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Theorem evls1sca 19607
 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 7512 . . . . . . 7 1𝑜 ∈ On
21a1i 11 . . . . . 6 (𝜑 → 1𝑜 ∈ On)
3 evls1sca.s . . . . . 6 (𝜑𝑆 ∈ CRing)
4 evls1sca.r . . . . . 6 (𝜑𝑅 ∈ (SubRing‘𝑆))
5 eqid 2621 . . . . . . 7 ((1𝑜 evalSub 𝑆)‘𝑅) = ((1𝑜 evalSub 𝑆)‘𝑅)
6 eqid 2621 . . . . . . 7 (1𝑜 mPoly 𝑈) = (1𝑜 mPoly 𝑈)
7 evls1sca.u . . . . . . 7 𝑈 = (𝑆s 𝑅)
8 eqid 2621 . . . . . . 7 (𝑆s (𝐵𝑚 1𝑜)) = (𝑆s (𝐵𝑚 1𝑜))
9 evls1sca.b . . . . . . 7 𝐵 = (Base‘𝑆)
105, 6, 7, 8, 9evlsrhm 19440 . . . . . 6 ((1𝑜 ∈ On ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1𝑜 evalSub 𝑆)‘𝑅) ∈ ((1𝑜 mPoly 𝑈) RingHom (𝑆s (𝐵𝑚 1𝑜))))
112, 3, 4, 10syl3anc 1323 . . . . 5 (𝜑 → ((1𝑜 evalSub 𝑆)‘𝑅) ∈ ((1𝑜 mPoly 𝑈) RingHom (𝑆s (𝐵𝑚 1𝑜))))
12 eqid 2621 . . . . . 6 (Base‘(1𝑜 mPoly 𝑈)) = (Base‘(1𝑜 mPoly 𝑈))
13 eqid 2621 . . . . . 6 (Base‘(𝑆s (𝐵𝑚 1𝑜))) = (Base‘(𝑆s (𝐵𝑚 1𝑜)))
1412, 13rhmf 18647 . . . . 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 2621 . . . . . . 7 (Scalar‘𝑊) = (Scalar‘𝑊)
187subrgring 18704 . . . . . . . . 9 (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
194, 18syl 17 . . . . . . . 8 (𝜑𝑈 ∈ Ring)
20 evls1sca.w . . . . . . . . 9 𝑊 = (Poly1𝑈)
2120ply1ring 19537 . . . . . . . 8 (𝑈 ∈ Ring → 𝑊 ∈ Ring)
2219, 21syl 17 . . . . . . 7 (𝜑𝑊 ∈ Ring)
2320ply1lmod 19541 . . . . . . . 8 (𝑈 ∈ Ring → 𝑊 ∈ LMod)
2419, 23syl 17 . . . . . . 7 (𝜑𝑊 ∈ LMod)
25 eqid 2621 . . . . . . 7 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
26 eqid 2621 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
2716, 17, 22, 24, 25, 26asclf 19256 . . . . . 6 (𝜑𝐴:(Base‘(Scalar‘𝑊))⟶(Base‘𝑊))
289subrgss 18702 . . . . . . . . . 10 (𝑅 ∈ (SubRing‘𝑆) → 𝑅𝐵)
294, 28syl 17 . . . . . . . . 9 (𝜑𝑅𝐵)
307, 9ressbas2 15852 . . . . . . . . 9 (𝑅𝐵𝑅 = (Base‘𝑈))
3129, 30syl 17 . . . . . . . 8 (𝜑𝑅 = (Base‘𝑈))
3220ply1sca 19542 . . . . . . . . . 10 (𝑈 ∈ Ring → 𝑈 = (Scalar‘𝑊))
3319, 32syl 17 . . . . . . . . 9 (𝜑𝑈 = (Scalar‘𝑊))
3433fveq2d 6152 . . . . . . . 8 (𝜑 → (Base‘𝑈) = (Base‘(Scalar‘𝑊)))
3531, 34eqtrd 2655 . . . . . . 7 (𝜑𝑅 = (Base‘(Scalar‘𝑊)))
36 eqid 2621 . . . . . . . . . 10 (PwSer1𝑈) = (PwSer1𝑈)
3720, 36, 26ply1bas 19484 . . . . . . . . 9 (Base‘𝑊) = (Base‘(1𝑜 mPoly 𝑈))
3837a1i 11 . . . . . . . 8 (𝜑 → (Base‘𝑊) = (Base‘(1𝑜 mPoly 𝑈)))
3938eqcomd 2627 . . . . . . 7 (𝜑 → (Base‘(1𝑜 mPoly 𝑈)) = (Base‘𝑊))
4035, 39feq23d 5997 . . . . . 6 (𝜑 → (𝐴:𝑅⟶(Base‘(1𝑜 mPoly 𝑈)) ↔ 𝐴:(Base‘(Scalar‘𝑊))⟶(Base‘𝑊)))
4127, 40mpbird 247 . . . . 5 (𝜑𝐴:𝑅⟶(Base‘(1𝑜 mPoly 𝑈)))
42 evls1sca.x . . . . 5 (𝜑𝑋𝑅)
4341, 42ffvelrnd 6316 . . . 4 (𝜑 → (𝐴𝑋) ∈ (Base‘(1𝑜 mPoly 𝑈)))
44 fvco3 6232 . . . 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 692 . . 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 2621 . . . . . . . . 9 (algSc‘𝑊) = (algSc‘𝑊)
4820, 47ply1ascl 19547 . . . . . . . 8 (algSc‘𝑊) = (algSc‘(1𝑜 mPoly 𝑈))
4946, 48syl6eq 2671 . . . . . . 7 (𝜑𝐴 = (algSc‘(1𝑜 mPoly 𝑈)))
5049fveq1d 6150 . . . . . 6 (𝜑 → (𝐴𝑋) = ((algSc‘(1𝑜 mPoly 𝑈))‘𝑋))
5150fveq2d 6152 . . . . 5 (𝜑 → (((1𝑜 evalSub 𝑆)‘𝑅)‘(𝐴𝑋)) = (((1𝑜 evalSub 𝑆)‘𝑅)‘((algSc‘(1𝑜 mPoly 𝑈))‘𝑋)))
52 eqid 2621 . . . . . 6 (algSc‘(1𝑜 mPoly 𝑈)) = (algSc‘(1𝑜 mPoly 𝑈))
535, 6, 7, 9, 52, 2, 3, 4, 42evlssca 19441 . . . . 5 (𝜑 → (((1𝑜 evalSub 𝑆)‘𝑅)‘((algSc‘(1𝑜 mPoly 𝑈))‘𝑋)) = ((𝐵𝑚 1𝑜) × {𝑋}))
5451, 53eqtrd 2655 . . . 4 (𝜑 → (((1𝑜 evalSub 𝑆)‘𝑅)‘(𝐴𝑋)) = ((𝐵𝑚 1𝑜) × {𝑋}))
5554fveq2d 6152 . . 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 2622 . . . . 5 (𝜑 → (𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) = (𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))))
57 coeq1 5239 . . . . . 6 (𝑥 = ((𝐵𝑚 1𝑜) × {𝑋}) → (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) = (((𝐵𝑚 1𝑜) × {𝑋}) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))
5857adantl 482 . . . . 5 ((𝜑𝑥 = ((𝐵𝑚 1𝑜) × {𝑋})) → (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) = (((𝐵𝑚 1𝑜) × {𝑋}) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))
5929, 42sseldd 3584 . . . . . . 7 (𝜑𝑋𝐵)
60 fconst6g 6051 . . . . . . 7 (𝑋𝐵 → ((𝐵𝑚 1𝑜) × {𝑋}):(𝐵𝑚 1𝑜)⟶𝐵)
6159, 60syl 17 . . . . . 6 (𝜑 → ((𝐵𝑚 1𝑜) × {𝑋}):(𝐵𝑚 1𝑜)⟶𝐵)
62 fvex 6158 . . . . . . . . 9 (Base‘𝑆) ∈ V
639, 62eqeltri 2694 . . . . . . . 8 𝐵 ∈ V
6463a1i 11 . . . . . . 7 (𝜑𝐵 ∈ V)
65 ovex 6632 . . . . . . . 8 (𝐵𝑚 1𝑜) ∈ V
6665a1i 11 . . . . . . 7 (𝜑 → (𝐵𝑚 1𝑜) ∈ V)
6764, 66elmapd 7816 . . . . . 6 (𝜑 → (((𝐵𝑚 1𝑜) × {𝑋}) ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↔ ((𝐵𝑚 1𝑜) × {𝑋}):(𝐵𝑚 1𝑜)⟶𝐵))
6861, 67mpbird 247 . . . . 5 (𝜑 → ((𝐵𝑚 1𝑜) × {𝑋}) ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)))
69 snex 4869 . . . . . . . 8 {𝑋} ∈ V
7065, 69xpex 6915 . . . . . . 7 ((𝐵𝑚 1𝑜) × {𝑋}) ∈ V
7170a1i 11 . . . . . 6 (𝜑 → ((𝐵𝑚 1𝑜) × {𝑋}) ∈ V)
72 mptexg 6438 . . . . . . 7 (𝐵 ∈ V → (𝑦𝐵 ↦ (1𝑜 × {𝑦})) ∈ V)
7364, 72syl 17 . . . . . 6 (𝜑 → (𝑦𝐵 ↦ (1𝑜 × {𝑦})) ∈ V)
74 coexg 7064 . . . . . 6 ((((𝐵𝑚 1𝑜) × {𝑋}) ∈ V ∧ (𝑦𝐵 ↦ (1𝑜 × {𝑦})) ∈ V) → (((𝐵𝑚 1𝑜) × {𝑋}) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) ∈ V)
7571, 73, 74syl2anc 692 . . . . 5 (𝜑 → (((𝐵𝑚 1𝑜) × {𝑋}) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) ∈ V)
7656, 58, 68, 75fvmptd 6245 . . . 4 (𝜑 → ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))‘((𝐵𝑚 1𝑜) × {𝑋})) = (((𝐵𝑚 1𝑜) × {𝑋}) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))
77 fconst6g 6051 . . . . . . 7 (𝑦𝐵 → (1𝑜 × {𝑦}):1𝑜𝐵)
7877adantl 482 . . . . . 6 ((𝜑𝑦𝐵) → (1𝑜 × {𝑦}):1𝑜𝐵)
7963, 1pm3.2i 471 . . . . . . . 8 (𝐵 ∈ V ∧ 1𝑜 ∈ On)
8079a1i 11 . . . . . . 7 ((𝜑𝑦𝐵) → (𝐵 ∈ V ∧ 1𝑜 ∈ On))
81 elmapg 7815 . . . . . . 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 247 . . . . 5 ((𝜑𝑦𝐵) → (1𝑜 × {𝑦}) ∈ (𝐵𝑚 1𝑜))
84 eqidd 2622 . . . . 5 (𝜑 → (𝑦𝐵 ↦ (1𝑜 × {𝑦})) = (𝑦𝐵 ↦ (1𝑜 × {𝑦})))
85 fconstmpt 5123 . . . . . 6 ((𝐵𝑚 1𝑜) × {𝑋}) = (𝑧 ∈ (𝐵𝑚 1𝑜) ↦ 𝑋)
8685a1i 11 . . . . 5 (𝜑 → ((𝐵𝑚 1𝑜) × {𝑋}) = (𝑧 ∈ (𝐵𝑚 1𝑜) ↦ 𝑋))
87 eqidd 2622 . . . . 5 (𝑧 = (1𝑜 × {𝑦}) → 𝑋 = 𝑋)
8883, 84, 86, 87fmptco 6351 . . . 4 (𝜑 → (((𝐵𝑚 1𝑜) × {𝑋}) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) = (𝑦𝐵𝑋))
8976, 88eqtrd 2655 . . 3 (𝜑 → ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))‘((𝐵𝑚 1𝑜) × {𝑋})) = (𝑦𝐵𝑋))
9045, 55, 893eqtrd 2659 . 2 (𝜑 → (((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅))‘(𝐴𝑋)) = (𝑦𝐵𝑋))
91 elpwg 4138 . . . . . 6 (𝑅 ∈ (SubRing‘𝑆) → (𝑅 ∈ 𝒫 𝐵𝑅𝐵))
9228, 91mpbird 247 . . . . 5 (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ∈ 𝒫 𝐵)
934, 92syl 17 . . . 4 (𝜑𝑅 ∈ 𝒫 𝐵)
94 evls1sca.q . . . . 5 𝑄 = (𝑆 evalSub1 𝑅)
95 eqid 2621 . . . . 5 (1𝑜 evalSub 𝑆) = (1𝑜 evalSub 𝑆)
9694, 95, 9evls1fval 19603 . . . 4 ((𝑆 ∈ CRing ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅)))
973, 93, 96syl2anc 692 . . 3 (𝜑𝑄 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅)))
9897fveq1d 6150 . 2 (𝜑 → (𝑄‘(𝐴𝑋)) = (((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅))‘(𝐴𝑋)))
99 fconstmpt 5123 . . 3 (𝐵 × {𝑋}) = (𝑦𝐵𝑋)
10099a1i 11 . 2 (𝜑 → (𝐵 × {𝑋}) = (𝑦𝐵𝑋))
10190, 98, 1003eqtr4d 2665 1 (𝜑 → (𝑄‘(𝐴𝑋)) = (𝐵 × {𝑋}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3186   ⊆ wss 3555  𝒫 cpw 4130  {csn 4148   ↦ cmpt 4673   × cxp 5072   ∘ ccom 5078  Oncon0 5682  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604  1𝑜c1o 7498   ↑𝑚 cmap 7802  Basecbs 15781   ↾s cress 15782  Scalarcsca 15865   ↑s cpws 16028  Ringcrg 18468  CRingccrg 18469   RingHom crh 18633  SubRingcsubrg 18697  LModclmod 18784  algSccascl 19230   mPoly cmpl 19272   evalSub ces 19423  PwSer1cps1 19464  Poly1cpl1 19466   evalSub1 ces1 19597 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-ofr 6851  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-sup 8292  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-fzo 12407  df-seq 12742  df-hash 13058  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-hom 15887  df-cco 15888  df-0g 16023  df-gsum 16024  df-prds 16029  df-pws 16031  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-submnd 17257  df-grp 17346  df-minusg 17347  df-sbg 17348  df-mulg 17462  df-subg 17512  df-ghm 17579  df-cntz 17671  df-cmn 18116  df-abl 18117  df-mgp 18411  df-ur 18423  df-srg 18427  df-ring 18470  df-cring 18471  df-rnghom 18636  df-subrg 18699  df-lmod 18786  df-lss 18852  df-lsp 18891  df-assa 19231  df-asp 19232  df-ascl 19233  df-psr 19275  df-mvr 19276  df-mpl 19277  df-opsr 19279  df-evls 19425  df-psr1 19469  df-ply1 19471  df-evls1 19599 This theorem is referenced by:  evls1scasrng  19622
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