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Theorem mplcoe3 20175
Description: Decompose a monomial in one variable into a power of a variable. (Contributed by Mario Carneiro, 7-Jan-2015.) (Proof shortened by AV, 18-Jul-2019.)
Hypotheses
Ref Expression
mplcoe1.p 𝑃 = (𝐼 mPoly 𝑅)
mplcoe1.d 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
mplcoe1.z 0 = (0g𝑅)
mplcoe1.o 1 = (1r𝑅)
mplcoe1.i (𝜑𝐼𝑊)
mplcoe2.g 𝐺 = (mulGrp‘𝑃)
mplcoe2.m = (.g𝐺)
mplcoe2.v 𝑉 = (𝐼 mVar 𝑅)
mplcoe3.r (𝜑𝑅 ∈ Ring)
mplcoe3.x (𝜑𝑋𝐼)
mplcoe3.n (𝜑𝑁 ∈ ℕ0)
Assertion
Ref Expression
mplcoe3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )) = (𝑁 (𝑉𝑋)))
Distinct variable groups:   ,𝑘   𝑦,𝑘, 1   𝑘,𝐺   𝑓,𝑘,𝑦,𝐼   𝑘,𝑁,𝑦   𝜑,𝑘,𝑦   𝑅,𝑓,𝑦   𝐷,𝑘,𝑦   𝑃,𝑘   𝑘,𝑉   0 ,𝑓,𝑘,𝑦   𝑓,𝑋,𝑘,𝑦   𝑘,𝑊,𝑦
Allowed substitution hints:   𝜑(𝑓)   𝐷(𝑓)   𝑃(𝑦,𝑓)   𝑅(𝑘)   1 (𝑓)   (𝑦,𝑓)   𝐺(𝑦,𝑓)   𝑁(𝑓)   𝑉(𝑦,𝑓)   𝑊(𝑓)

Proof of Theorem mplcoe3
Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mplcoe3.n . 2 (𝜑𝑁 ∈ ℕ0)
2 ifeq1 4467 . . . . . . . . . . 11 (𝑥 = 0 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 0, 0))
3 ifid 4502 . . . . . . . . . . 11 if(𝑘 = 𝑋, 0, 0) = 0
42, 3syl6eq 2869 . . . . . . . . . 10 (𝑥 = 0 → if(𝑘 = 𝑋, 𝑥, 0) = 0)
54mpteq2dv 5153 . . . . . . . . 9 (𝑥 = 0 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ 0))
6 fconstmpt 5607 . . . . . . . . 9 (𝐼 × {0}) = (𝑘𝐼 ↦ 0)
75, 6syl6eqr 2871 . . . . . . . 8 (𝑥 = 0 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝐼 × {0}))
87eqeq2d 2829 . . . . . . 7 (𝑥 = 0 → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝐼 × {0})))
98ifbid 4485 . . . . . 6 (𝑥 = 0 → if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝐼 × {0}), 1 , 0 ))
109mpteq2dv 5153 . . . . 5 (𝑥 = 0 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
11 oveq1 7152 . . . . 5 (𝑥 = 0 → (𝑥 (𝑉𝑋)) = (0 (𝑉𝑋)))
1210, 11eqeq12d 2834 . . . 4 (𝑥 = 0 → ((𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑥 (𝑉𝑋)) ↔ (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (0 (𝑉𝑋))))
1312imbi2d 342 . . 3 (𝑥 = 0 → ((𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑥 (𝑉𝑋))) ↔ (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (0 (𝑉𝑋)))))
14 ifeq1 4467 . . . . . . . . 9 (𝑥 = 𝑛 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑛, 0))
1514mpteq2dv 5153 . . . . . . . 8 (𝑥 = 𝑛 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
1615eqeq2d 2829 . . . . . . 7 (𝑥 = 𝑛 → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0))))
1716ifbid 4485 . . . . . 6 (𝑥 = 𝑛 → if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))
1817mpteq2dv 5153 . . . . 5 (𝑥 = 𝑛 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 )))
19 oveq1 7152 . . . . 5 (𝑥 = 𝑛 → (𝑥 (𝑉𝑋)) = (𝑛 (𝑉𝑋)))
2018, 19eqeq12d 2834 . . . 4 (𝑥 = 𝑛 → ((𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑥 (𝑉𝑋)) ↔ (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 )) = (𝑛 (𝑉𝑋))))
2120imbi2d 342 . . 3 (𝑥 = 𝑛 → ((𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑥 (𝑉𝑋))) ↔ (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 )) = (𝑛 (𝑉𝑋)))))
22 ifeq1 4467 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
2322mpteq2dv 5153 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)))
2423eqeq2d 2829 . . . . . . 7 (𝑥 = (𝑛 + 1) → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0))))
2524ifbid 4485 . . . . . 6 (𝑥 = (𝑛 + 1) → if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 ))
2625mpteq2dv 5153 . . . . 5 (𝑥 = (𝑛 + 1) → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )))
27 oveq1 7152 . . . . 5 (𝑥 = (𝑛 + 1) → (𝑥 (𝑉𝑋)) = ((𝑛 + 1) (𝑉𝑋)))
2826, 27eqeq12d 2834 . . . 4 (𝑥 = (𝑛 + 1) → ((𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑥 (𝑉𝑋)) ↔ (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )) = ((𝑛 + 1) (𝑉𝑋))))
2928imbi2d 342 . . 3 (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑥 (𝑉𝑋))) ↔ (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )) = ((𝑛 + 1) (𝑉𝑋)))))
30 ifeq1 4467 . . . . . . . . 9 (𝑥 = 𝑁 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑁, 0))
3130mpteq2dv 5153 . . . . . . . 8 (𝑥 = 𝑁 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)))
3231eqeq2d 2829 . . . . . . 7 (𝑥 = 𝑁 → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0))))
3332ifbid 4485 . . . . . 6 (𝑥 = 𝑁 → if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 ))
3433mpteq2dv 5153 . . . . 5 (𝑥 = 𝑁 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )))
35 oveq1 7152 . . . . 5 (𝑥 = 𝑁 → (𝑥 (𝑉𝑋)) = (𝑁 (𝑉𝑋)))
3634, 35eqeq12d 2834 . . . 4 (𝑥 = 𝑁 → ((𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑥 (𝑉𝑋)) ↔ (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )) = (𝑁 (𝑉𝑋))))
3736imbi2d 342 . . 3 (𝑥 = 𝑁 → ((𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑥 (𝑉𝑋))) ↔ (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )) = (𝑁 (𝑉𝑋)))))
38 mplcoe1.p . . . . . 6 𝑃 = (𝐼 mPoly 𝑅)
39 mplcoe2.v . . . . . 6 𝑉 = (𝐼 mVar 𝑅)
40 eqid 2818 . . . . . 6 (Base‘𝑃) = (Base‘𝑃)
41 mplcoe1.i . . . . . 6 (𝜑𝐼𝑊)
42 mplcoe3.r . . . . . 6 (𝜑𝑅 ∈ Ring)
43 mplcoe3.x . . . . . 6 (𝜑𝑋𝐼)
4438, 39, 40, 41, 42, 43mvrcl 20157 . . . . 5 (𝜑 → (𝑉𝑋) ∈ (Base‘𝑃))
45 mplcoe2.g . . . . . . 7 𝐺 = (mulGrp‘𝑃)
4645, 40mgpbas 19174 . . . . . 6 (Base‘𝑃) = (Base‘𝐺)
47 eqid 2818 . . . . . . 7 (1r𝑃) = (1r𝑃)
4845, 47ringidval 19182 . . . . . 6 (1r𝑃) = (0g𝐺)
49 mplcoe2.m . . . . . 6 = (.g𝐺)
5046, 48, 49mulg0 18169 . . . . 5 ((𝑉𝑋) ∈ (Base‘𝑃) → (0 (𝑉𝑋)) = (1r𝑃))
5144, 50syl 17 . . . 4 (𝜑 → (0 (𝑉𝑋)) = (1r𝑃))
52 mplcoe1.d . . . . 5 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
53 mplcoe1.z . . . . 5 0 = (0g𝑅)
54 mplcoe1.o . . . . 5 1 = (1r𝑅)
5538, 52, 53, 54, 47, 41, 42mpl1 20152 . . . 4 (𝜑 → (1r𝑃) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
5651, 55eqtr2d 2854 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (0 (𝑉𝑋)))
57 oveq1 7152 . . . . . 6 ((𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 )) = (𝑛 (𝑉𝑋)) → ((𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))(.r𝑃)(𝑉𝑋)) = ((𝑛 (𝑉𝑋))(.r𝑃)(𝑉𝑋)))
5841adantr 481 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → 𝐼𝑊)
5942adantr 481 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
6052snifpsrbag 20074 . . . . . . . . . 10 ((𝐼𝑊𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
6141, 60sylan 580 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
62 eqid 2818 . . . . . . . . 9 (.r𝑃) = (.r𝑃)
63 1nn0 11901 . . . . . . . . . . 11 1 ∈ ℕ0
6463a1i 11 . . . . . . . . . 10 (𝑛 ∈ ℕ0 → 1 ∈ ℕ0)
6552snifpsrbag 20074 . . . . . . . . . 10 ((𝐼𝑊 ∧ 1 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
6641, 64, 65syl2an 595 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
6738, 40, 53, 54, 52, 58, 59, 61, 62, 66mplmonmul 20173 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → ((𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))(.r𝑃)(𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 ))) = (𝑦𝐷 ↦ if(𝑦 = ((𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∘f + (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0))), 1 , 0 )))
6843adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → 𝑋𝐼)
6939, 52, 53, 54, 58, 59, 68mvrval 20129 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → (𝑉𝑋) = (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )))
7069eqcomd 2824 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )) = (𝑉𝑋))
7170oveq2d 7161 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → ((𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))(.r𝑃)(𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 ))) = ((𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))(.r𝑃)(𝑉𝑋)))
72 simplr 765 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ0) ∧ 𝑘𝐼) → 𝑛 ∈ ℕ0)
73 0nn0 11900 . . . . . . . . . . . . . 14 0 ∈ ℕ0
74 ifcl 4507 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑘 = 𝑋, 𝑛, 0) ∈ ℕ0)
7572, 73, 74sylancl 586 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ0) ∧ 𝑘𝐼) → if(𝑘 = 𝑋, 𝑛, 0) ∈ ℕ0)
7663, 73ifcli 4509 . . . . . . . . . . . . . 14 if(𝑘 = 𝑋, 1, 0) ∈ ℕ0
7776a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ0) ∧ 𝑘𝐼) → if(𝑘 = 𝑋, 1, 0) ∈ ℕ0)
78 eqidd 2819 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
79 eqidd 2819 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)))
8058, 75, 77, 78, 79offval2 7415 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ0) → ((𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∘f + (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0))) = (𝑘𝐼 ↦ (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0))))
81 iftrue 4469 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 𝑛)
82 iftrue 4469 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 1)
8381, 82oveq12d 7163 . . . . . . . . . . . . . . 15 (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (𝑛 + 1))
84 iftrue 4469 . . . . . . . . . . . . . . 15 (𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = (𝑛 + 1))
8583, 84eqtr4d 2856 . . . . . . . . . . . . . 14 (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
86 00id 10803 . . . . . . . . . . . . . . 15 (0 + 0) = 0
87 iffalse 4472 . . . . . . . . . . . . . . . 16 𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 0)
88 iffalse 4472 . . . . . . . . . . . . . . . 16 𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 0)
8987, 88oveq12d 7163 . . . . . . . . . . . . . . 15 𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (0 + 0))
90 iffalse 4472 . . . . . . . . . . . . . . 15 𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = 0)
9186, 89, 903eqtr4a 2879 . . . . . . . . . . . . . 14 𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
9285, 91pm2.61i 183 . . . . . . . . . . . . 13 (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0)
9392mpteq2i 5149 . . . . . . . . . . . 12 (𝑘𝐼 ↦ (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0))) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0))
9480, 93syl6eq 2869 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → ((𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∘f + (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0))) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)))
9594eqeq2d 2829 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → (𝑦 = ((𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∘f + (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0))) ↔ 𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0))))
9695ifbid 4485 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → if(𝑦 = ((𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∘f + (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0))), 1 , 0 ) = if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 ))
9796mpteq2dv 5153 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → (𝑦𝐷 ↦ if(𝑦 = ((𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∘f + (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0))), 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )))
9867, 71, 973eqtr3rd 2862 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )) = ((𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))(.r𝑃)(𝑉𝑋)))
9938mplring 20160 . . . . . . . . . . 11 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ Ring)
10041, 42, 99syl2anc 584 . . . . . . . . . 10 (𝜑𝑃 ∈ Ring)
10145ringmgp 19232 . . . . . . . . . 10 (𝑃 ∈ Ring → 𝐺 ∈ Mnd)
102100, 101syl 17 . . . . . . . . 9 (𝜑𝐺 ∈ Mnd)
103102adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝐺 ∈ Mnd)
104 simpr 485 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
10544adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → (𝑉𝑋) ∈ (Base‘𝑃))
10645, 62mgpplusg 19172 . . . . . . . . 9 (.r𝑃) = (+g𝐺)
10746, 49, 106mulgnn0p1 18177 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑛 ∈ ℕ0 ∧ (𝑉𝑋) ∈ (Base‘𝑃)) → ((𝑛 + 1) (𝑉𝑋)) = ((𝑛 (𝑉𝑋))(.r𝑃)(𝑉𝑋)))
108103, 104, 105, 107syl3anc 1363 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 + 1) (𝑉𝑋)) = ((𝑛 (𝑉𝑋))(.r𝑃)(𝑉𝑋)))
10998, 108eqeq12d 2834 . . . . . 6 ((𝜑𝑛 ∈ ℕ0) → ((𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )) = ((𝑛 + 1) (𝑉𝑋)) ↔ ((𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))(.r𝑃)(𝑉𝑋)) = ((𝑛 (𝑉𝑋))(.r𝑃)(𝑉𝑋))))
11057, 109syl5ibr 247 . . . . 5 ((𝜑𝑛 ∈ ℕ0) → ((𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 )) = (𝑛 (𝑉𝑋)) → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )) = ((𝑛 + 1) (𝑉𝑋))))
111110expcom 414 . . . 4 (𝑛 ∈ ℕ0 → (𝜑 → ((𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 )) = (𝑛 (𝑉𝑋)) → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )) = ((𝑛 + 1) (𝑉𝑋)))))
112111a2d 29 . . 3 (𝑛 ∈ ℕ0 → ((𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 )) = (𝑛 (𝑉𝑋))) → (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )) = ((𝑛 + 1) (𝑉𝑋)))))
11313, 21, 29, 37, 56, 112nn0ind 12065 . 2 (𝑁 ∈ ℕ0 → (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )) = (𝑁 (𝑉𝑋))))
1141, 113mpcom 38 1 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )) = (𝑁 (𝑉𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  {crab 3139  ifcif 4463  {csn 4557  cmpt 5137   × cxp 5546  ccnv 5547  cima 5551  cfv 6348  (class class class)co 7145  f cof 7396  m cmap 8395  Fincfn 8497  0cc0 10525  1c1 10526   + caddc 10528  cn 11626  0cn0 11885  Basecbs 16471  .rcmulr 16554  0gc0g 16701  Mndcmnd 17899  .gcmg 18162  mulGrpcmgp 19168  1rcur 19180  Ringcrg 19226   mVar cmvr 20060   mPoly cmpl 20061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-ofr 7399  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-seq 13358  df-hash 13679  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-sca 16569  df-vsca 16570  df-tset 16572  df-0g 16703  df-gsum 16704  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mhm 17944  df-submnd 17945  df-grp 18044  df-minusg 18045  df-mulg 18163  df-subg 18214  df-ghm 18294  df-cntz 18385  df-cmn 18837  df-abl 18838  df-mgp 19169  df-ur 19181  df-ring 19228  df-subrg 19462  df-psr 20064  df-mvr 20065  df-mpl 20066
This theorem is referenced by:  mplcoe5  20177  coe1tm  20369
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