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Theorem mplcoe3 22041
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 4527 . . . . . . . . . . 11 (𝑥 = 0 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 0, 0))
3 ifid 4563 . . . . . . . . . . 11 if(𝑘 = 𝑋, 0, 0) = 0
42, 3eqtrdi 2782 . . . . . . . . . 10 (𝑥 = 0 → if(𝑘 = 𝑋, 𝑥, 0) = 0)
54mpteq2dv 5247 . . . . . . . . 9 (𝑥 = 0 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ 0))
6 fconstmpt 5736 . . . . . . . . 9 (𝐼 × {0}) = (𝑘𝐼 ↦ 0)
75, 6eqtr4di 2784 . . . . . . . 8 (𝑥 = 0 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝐼 × {0}))
87eqeq2d 2737 . . . . . . 7 (𝑥 = 0 → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝐼 × {0})))
98ifbid 4546 . . . . . 6 (𝑥 = 0 → if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝐼 × {0}), 1 , 0 ))
109mpteq2dv 5247 . . . . 5 (𝑥 = 0 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
11 oveq1 7423 . . . . 5 (𝑥 = 0 → (𝑥 (𝑉𝑋)) = (0 (𝑉𝑋)))
1210, 11eqeq12d 2742 . . . 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 339 . . 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 4527 . . . . . . . . 9 (𝑥 = 𝑛 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑛, 0))
1514mpteq2dv 5247 . . . . . . . 8 (𝑥 = 𝑛 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
1615eqeq2d 2737 . . . . . . 7 (𝑥 = 𝑛 → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0))))
1716ifbid 4546 . . . . . 6 (𝑥 = 𝑛 → if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))
1817mpteq2dv 5247 . . . . 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 7423 . . . . 5 (𝑥 = 𝑛 → (𝑥 (𝑉𝑋)) = (𝑛 (𝑉𝑋)))
2018, 19eqeq12d 2742 . . . 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 339 . . 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 4527 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
2322mpteq2dv 5247 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)))
2423eqeq2d 2737 . . . . . . 7 (𝑥 = (𝑛 + 1) → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0))))
2524ifbid 4546 . . . . . 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 5247 . . . . 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 7423 . . . . 5 (𝑥 = (𝑛 + 1) → (𝑥 (𝑉𝑋)) = ((𝑛 + 1) (𝑉𝑋)))
2826, 27eqeq12d 2742 . . . 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 339 . . 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 4527 . . . . . . . . 9 (𝑥 = 𝑁 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑁, 0))
3130mpteq2dv 5247 . . . . . . . 8 (𝑥 = 𝑁 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)))
3231eqeq2d 2737 . . . . . . 7 (𝑥 = 𝑁 → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0))))
3332ifbid 4546 . . . . . 6 (𝑥 = 𝑁 → if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 ))
3433mpteq2dv 5247 . . . . 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 7423 . . . . 5 (𝑥 = 𝑁 → (𝑥 (𝑉𝑋)) = (𝑁 (𝑉𝑋)))
3634, 35eqeq12d 2742 . . . 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 339 . . 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 2726 . . . . . 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 21997 . . . . 5 (𝜑 → (𝑉𝑋) ∈ (Base‘𝑃))
45 mplcoe2.g . . . . . . 7 𝐺 = (mulGrp‘𝑃)
4645, 40mgpbas 20119 . . . . . 6 (Base‘𝑃) = (Base‘𝐺)
47 eqid 2726 . . . . . . 7 (1r𝑃) = (1r𝑃)
4845, 47ringidval 20162 . . . . . 6 (1r𝑃) = (0g𝐺)
49 mplcoe2.m . . . . . 6 = (.g𝐺)
5046, 48, 49mulg0 19064 . . . . 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 22017 . . . 4 (𝜑 → (1r𝑃) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
5651, 55eqtr2d 2767 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (0 (𝑉𝑋)))
57 oveq1 7423 . . . . . 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 479 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → 𝐼𝑊)
5942adantr 479 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
6052snifpsrbag 21915 . . . . . . . . . 10 ((𝐼𝑊𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
6141, 60sylan 578 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
62 eqid 2726 . . . . . . . . 9 (.r𝑃) = (.r𝑃)
63 1nn0 12534 . . . . . . . . . . 11 1 ∈ ℕ0
6463a1i 11 . . . . . . . . . 10 (𝑛 ∈ ℕ0 → 1 ∈ ℕ0)
6552snifpsrbag 21915 . . . . . . . . . 10 ((𝐼𝑊 ∧ 1 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
6641, 64, 65syl2an 594 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
6738, 40, 53, 54, 52, 58, 59, 61, 62, 66mplmonmul 22039 . . . . . . . 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 479 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → 𝑋𝐼)
6939, 52, 53, 54, 58, 59, 68mvrval 21987 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → (𝑉𝑋) = (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )))
7069eqcomd 2732 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )) = (𝑉𝑋))
7170oveq2d 7432 . . . . . . . 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 767 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ0) ∧ 𝑘𝐼) → 𝑛 ∈ ℕ0)
73 0nn0 12533 . . . . . . . . . . . . . 14 0 ∈ ℕ0
74 ifcl 4568 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑘 = 𝑋, 𝑛, 0) ∈ ℕ0)
7572, 73, 74sylancl 584 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ0) ∧ 𝑘𝐼) → if(𝑘 = 𝑋, 𝑛, 0) ∈ ℕ0)
7663, 73ifcli 4570 . . . . . . . . . . . . . 14 if(𝑘 = 𝑋, 1, 0) ∈ ℕ0
7776a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ0) ∧ 𝑘𝐼) → if(𝑘 = 𝑋, 1, 0) ∈ ℕ0)
78 eqidd 2727 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
79 eqidd 2727 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)))
8058, 75, 77, 78, 79offval2 7702 . . . . . . . . . . . 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 4529 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 𝑛)
82 iftrue 4529 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 1)
8381, 82oveq12d 7434 . . . . . . . . . . . . . . 15 (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (𝑛 + 1))
84 iftrue 4529 . . . . . . . . . . . . . . 15 (𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = (𝑛 + 1))
8583, 84eqtr4d 2769 . . . . . . . . . . . . . 14 (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
86 00id 11430 . . . . . . . . . . . . . . 15 (0 + 0) = 0
87 iffalse 4532 . . . . . . . . . . . . . . . 16 𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 0)
88 iffalse 4532 . . . . . . . . . . . . . . . 16 𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 0)
8987, 88oveq12d 7434 . . . . . . . . . . . . . . 15 𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (0 + 0))
90 iffalse 4532 . . . . . . . . . . . . . . 15 𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = 0)
9186, 89, 903eqtr4a 2792 . . . . . . . . . . . . . 14 𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
9285, 91pm2.61i 182 . . . . . . . . . . . . 13 (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0)
9392mpteq2i 5250 . . . . . . . . . . . 12 (𝑘𝐼 ↦ (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0))) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0))
9480, 93eqtrdi 2782 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → ((𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∘f + (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0))) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)))
9594eqeq2d 2737 . . . . . . . . . 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 4546 . . . . . . . . 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 5247 . . . . . . . 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 2775 . . . . . . 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)))
9938, 41, 42mplringd 22028 . . . . . . . . . 10 (𝜑𝑃 ∈ Ring)
10045ringmgp 20218 . . . . . . . . . 10 (𝑃 ∈ Ring → 𝐺 ∈ Mnd)
10199, 100syl 17 . . . . . . . . 9 (𝜑𝐺 ∈ Mnd)
102101adantr 479 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝐺 ∈ Mnd)
103 simpr 483 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
10444adantr 479 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → (𝑉𝑋) ∈ (Base‘𝑃))
10545, 62mgpplusg 20117 . . . . . . . . 9 (.r𝑃) = (+g𝐺)
10646, 49, 105mulgnn0p1 19075 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑛 ∈ ℕ0 ∧ (𝑉𝑋) ∈ (Base‘𝑃)) → ((𝑛 + 1) (𝑉𝑋)) = ((𝑛 (𝑉𝑋))(.r𝑃)(𝑉𝑋)))
107102, 103, 104, 106syl3anc 1368 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 + 1) (𝑉𝑋)) = ((𝑛 (𝑉𝑋))(.r𝑃)(𝑉𝑋)))
10898, 107eqeq12d 2742 . . . . . 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))))
10957, 108imbitrrid 245 . . . . 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))))
110109expcom 412 . . . 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)))))
111110a2d 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)))))
11213, 21, 29, 37, 56, 111nn0ind 12703 . 2 (𝑁 ∈ ℕ0 → (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )) = (𝑁 (𝑉𝑋))))
1131, 112mpcom 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 394   = wceq 1534  wcel 2099  {crab 3419  ifcif 4523  {csn 4623  cmpt 5228   × cxp 5672  ccnv 5673  cima 5677  cfv 6546  (class class class)co 7416  f cof 7680  m cmap 8847  Fincfn 8966  0cc0 11149  1c1 11150   + caddc 11152  cn 12258  0cn0 12518  Basecbs 17208  .rcmulr 17262  0gc0g 17449  Mndcmnd 18722  .gcmg 19057  mulGrpcmgp 20113  1rcur 20160  Ringcrg 20212   mVar cmvr 21898   mPoly cmpl 21899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-iin 4996  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-isom 6555  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-ofr 7683  df-om 7869  df-1st 7995  df-2nd 7996  df-supp 8167  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-2o 8489  df-er 8726  df-map 8849  df-pm 8850  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fsupp 9399  df-sup 9478  df-oi 9546  df-card 9975  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-nn 12259  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12519  df-z 12605  df-dec 12724  df-uz 12869  df-fz 13533  df-fzo 13676  df-seq 14016  df-hash 14343  df-struct 17144  df-sets 17161  df-slot 17179  df-ndx 17191  df-base 17209  df-ress 17238  df-plusg 17274  df-mulr 17275  df-sca 17277  df-vsca 17278  df-ip 17279  df-tset 17280  df-ple 17281  df-ds 17283  df-hom 17285  df-cco 17286  df-0g 17451  df-gsum 17452  df-prds 17457  df-pws 17459  df-mre 17594  df-mrc 17595  df-acs 17597  df-mgm 18628  df-sgrp 18707  df-mnd 18723  df-mhm 18768  df-submnd 18769  df-grp 18926  df-minusg 18927  df-mulg 19058  df-subg 19113  df-ghm 19203  df-cntz 19307  df-cmn 19776  df-abl 19777  df-mgp 20114  df-rng 20132  df-ur 20161  df-ring 20214  df-subrng 20524  df-subrg 20549  df-psr 21902  df-mvr 21903  df-mpl 21904
This theorem is referenced by:  mplcoe5  22043  coe1tm  22260
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