))
6867ralimdva 2945 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 → ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛)))
6968imp 444 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) → ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛))
706, 8, 4ply1sclid 19479 . . . . . . . 8 ((𝑅 ∈ Ring ∧ ((coe1𝑀)‘0) ∈ 𝐾) → ((coe1𝑀)‘0) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘0))
7123, 70syldan 486 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ((coe1𝑀)‘0) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘0))
7271adantr 480 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) → ((coe1𝑀)‘0) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘0))
73 df-n0 11170 . . . . . . . 8 0 = (ℕ ∪ {0})
7473raleqi 3119 . . . . . . 7 (∀𝑛 ∈ ℕ0 ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛) ↔ ∀𝑛 ∈ (ℕ ∪ {0})((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛))
75 c0ex 9913 . . . . . . . 8 0 ∈ V
76 fveq2 6103 . . . . . . . . . 10 (𝑛 = 0 → ((coe1𝑀)‘𝑛) = ((coe1𝑀)‘0))
77 fveq2 6103 . . . . . . . . . 10 (𝑛 = 0 → ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘0))
7876, 77eqeq12d 2625 . . . . . . . . 9 (𝑛 = 0 → (((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛) ↔ ((coe1𝑀)‘0) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘0)))
7978ralunsn 4360 . . . . . . . 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))))
8075, 79mp1i 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))))
8174, 80syl5bb 271 . . . . . 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))))
8269, 72, 81mpbir2and 959 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) → ∀𝑛 ∈ ℕ0 ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛))
83 eqid 2610 . . . . . 6 (coe1‘(𝐴‘((coe1𝑀)‘0))) = (coe1‘(𝐴‘((coe1𝑀)‘0)))
846, 7, 21, 83eqcoe1ply1eq 19488 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑀𝐵 ∧ (𝐴‘((coe1𝑀)‘0)) ∈ 𝐵) → (∀𝑛 ∈ ℕ0 ((coe1𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1𝑀)‘0)))‘𝑛) → 𝑀 = (𝐴‘((coe1𝑀)‘0))))
8544, 82, 84sylc 63 . . . 4 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) → 𝑀 = (𝐴‘((coe1𝑀)‘0)))
8624, 27, 85rspcedvd 3289 . . 3 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ) → ∃𝑠𝐾 𝑀 = (𝐴𝑠))
8786ex 449 . 2 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 → ∃𝑠𝐾 𝑀 = (𝐴𝑠)))
8818, 87impbid 201 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∪ cun 3538  ifcif 4036  {csn 4125   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  0cc0 9815  ℕcn 10897  ℕ0cn0 11169  Basecbs 15695  Scalarcsca 15771  0gc0g 15923  Ringcrg 18370  algSccascl 19132  Poly1cpl1 19368  coe1cco1 19369 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-tset 15787  df-ple 15788  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-srg 18329  df-ring 18372  df-subrg 18601  df-lmod 18688  df-lss 18754  df-ascl 19135  df-psr 19177  df-mvr 19178  df-mpl 19179  df-opsr 19181  df-psr1 19371  df-vr1 19372  df-ply1 19373  df-coe1 19374 This theorem is referenced by:  cpmatel2  20337
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