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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  cn 10897  0cn0 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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