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Theorem evls1fpws 22374
Description: Evaluation of a univariate subring polynomial as a function in a power series. (Contributed by Thierry Arnoux, 23-Jan-2025.)
Hypotheses
Ref Expression
ressply1evl2.q 𝑄 = (𝑆 evalSub1 𝑅)
ressply1evl2.k 𝐾 = (Base‘𝑆)
ressply1evl2.w 𝑊 = (Poly1𝑈)
ressply1evl2.u 𝑈 = (𝑆s 𝑅)
ressply1evl2.b 𝐵 = (Base‘𝑊)
evls1fpws.s (𝜑𝑆 ∈ CRing)
evls1fpws.r (𝜑𝑅 ∈ (SubRing‘𝑆))
evls1fpws.y (𝜑𝑀𝐵)
evls1fpws.1 · = (.r𝑆)
evls1fpws.2 = (.g‘(mulGrp‘𝑆))
evls1fpws.a 𝐴 = (coe1𝑀)
Assertion
Ref Expression
evls1fpws (𝜑 → (𝑄𝑀) = (𝑥𝐾 ↦ (𝑆 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑥))))))
Distinct variable groups:   · ,𝑘,𝑥   𝐴,𝑘,𝑥   𝐵,𝑘   𝑘,𝐾,𝑥   𝑘,𝑀   𝑄,𝑘,𝑥   𝑆,𝑘,𝑥   𝑈,𝑘,𝑥   𝑘,𝑊,𝑥   𝜑,𝑘,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝑅(𝑥,𝑘)   (𝑥,𝑘)   𝑀(𝑥)

Proof of Theorem evls1fpws
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evls1fpws.r . . . . 5 (𝜑𝑅 ∈ (SubRing‘𝑆))
2 ressply1evl2.u . . . . . 6 𝑈 = (𝑆s 𝑅)
32subrgring 20575 . . . . 5 (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
41, 3syl 17 . . . 4 (𝜑𝑈 ∈ Ring)
5 evls1fpws.y . . . 4 (𝜑𝑀𝐵)
6 ressply1evl2.w . . . . 5 𝑊 = (Poly1𝑈)
7 eqid 2736 . . . . 5 (var1𝑈) = (var1𝑈)
8 ressply1evl2.b . . . . 5 𝐵 = (Base‘𝑊)
9 eqid 2736 . . . . 5 ( ·𝑠𝑊) = ( ·𝑠𝑊)
10 eqid 2736 . . . . 5 (mulGrp‘𝑊) = (mulGrp‘𝑊)
11 eqid 2736 . . . . 5 (.g‘(mulGrp‘𝑊)) = (.g‘(mulGrp‘𝑊))
12 evls1fpws.a . . . . 5 𝐴 = (coe1𝑀)
136, 7, 8, 9, 10, 11, 12ply1coe 22303 . . . 4 ((𝑈 ∈ Ring ∧ 𝑀𝐵) → 𝑀 = (𝑊 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘)( ·𝑠𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈))))))
144, 5, 13syl2anc 584 . . 3 (𝜑𝑀 = (𝑊 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘)( ·𝑠𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈))))))
1514fveq2d 6909 . 2 (𝜑 → (𝑄𝑀) = (𝑄‘(𝑊 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘)( ·𝑠𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))))))
16 ressply1evl2.q . . . 4 𝑄 = (𝑆 evalSub1 𝑅)
17 ressply1evl2.k . . . 4 𝐾 = (Base‘𝑆)
18 eqid 2736 . . . 4 (0g𝑊) = (0g𝑊)
19 eqid 2736 . . . 4 (𝑆s 𝐾) = (𝑆s 𝐾)
20 evls1fpws.s . . . 4 (𝜑𝑆 ∈ CRing)
216ply1lmod 22254 . . . . . . 7 (𝑈 ∈ Ring → 𝑊 ∈ LMod)
224, 21syl 17 . . . . . 6 (𝜑𝑊 ∈ LMod)
2322adantr 480 . . . . 5 ((𝜑𝑘 ∈ ℕ0) → 𝑊 ∈ LMod)
24 eqid 2736 . . . . . . . 8 (Base‘𝑈) = (Base‘𝑈)
2512, 8, 6, 24coe1fvalcl 22215 . . . . . . 7 ((𝑀𝐵𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ (Base‘𝑈))
265, 25sylan 580 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ (Base‘𝑈))
276ply1sca 22255 . . . . . . . . 9 (𝑈 ∈ Ring → 𝑈 = (Scalar‘𝑊))
284, 27syl 17 . . . . . . . 8 (𝜑𝑈 = (Scalar‘𝑊))
2928fveq2d 6909 . . . . . . 7 (𝜑 → (Base‘𝑈) = (Base‘(Scalar‘𝑊)))
3029adantr 480 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (Base‘𝑈) = (Base‘(Scalar‘𝑊)))
3126, 30eleqtrd 2842 . . . . 5 ((𝜑𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ (Base‘(Scalar‘𝑊)))
3210, 8mgpbas 20143 . . . . . 6 𝐵 = (Base‘(mulGrp‘𝑊))
336ply1ring 22250 . . . . . . . . 9 (𝑈 ∈ Ring → 𝑊 ∈ Ring)
344, 33syl 17 . . . . . . . 8 (𝜑𝑊 ∈ Ring)
3534adantr 480 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → 𝑊 ∈ Ring)
3610ringmgp 20237 . . . . . . 7 (𝑊 ∈ Ring → (mulGrp‘𝑊) ∈ Mnd)
3735, 36syl 17 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (mulGrp‘𝑊) ∈ Mnd)
38 simpr 484 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
394adantr 480 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → 𝑈 ∈ Ring)
407, 6, 8vr1cl 22220 . . . . . . 7 (𝑈 ∈ Ring → (var1𝑈) ∈ 𝐵)
4139, 40syl 17 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (var1𝑈) ∈ 𝐵)
4232, 11, 37, 38, 41mulgnn0cld 19114 . . . . 5 ((𝜑𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)) ∈ 𝐵)
43 eqid 2736 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
44 eqid 2736 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
458, 43, 9, 44lmodvscl 20877 . . . . 5 ((𝑊 ∈ LMod ∧ (𝐴𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)) ∈ 𝐵) → ((𝐴𝑘)( ·𝑠𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈))) ∈ 𝐵)
4623, 31, 42, 45syl3anc 1372 . . . 4 ((𝜑𝑘 ∈ ℕ0) → ((𝐴𝑘)( ·𝑠𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈))) ∈ 𝐵)
47 ssidd 4006 . . . 4 (𝜑 → ℕ0 ⊆ ℕ0)
48 fvexd 6920 . . . . 5 (𝜑 → (0g𝑊) ∈ V)
49 fveq2 6905 . . . . . 6 (𝑘 = 𝑗 → (𝐴𝑘) = (𝐴𝑗))
50 oveq1 7439 . . . . . 6 (𝑘 = 𝑗 → (𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)) = (𝑗(.g‘(mulGrp‘𝑊))(var1𝑈)))
5149, 50oveq12d 7450 . . . . 5 (𝑘 = 𝑗 → ((𝐴𝑘)( ·𝑠𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈))) = ((𝐴𝑗)( ·𝑠𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1𝑈))))
52 eqid 2736 . . . . . . . 8 (0g𝑈) = (0g𝑈)
5312, 8, 6, 52coe1ae0 22219 . . . . . . 7 (𝑀𝐵 → ∃𝑖 ∈ ℕ0𝑗 ∈ ℕ0 (𝑖 < 𝑗 → (𝐴𝑗) = (0g𝑈)))
545, 53syl 17 . . . . . 6 (𝜑 → ∃𝑖 ∈ ℕ0𝑗 ∈ ℕ0 (𝑖 < 𝑗 → (𝐴𝑗) = (0g𝑈)))
55 simpr 484 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴𝑗) = (0g𝑈)) → (𝐴𝑗) = (0g𝑈))
5628ad3antrrr 730 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴𝑗) = (0g𝑈)) → 𝑈 = (Scalar‘𝑊))
5756fveq2d 6909 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴𝑗) = (0g𝑈)) → (0g𝑈) = (0g‘(Scalar‘𝑊)))
5855, 57eqtrd 2776 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴𝑗) = (0g𝑈)) → (𝐴𝑗) = (0g‘(Scalar‘𝑊)))
5958oveq1d 7447 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴𝑗) = (0g𝑈)) → ((𝐴𝑗)( ·𝑠𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1𝑈))) = ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1𝑈))))
6022ad3antrrr 730 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴𝑗) = (0g𝑈)) → 𝑊 ∈ LMod)
6134, 36syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (mulGrp‘𝑊) ∈ Mnd)
6261adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ0) → (mulGrp‘𝑊) ∈ Mnd)
63 simpr 484 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ0) → 𝑗 ∈ ℕ0)
644, 40syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (var1𝑈) ∈ 𝐵)
6564adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ0) → (var1𝑈) ∈ 𝐵)
6632, 11, 62, 63, 65mulgnn0cld 19114 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ0) → (𝑗(.g‘(mulGrp‘𝑊))(var1𝑈)) ∈ 𝐵)
6766ad4ant13 751 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴𝑗) = (0g𝑈)) → (𝑗(.g‘(mulGrp‘𝑊))(var1𝑈)) ∈ 𝐵)
68 eqid 2736 . . . . . . . . . . . . 13 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
698, 43, 9, 68, 18lmod0vs 20894 . . . . . . . . . . . 12 ((𝑊 ∈ LMod ∧ (𝑗(.g‘(mulGrp‘𝑊))(var1𝑈)) ∈ 𝐵) → ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1𝑈))) = (0g𝑊))
7060, 67, 69syl2anc 584 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴𝑗) = (0g𝑈)) → ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1𝑈))) = (0g𝑊))
7159, 70eqtrd 2776 . . . . . . . . . 10 ((((𝜑𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴𝑗) = (0g𝑈)) → ((𝐴𝑗)( ·𝑠𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1𝑈))) = (0g𝑊))
7271ex 412 . . . . . . . . 9 (((𝜑𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) → ((𝐴𝑗) = (0g𝑈) → ((𝐴𝑗)( ·𝑠𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1𝑈))) = (0g𝑊)))
7372imim2d 57 . . . . . . . 8 (((𝜑𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) → ((𝑖 < 𝑗 → (𝐴𝑗) = (0g𝑈)) → (𝑖 < 𝑗 → ((𝐴𝑗)( ·𝑠𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1𝑈))) = (0g𝑊))))
7473ralimdva 3166 . . . . . . 7 ((𝜑𝑖 ∈ ℕ0) → (∀𝑗 ∈ ℕ0 (𝑖 < 𝑗 → (𝐴𝑗) = (0g𝑈)) → ∀𝑗 ∈ ℕ0 (𝑖 < 𝑗 → ((𝐴𝑗)( ·𝑠𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1𝑈))) = (0g𝑊))))
7574reximdva 3167 . . . . . 6 (𝜑 → (∃𝑖 ∈ ℕ0𝑗 ∈ ℕ0 (𝑖 < 𝑗 → (𝐴𝑗) = (0g𝑈)) → ∃𝑖 ∈ ℕ0𝑗 ∈ ℕ0 (𝑖 < 𝑗 → ((𝐴𝑗)( ·𝑠𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1𝑈))) = (0g𝑊))))
7654, 75mpd 15 . . . . 5 (𝜑 → ∃𝑖 ∈ ℕ0𝑗 ∈ ℕ0 (𝑖 < 𝑗 → ((𝐴𝑗)( ·𝑠𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1𝑈))) = (0g𝑊)))
7748, 46, 51, 76mptnn0fsuppd 14040 . . . 4 (𝜑 → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘)( ·𝑠𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))) finSupp (0g𝑊))
7816, 17, 6, 18, 2, 19, 8, 20, 1, 46, 47, 77evls1gsumadd 22329 . . 3 (𝜑 → (𝑄‘(𝑊 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘)( ·𝑠𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))))) = ((𝑆s 𝐾) Σg (𝑘 ∈ ℕ0 ↦ (𝑄‘((𝐴𝑘)( ·𝑠𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))))))
7916, 17, 19, 2, 6evls1rhm 22327 . . . . . . . . 9 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom (𝑆s 𝐾)))
8020, 1, 79syl2anc 584 . . . . . . . 8 (𝜑𝑄 ∈ (𝑊 RingHom (𝑆s 𝐾)))
8180adantr 480 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → 𝑄 ∈ (𝑊 RingHom (𝑆s 𝐾)))
82 eqid 2736 . . . . . . . . . 10 (algSc‘𝑊) = (algSc‘𝑊)
8382, 43, 34, 22, 44, 8asclf 21903 . . . . . . . . 9 (𝜑 → (algSc‘𝑊):(Base‘(Scalar‘𝑊))⟶𝐵)
8483adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → (algSc‘𝑊):(Base‘(Scalar‘𝑊))⟶𝐵)
8584, 31ffvelcdmd 7104 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → ((algSc‘𝑊)‘(𝐴𝑘)) ∈ 𝐵)
86 eqid 2736 . . . . . . . 8 (.r𝑊) = (.r𝑊)
87 eqid 2736 . . . . . . . 8 (.r‘(𝑆s 𝐾)) = (.r‘(𝑆s 𝐾))
888, 86, 87rhmmul 20487 . . . . . . 7 ((𝑄 ∈ (𝑊 RingHom (𝑆s 𝐾)) ∧ ((algSc‘𝑊)‘(𝐴𝑘)) ∈ 𝐵 ∧ (𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)) ∈ 𝐵) → (𝑄‘(((algSc‘𝑊)‘(𝐴𝑘))(.r𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))) = ((𝑄‘((algSc‘𝑊)‘(𝐴𝑘)))(.r‘(𝑆s 𝐾))(𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))))
8981, 85, 42, 88syl3anc 1372 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (𝑄‘(((algSc‘𝑊)‘(𝐴𝑘))(.r𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))) = ((𝑄‘((algSc‘𝑊)‘(𝐴𝑘)))(.r‘(𝑆s 𝐾))(𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))))
902subrgcrng 20576 . . . . . . . . . . 11 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
9120, 1, 90syl2anc 584 . . . . . . . . . 10 (𝜑𝑈 ∈ CRing)
926ply1assa 22202 . . . . . . . . . 10 (𝑈 ∈ CRing → 𝑊 ∈ AssAlg)
9391, 92syl 17 . . . . . . . . 9 (𝜑𝑊 ∈ AssAlg)
9493adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → 𝑊 ∈ AssAlg)
9582, 43, 44, 8, 86, 9asclmul1 21907 . . . . . . . 8 ((𝑊 ∈ AssAlg ∧ (𝐴𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)) ∈ 𝐵) → (((algSc‘𝑊)‘(𝐴𝑘))(.r𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈))) = ((𝐴𝑘)( ·𝑠𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈))))
9694, 31, 42, 95syl3anc 1372 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → (((algSc‘𝑊)‘(𝐴𝑘))(.r𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈))) = ((𝐴𝑘)( ·𝑠𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈))))
9796fveq2d 6909 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (𝑄‘(((algSc‘𝑊)‘(𝐴𝑘))(.r𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))) = (𝑄‘((𝐴𝑘)( ·𝑠𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))))
98 eqid 2736 . . . . . . . 8 (Base‘(𝑆s 𝐾)) = (Base‘(𝑆s 𝐾))
9920adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → 𝑆 ∈ CRing)
10017fvexi 6919 . . . . . . . . 9 𝐾 ∈ V
101100a1i 11 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → 𝐾 ∈ V)
1028, 98rhmf 20486 . . . . . . . . . 10 (𝑄 ∈ (𝑊 RingHom (𝑆s 𝐾)) → 𝑄:𝐵⟶(Base‘(𝑆s 𝐾)))
10381, 102syl 17 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ0) → 𝑄:𝐵⟶(Base‘(𝑆s 𝐾)))
104103, 85ffvelcdmd 7104 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → (𝑄‘((algSc‘𝑊)‘(𝐴𝑘))) ∈ (Base‘(𝑆s 𝐾)))
105103, 42ffvelcdmd 7104 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → (𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈))) ∈ (Base‘(𝑆s 𝐾)))
106 evls1fpws.1 . . . . . . . 8 · = (.r𝑆)
10719, 98, 99, 101, 104, 105, 106, 87pwsmulrval 17537 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → ((𝑄‘((algSc‘𝑊)‘(𝐴𝑘)))(.r‘(𝑆s 𝐾))(𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))) = ((𝑄‘((algSc‘𝑊)‘(𝐴𝑘))) ∘f · (𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))))
10819, 17, 98, 99, 101, 104pwselbas 17535 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ0) → (𝑄‘((algSc‘𝑊)‘(𝐴𝑘))):𝐾𝐾)
109108ffnd 6736 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → (𝑄‘((algSc‘𝑊)‘(𝐴𝑘))) Fn 𝐾)
11019, 17, 98, 99, 101, 105pwselbas 17535 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ0) → (𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈))):𝐾𝐾)
111110ffnd 6736 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → (𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈))) Fn 𝐾)
112 inidm 4226 . . . . . . . 8 (𝐾𝐾) = 𝐾
11320ad2antrr 726 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑥𝐾) → 𝑆 ∈ CRing)
1141ad2antrr 726 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑥𝐾) → 𝑅 ∈ (SubRing‘𝑆))
11517subrgss 20573 . . . . . . . . . . . . . 14 (𝑅 ∈ (SubRing‘𝑆) → 𝑅𝐾)
1161, 115syl 17 . . . . . . . . . . . . 13 (𝜑𝑅𝐾)
1172, 17ressbas2 17284 . . . . . . . . . . . . 13 (𝑅𝐾𝑅 = (Base‘𝑈))
118116, 117syl 17 . . . . . . . . . . . 12 (𝜑𝑅 = (Base‘𝑈))
119118adantr 480 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ0) → 𝑅 = (Base‘𝑈))
12026, 119eleqtrrd 2843 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ 𝑅)
121120adantr 480 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑥𝐾) → (𝐴𝑘) ∈ 𝑅)
122 simpr 484 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑥𝐾) → 𝑥𝐾)
12316, 6, 2, 17, 82, 113, 114, 121, 122evls1scafv 22371 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑥𝐾) → ((𝑄‘((algSc‘𝑊)‘(𝐴𝑘)))‘𝑥) = (𝐴𝑘))
124 evls1fpws.2 . . . . . . . . 9 = (.g‘(mulGrp‘𝑆))
125 simplr 768 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑥𝐾) → 𝑘 ∈ ℕ0)
12616, 2, 6, 7, 17, 11, 124, 113, 114, 125, 122evls1varpwval 22373 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑥𝐾) → ((𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))‘𝑥) = (𝑘 𝑥))
127109, 111, 101, 101, 112, 123, 126offval 7707 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → ((𝑄‘((algSc‘𝑊)‘(𝐴𝑘))) ∘f · (𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))) = (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥))))
128107, 127eqtrd 2776 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → ((𝑄‘((algSc‘𝑊)‘(𝐴𝑘)))(.r‘(𝑆s 𝐾))(𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))) = (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥))))
12989, 97, 1283eqtr3d 2784 . . . . 5 ((𝜑𝑘 ∈ ℕ0) → (𝑄‘((𝐴𝑘)( ·𝑠𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))) = (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥))))
130129mpteq2dva 5241 . . . 4 (𝜑 → (𝑘 ∈ ℕ0 ↦ (𝑄‘((𝐴𝑘)( ·𝑠𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈))))) = (𝑘 ∈ ℕ0 ↦ (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥)))))
131130oveq2d 7448 . . 3 (𝜑 → ((𝑆s 𝐾) Σg (𝑘 ∈ ℕ0 ↦ (𝑄‘((𝐴𝑘)( ·𝑠𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))))) = ((𝑆s 𝐾) Σg (𝑘 ∈ ℕ0 ↦ (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥))))))
132 eqid 2736 . . . 4 (0g‘(𝑆s 𝐾)) = (0g‘(𝑆s 𝐾))
133100a1i 11 . . . 4 (𝜑𝐾 ∈ V)
134 nn0ex 12534 . . . . 5 0 ∈ V
135134a1i 11 . . . 4 (𝜑 → ℕ0 ∈ V)
13620crngringd 20244 . . . . 5 (𝜑𝑆 ∈ Ring)
137136ringcmnd 20282 . . . 4 (𝜑𝑆 ∈ CMnd)
138136ad2antrr 726 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑥𝐾) → 𝑆 ∈ Ring)
1391adantr 480 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ0) → 𝑅 ∈ (SubRing‘𝑆))
140139, 115syl 17 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ0) → 𝑅𝐾)
141140, 120sseldd 3983 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ 𝐾)
142141adantr 480 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑥𝐾) → (𝐴𝑘) ∈ 𝐾)
143 eqid 2736 . . . . . . . . . 10 (mulGrp‘𝑆) = (mulGrp‘𝑆)
144143, 17mgpbas 20143 . . . . . . . . 9 𝐾 = (Base‘(mulGrp‘𝑆))
145143ringmgp 20237 . . . . . . . . . . 11 (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd)
146136, 145syl 17 . . . . . . . . . 10 (𝜑 → (mulGrp‘𝑆) ∈ Mnd)
147146ad2antrr 726 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑥𝐾) → (mulGrp‘𝑆) ∈ Mnd)
148144, 124, 147, 125, 122mulgnn0cld 19114 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑥𝐾) → (𝑘 𝑥) ∈ 𝐾)
14917, 106, 138, 142, 148ringcld 20258 . . . . . . 7 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑥𝐾) → ((𝐴𝑘) · (𝑘 𝑥)) ∈ 𝐾)
1501493impa 1109 . . . . . 6 ((𝜑𝑘 ∈ ℕ0𝑥𝐾) → ((𝐴𝑘) · (𝑘 𝑥)) ∈ 𝐾)
1511503com23 1126 . . . . 5 ((𝜑𝑥𝐾𝑘 ∈ ℕ0) → ((𝐴𝑘) · (𝑘 𝑥)) ∈ 𝐾)
1521513expb 1120 . . . 4 ((𝜑 ∧ (𝑥𝐾𝑘 ∈ ℕ0)) → ((𝐴𝑘) · (𝑘 𝑥)) ∈ 𝐾)
153135mptexd 7245 . . . . 5 (𝜑 → (𝑘 ∈ ℕ0 ↦ (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥)))) ∈ V)
154 funmpt 6603 . . . . . 6 Fun (𝑘 ∈ ℕ0 ↦ (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥))))
155154a1i 11 . . . . 5 (𝜑 → Fun (𝑘 ∈ ℕ0 ↦ (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥)))))
156 fvexd 6920 . . . . 5 (𝜑 → (0g‘(𝑆s 𝐾)) ∈ V)
15712, 8, 6, 52coe1sfi 22216 . . . . . . 7 (𝑀𝐵𝐴 finSupp (0g𝑈))
1585, 157syl 17 . . . . . 6 (𝜑𝐴 finSupp (0g𝑈))
159158fsuppimpd 9410 . . . . 5 (𝜑 → (𝐴 supp (0g𝑈)) ∈ Fin)
16012, 8, 6, 24coe1f 22214 . . . . . . . . . . . . . . . . 17 (𝑀𝐵𝐴:ℕ0⟶(Base‘𝑈))
1615, 160syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐴:ℕ0⟶(Base‘𝑈))
162161ffnd 6736 . . . . . . . . . . . . . . 15 (𝜑𝐴 Fn ℕ0)
163162adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) → 𝐴 Fn ℕ0)
164134a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) → ℕ0 ∈ V)
165 fvexd 6920 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) → (0g𝑈) ∈ V)
166 simpr 484 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) → 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈))))
167163, 164, 165, 166fvdifsupp 8197 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) → (𝐴𝑘) = (0g𝑈))
168 eqid 2736 . . . . . . . . . . . . . . . 16 (0g𝑆) = (0g𝑆)
1692, 168subrg0 20580 . . . . . . . . . . . . . . 15 (𝑅 ∈ (SubRing‘𝑆) → (0g𝑆) = (0g𝑈))
1701, 169syl 17 . . . . . . . . . . . . . 14 (𝜑 → (0g𝑆) = (0g𝑈))
171170adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) → (0g𝑆) = (0g𝑈))
172167, 171eqtr4d 2779 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) → (𝐴𝑘) = (0g𝑆))
173172adantr 480 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) ∧ 𝑥𝐾) → (𝐴𝑘) = (0g𝑆))
174173oveq1d 7447 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) ∧ 𝑥𝐾) → ((𝐴𝑘) · (𝑘 𝑥)) = ((0g𝑆) · (𝑘 𝑥)))
175136ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) ∧ 𝑥𝐾) → 𝑆 ∈ Ring)
176175, 145syl 17 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) ∧ 𝑥𝐾) → (mulGrp‘𝑆) ∈ Mnd)
177 simplr 768 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) ∧ 𝑥𝐾) → 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈))))
178177eldifad 3962 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) ∧ 𝑥𝐾) → 𝑘 ∈ ℕ0)
179 simpr 484 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) ∧ 𝑥𝐾) → 𝑥𝐾)
180144, 124, 176, 178, 179mulgnn0cld 19114 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) ∧ 𝑥𝐾) → (𝑘 𝑥) ∈ 𝐾)
18117, 106, 168ringlz 20291 . . . . . . . . . . 11 ((𝑆 ∈ Ring ∧ (𝑘 𝑥) ∈ 𝐾) → ((0g𝑆) · (𝑘 𝑥)) = (0g𝑆))
182175, 180, 181syl2anc 584 . . . . . . . . . 10 (((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) ∧ 𝑥𝐾) → ((0g𝑆) · (𝑘 𝑥)) = (0g𝑆))
183174, 182eqtrd 2776 . . . . . . . . 9 (((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) ∧ 𝑥𝐾) → ((𝐴𝑘) · (𝑘 𝑥)) = (0g𝑆))
184183mpteq2dva 5241 . . . . . . . 8 ((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) → (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥))) = (𝑥𝐾 ↦ (0g𝑆)))
185 fconstmpt 5746 . . . . . . . 8 (𝐾 × {(0g𝑆)}) = (𝑥𝐾 ↦ (0g𝑆))
186184, 185eqtr4di 2794 . . . . . . 7 ((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) → (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥))) = (𝐾 × {(0g𝑆)}))
187137cmnmndd 19823 . . . . . . . . 9 (𝜑𝑆 ∈ Mnd)
18819, 168pws0g 18787 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ 𝐾 ∈ V) → (𝐾 × {(0g𝑆)}) = (0g‘(𝑆s 𝐾)))
189187, 133, 188syl2anc 584 . . . . . . . 8 (𝜑 → (𝐾 × {(0g𝑆)}) = (0g‘(𝑆s 𝐾)))
190189adantr 480 . . . . . . 7 ((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) → (𝐾 × {(0g𝑆)}) = (0g‘(𝑆s 𝐾)))
191186, 190eqtrd 2776 . . . . . 6 ((𝜑𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g𝑈)))) → (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥))) = (0g‘(𝑆s 𝐾)))
192191, 135suppss2 8226 . . . . 5 (𝜑 → ((𝑘 ∈ ℕ0 ↦ (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥)))) supp (0g‘(𝑆s 𝐾))) ⊆ (𝐴 supp (0g𝑈)))
193 suppssfifsupp 9421 . . . . 5 ((((𝑘 ∈ ℕ0 ↦ (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥)))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥)))) ∧ (0g‘(𝑆s 𝐾)) ∈ V) ∧ ((𝐴 supp (0g𝑈)) ∈ Fin ∧ ((𝑘 ∈ ℕ0 ↦ (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥)))) supp (0g‘(𝑆s 𝐾))) ⊆ (𝐴 supp (0g𝑈)))) → (𝑘 ∈ ℕ0 ↦ (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥)))) finSupp (0g‘(𝑆s 𝐾)))
194153, 155, 156, 159, 192, 193syl32anc 1379 . . . 4 (𝜑 → (𝑘 ∈ ℕ0 ↦ (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥)))) finSupp (0g‘(𝑆s 𝐾)))
19519, 17, 132, 133, 135, 137, 152, 194pwsgsum 20001 . . 3 (𝜑 → ((𝑆s 𝐾) Σg (𝑘 ∈ ℕ0 ↦ (𝑥𝐾 ↦ ((𝐴𝑘) · (𝑘 𝑥))))) = (𝑥𝐾 ↦ (𝑆 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑥))))))
19678, 131, 1953eqtrd 2780 . 2 (𝜑 → (𝑄‘(𝑊 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘)( ·𝑠𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1𝑈)))))) = (𝑥𝐾 ↦ (𝑆 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑥))))))
19715, 196eqtrd 2776 1 (𝜑 → (𝑄𝑀) = (𝑥𝐾 ↦ (𝑆 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑥))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  wrex 3069  Vcvv 3479  cdif 3947  wss 3950  {csn 4625   class class class wbr 5142  cmpt 5224   × cxp 5682  Fun wfun 6554   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  f cof 7696   supp csupp 8186  Fincfn 8986   finSupp cfsupp 9402   < clt 11296  0cn0 12528  Basecbs 17248  s cress 17275  .rcmulr 17299  Scalarcsca 17301   ·𝑠 cvsca 17302  0gc0g 17485   Σg cgsu 17486  s cpws 17492  Mndcmnd 18748  .gcmg 19086  mulGrpcmgp 20138  Ringcrg 20231  CRingccrg 20232   RingHom crh 20470  SubRingcsubrg 20570  LModclmod 20859  AssAlgcasa 21871  algSccascl 21873  var1cv1 22178  Poly1cpl1 22179  coe1cco1 22180   evalSub1 ces1 22318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-ofr 7699  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-pm 8870  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-sup 9483  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-ghm 19232  df-cntz 19336  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-srg 20185  df-ring 20233  df-cring 20234  df-rhm 20473  df-subrng 20547  df-subrg 20571  df-lmod 20861  df-lss 20931  df-lsp 20971  df-assa 21874  df-asp 21875  df-ascl 21876  df-psr 21930  df-mvr 21931  df-mpl 21932  df-opsr 21934  df-evls 22099  df-evl 22100  df-psr1 22182  df-vr1 22183  df-ply1 22184  df-coe1 22185  df-evls1 22320  df-evl1 22321
This theorem is referenced by:  ressply1evl  22375  evl1fpws  33591  evls1fldgencl  33721
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