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Theorem evls1sca 19509
Description: Univariate polynomial evaluation maps scalars to constant functions. (Contributed by AV, 8-Sep-2019.)
Hypotheses
Ref Expression
evls1sca.q 𝑄 = (𝑆 evalSub1 𝑅)
evls1sca.w 𝑊 = (Poly1𝑈)
evls1sca.u 𝑈 = (𝑆s 𝑅)
evls1sca.b 𝐵 = (Base‘𝑆)
evls1sca.a 𝐴 = (algSc‘𝑊)
evls1sca.s (𝜑𝑆 ∈ CRing)
evls1sca.r (𝜑𝑅 ∈ (SubRing‘𝑆))
evls1sca.x (𝜑𝑋𝑅)
Assertion
Ref Expression
evls1sca (𝜑 → (𝑄‘(𝐴𝑋)) = (𝐵 × {𝑋}))

Proof of Theorem evls1sca
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1on 7454 . . . . . . 7 1𝑜 ∈ On
21a1i 11 . . . . . 6 (𝜑 → 1𝑜 ∈ On)
3 evls1sca.s . . . . . 6 (𝜑𝑆 ∈ CRing)
4 evls1sca.r . . . . . 6 (𝜑𝑅 ∈ (SubRing‘𝑆))
5 eqid 2610 . . . . . . 7 ((1𝑜 evalSub 𝑆)‘𝑅) = ((1𝑜 evalSub 𝑆)‘𝑅)
6 eqid 2610 . . . . . . 7 (1𝑜 mPoly 𝑈) = (1𝑜 mPoly 𝑈)
7 evls1sca.u . . . . . . 7 𝑈 = (𝑆s 𝑅)
8 eqid 2610 . . . . . . 7 (𝑆s (𝐵𝑚 1𝑜)) = (𝑆s (𝐵𝑚 1𝑜))
9 evls1sca.b . . . . . . 7 𝐵 = (Base‘𝑆)
105, 6, 7, 8, 9evlsrhm 19342 . . . . . 6 ((1𝑜 ∈ On ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1𝑜 evalSub 𝑆)‘𝑅) ∈ ((1𝑜 mPoly 𝑈) RingHom (𝑆s (𝐵𝑚 1𝑜))))
112, 3, 4, 10syl3anc 1318 . . . . 5 (𝜑 → ((1𝑜 evalSub 𝑆)‘𝑅) ∈ ((1𝑜 mPoly 𝑈) RingHom (𝑆s (𝐵𝑚 1𝑜))))
12 eqid 2610 . . . . . 6 (Base‘(1𝑜 mPoly 𝑈)) = (Base‘(1𝑜 mPoly 𝑈))
13 eqid 2610 . . . . . 6 (Base‘(𝑆s (𝐵𝑚 1𝑜))) = (Base‘(𝑆s (𝐵𝑚 1𝑜)))
1412, 13rhmf 18549 . . . . 5 (((1𝑜 evalSub 𝑆)‘𝑅) ∈ ((1𝑜 mPoly 𝑈) RingHom (𝑆s (𝐵𝑚 1𝑜))) → ((1𝑜 evalSub 𝑆)‘𝑅):(Base‘(1𝑜 mPoly 𝑈))⟶(Base‘(𝑆s (𝐵𝑚 1𝑜))))
1511, 14syl 17 . . . 4 (𝜑 → ((1𝑜 evalSub 𝑆)‘𝑅):(Base‘(1𝑜 mPoly 𝑈))⟶(Base‘(𝑆s (𝐵𝑚 1𝑜))))
16 evls1sca.a . . . . . . 7 𝐴 = (algSc‘𝑊)
17 eqid 2610 . . . . . . 7 (Scalar‘𝑊) = (Scalar‘𝑊)
187subrgring 18606 . . . . . . . . 9 (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
194, 18syl 17 . . . . . . . 8 (𝜑𝑈 ∈ Ring)
20 evls1sca.w . . . . . . . . 9 𝑊 = (Poly1𝑈)
2120ply1ring 19439 . . . . . . . 8 (𝑈 ∈ Ring → 𝑊 ∈ Ring)
2219, 21syl 17 . . . . . . 7 (𝜑𝑊 ∈ Ring)
2320ply1lmod 19443 . . . . . . . 8 (𝑈 ∈ Ring → 𝑊 ∈ LMod)
2419, 23syl 17 . . . . . . 7 (𝜑𝑊 ∈ LMod)
25 eqid 2610 . . . . . . 7 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
26 eqid 2610 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
2716, 17, 22, 24, 25, 26asclf 19158 . . . . . 6 (𝜑𝐴:(Base‘(Scalar‘𝑊))⟶(Base‘𝑊))
289subrgss 18604 . . . . . . . . . 10 (𝑅 ∈ (SubRing‘𝑆) → 𝑅𝐵)
294, 28syl 17 . . . . . . . . 9 (𝜑𝑅𝐵)
307, 9ressbas2 15758 . . . . . . . . 9 (𝑅𝐵𝑅 = (Base‘𝑈))
3129, 30syl 17 . . . . . . . 8 (𝜑𝑅 = (Base‘𝑈))
3220ply1sca 19444 . . . . . . . . . 10 (𝑈 ∈ Ring → 𝑈 = (Scalar‘𝑊))
3319, 32syl 17 . . . . . . . . 9 (𝜑𝑈 = (Scalar‘𝑊))
3433fveq2d 6107 . . . . . . . 8 (𝜑 → (Base‘𝑈) = (Base‘(Scalar‘𝑊)))
3531, 34eqtrd 2644 . . . . . . 7 (𝜑𝑅 = (Base‘(Scalar‘𝑊)))
36 eqid 2610 . . . . . . . . . 10 (PwSer1𝑈) = (PwSer1𝑈)
3720, 36, 26ply1bas 19386 . . . . . . . . 9 (Base‘𝑊) = (Base‘(1𝑜 mPoly 𝑈))
3837a1i 11 . . . . . . . 8 (𝜑 → (Base‘𝑊) = (Base‘(1𝑜 mPoly 𝑈)))
3938eqcomd 2616 . . . . . . 7 (𝜑 → (Base‘(1𝑜 mPoly 𝑈)) = (Base‘𝑊))
4035, 39feq23d 5953 . . . . . 6 (𝜑 → (𝐴:𝑅⟶(Base‘(1𝑜 mPoly 𝑈)) ↔ 𝐴:(Base‘(Scalar‘𝑊))⟶(Base‘𝑊)))
4127, 40mpbird 246 . . . . 5 (𝜑𝐴:𝑅⟶(Base‘(1𝑜 mPoly 𝑈)))
42 evls1sca.x . . . . 5 (𝜑𝑋𝑅)
4341, 42ffvelrnd 6268 . . . 4 (𝜑 → (𝐴𝑋) ∈ (Base‘(1𝑜 mPoly 𝑈)))
44 fvco3 6185 . . . 4 ((((1𝑜 evalSub 𝑆)‘𝑅):(Base‘(1𝑜 mPoly 𝑈))⟶(Base‘(𝑆s (𝐵𝑚 1𝑜))) ∧ (𝐴𝑋) ∈ (Base‘(1𝑜 mPoly 𝑈))) → (((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅))‘(𝐴𝑋)) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))‘(((1𝑜 evalSub 𝑆)‘𝑅)‘(𝐴𝑋))))
4515, 43, 44syl2anc 691 . . 3 (𝜑 → (((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅))‘(𝐴𝑋)) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))‘(((1𝑜 evalSub 𝑆)‘𝑅)‘(𝐴𝑋))))
4616a1i 11 . . . . . . . 8 (𝜑𝐴 = (algSc‘𝑊))
47 eqid 2610 . . . . . . . . 9 (algSc‘𝑊) = (algSc‘𝑊)
4820, 47ply1ascl 19449 . . . . . . . 8 (algSc‘𝑊) = (algSc‘(1𝑜 mPoly 𝑈))
4946, 48syl6eq 2660 . . . . . . 7 (𝜑𝐴 = (algSc‘(1𝑜 mPoly 𝑈)))
5049fveq1d 6105 . . . . . 6 (𝜑 → (𝐴𝑋) = ((algSc‘(1𝑜 mPoly 𝑈))‘𝑋))
5150fveq2d 6107 . . . . 5 (𝜑 → (((1𝑜 evalSub 𝑆)‘𝑅)‘(𝐴𝑋)) = (((1𝑜 evalSub 𝑆)‘𝑅)‘((algSc‘(1𝑜 mPoly 𝑈))‘𝑋)))
52 eqid 2610 . . . . . 6 (algSc‘(1𝑜 mPoly 𝑈)) = (algSc‘(1𝑜 mPoly 𝑈))
535, 6, 7, 9, 52, 2, 3, 4, 42evlssca 19343 . . . . 5 (𝜑 → (((1𝑜 evalSub 𝑆)‘𝑅)‘((algSc‘(1𝑜 mPoly 𝑈))‘𝑋)) = ((𝐵𝑚 1𝑜) × {𝑋}))
5451, 53eqtrd 2644 . . . 4 (𝜑 → (((1𝑜 evalSub 𝑆)‘𝑅)‘(𝐴𝑋)) = ((𝐵𝑚 1𝑜) × {𝑋}))
5554fveq2d 6107 . . 3 (𝜑 → ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))‘(((1𝑜 evalSub 𝑆)‘𝑅)‘(𝐴𝑋))) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))‘((𝐵𝑚 1𝑜) × {𝑋})))
56 eqidd 2611 . . . . 5 (𝜑 → (𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) = (𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))))
57 coeq1 5201 . . . . . 6 (𝑥 = ((𝐵𝑚 1𝑜) × {𝑋}) → (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) = (((𝐵𝑚 1𝑜) × {𝑋}) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))
5857adantl 481 . . . . 5 ((𝜑𝑥 = ((𝐵𝑚 1𝑜) × {𝑋})) → (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) = (((𝐵𝑚 1𝑜) × {𝑋}) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))
5929, 42sseldd 3569 . . . . . . 7 (𝜑𝑋𝐵)
60 fconst6g 6007 . . . . . . 7 (𝑋𝐵 → ((𝐵𝑚 1𝑜) × {𝑋}):(𝐵𝑚 1𝑜)⟶𝐵)
6159, 60syl 17 . . . . . 6 (𝜑 → ((𝐵𝑚 1𝑜) × {𝑋}):(𝐵𝑚 1𝑜)⟶𝐵)
62 fvex 6113 . . . . . . . . 9 (Base‘𝑆) ∈ V
639, 62eqeltri 2684 . . . . . . . 8 𝐵 ∈ V
6463a1i 11 . . . . . . 7 (𝜑𝐵 ∈ V)
65 ovex 6577 . . . . . . . 8 (𝐵𝑚 1𝑜) ∈ V
6665a1i 11 . . . . . . 7 (𝜑 → (𝐵𝑚 1𝑜) ∈ V)
6764, 66elmapd 7758 . . . . . 6 (𝜑 → (((𝐵𝑚 1𝑜) × {𝑋}) ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↔ ((𝐵𝑚 1𝑜) × {𝑋}):(𝐵𝑚 1𝑜)⟶𝐵))
6861, 67mpbird 246 . . . . 5 (𝜑 → ((𝐵𝑚 1𝑜) × {𝑋}) ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)))
69 snex 4835 . . . . . . . 8 {𝑋} ∈ V
7065, 69xpex 6860 . . . . . . 7 ((𝐵𝑚 1𝑜) × {𝑋}) ∈ V
7170a1i 11 . . . . . 6 (𝜑 → ((𝐵𝑚 1𝑜) × {𝑋}) ∈ V)
72 mptexg 6389 . . . . . . 7 (𝐵 ∈ V → (𝑦𝐵 ↦ (1𝑜 × {𝑦})) ∈ V)
7364, 72syl 17 . . . . . 6 (𝜑 → (𝑦𝐵 ↦ (1𝑜 × {𝑦})) ∈ V)
74 coexg 7010 . . . . . 6 ((((𝐵𝑚 1𝑜) × {𝑋}) ∈ V ∧ (𝑦𝐵 ↦ (1𝑜 × {𝑦})) ∈ V) → (((𝐵𝑚 1𝑜) × {𝑋}) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) ∈ V)
7571, 73, 74syl2anc 691 . . . . 5 (𝜑 → (((𝐵𝑚 1𝑜) × {𝑋}) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) ∈ V)
7656, 58, 68, 75fvmptd 6197 . . . 4 (𝜑 → ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))‘((𝐵𝑚 1𝑜) × {𝑋})) = (((𝐵𝑚 1𝑜) × {𝑋}) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))
77 fconst6g 6007 . . . . . . 7 (𝑦𝐵 → (1𝑜 × {𝑦}):1𝑜𝐵)
7877adantl 481 . . . . . 6 ((𝜑𝑦𝐵) → (1𝑜 × {𝑦}):1𝑜𝐵)
7963, 1pm3.2i 470 . . . . . . . 8 (𝐵 ∈ V ∧ 1𝑜 ∈ On)
8079a1i 11 . . . . . . 7 ((𝜑𝑦𝐵) → (𝐵 ∈ V ∧ 1𝑜 ∈ On))
81 elmapg 7757 . . . . . . 7 ((𝐵 ∈ V ∧ 1𝑜 ∈ On) → ((1𝑜 × {𝑦}) ∈ (𝐵𝑚 1𝑜) ↔ (1𝑜 × {𝑦}):1𝑜𝐵))
8280, 81syl 17 . . . . . 6 ((𝜑𝑦𝐵) → ((1𝑜 × {𝑦}) ∈ (𝐵𝑚 1𝑜) ↔ (1𝑜 × {𝑦}):1𝑜𝐵))
8378, 82mpbird 246 . . . . 5 ((𝜑𝑦𝐵) → (1𝑜 × {𝑦}) ∈ (𝐵𝑚 1𝑜))
84 eqidd 2611 . . . . 5 (𝜑 → (𝑦𝐵 ↦ (1𝑜 × {𝑦})) = (𝑦𝐵 ↦ (1𝑜 × {𝑦})))
85 fconstmpt 5085 . . . . . 6 ((𝐵𝑚 1𝑜) × {𝑋}) = (𝑧 ∈ (𝐵𝑚 1𝑜) ↦ 𝑋)
8685a1i 11 . . . . 5 (𝜑 → ((𝐵𝑚 1𝑜) × {𝑋}) = (𝑧 ∈ (𝐵𝑚 1𝑜) ↦ 𝑋))
87 eqidd 2611 . . . . 5 (𝑧 = (1𝑜 × {𝑦}) → 𝑋 = 𝑋)
8883, 84, 86, 87fmptco 6303 . . . 4 (𝜑 → (((𝐵𝑚 1𝑜) × {𝑋}) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) = (𝑦𝐵𝑋))
8976, 88eqtrd 2644 . . 3 (𝜑 → ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))‘((𝐵𝑚 1𝑜) × {𝑋})) = (𝑦𝐵𝑋))
9045, 55, 893eqtrd 2648 . 2 (𝜑 → (((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅))‘(𝐴𝑋)) = (𝑦𝐵𝑋))
91 elpwg 4116 . . . . . 6 (𝑅 ∈ (SubRing‘𝑆) → (𝑅 ∈ 𝒫 𝐵𝑅𝐵))
9228, 91mpbird 246 . . . . 5 (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ∈ 𝒫 𝐵)
934, 92syl 17 . . . 4 (𝜑𝑅 ∈ 𝒫 𝐵)
94 evls1sca.q . . . . 5 𝑄 = (𝑆 evalSub1 𝑅)
95 eqid 2610 . . . . 5 (1𝑜 evalSub 𝑆) = (1𝑜 evalSub 𝑆)
9694, 95, 9evls1fval 19505 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅)))
973, 93, 96syl2anc 691 . . 3 (𝜑𝑄 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅)))
9897fveq1d 6105 . 2 (𝜑 → (𝑄‘(𝐴𝑋)) = (((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅))‘(𝐴𝑋)))
99 fconstmpt 5085 . . 3 (𝐵 × {𝑋}) = (𝑦𝐵𝑋)
10099a1i 11 . 2 (𝜑 → (𝐵 × {𝑋}) = (𝑦𝐵𝑋))
10190, 98, 1003eqtr4d 2654 1 (𝜑 → (𝑄‘(𝐴𝑋)) = (𝐵 × {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  wss 3540  𝒫 cpw 4108  {csn 4125  cmpt 4643   × cxp 5036  ccom 5042  Oncon0 5640  wf 5800  cfv 5804  (class class class)co 6549  1𝑜c1o 7440  𝑚 cmap 7744  Basecbs 15695  s cress 15696  Scalarcsca 15771  s cpws 15930  Ringcrg 18370  CRingccrg 18371   RingHom crh 18535  SubRingcsubrg 18599  LModclmod 18686  algSccascl 19132   mPoly cmpl 19174   evalSub ces 19325  PwSer1cps1 19366  Poly1cpl1 19368   evalSub1 ces1 19499
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-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-sup 8231  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-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-0g 15925  df-gsum 15926  df-prds 15931  df-pws 15933  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-cring 18373  df-rnghom 18538  df-subrg 18601  df-lmod 18688  df-lss 18754  df-lsp 18793  df-assa 19133  df-asp 19134  df-ascl 19135  df-psr 19177  df-mvr 19178  df-mpl 19179  df-opsr 19181  df-evls 19327  df-psr1 19371  df-ply1 19373  df-evls1 19501
This theorem is referenced by:  evls1scasrng  19524
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