MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mplcoe3 Structured version   Visualization version   GIF version

Theorem mplcoe3 21345
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 4482 . . . . . . . . . . 11 (𝑥 = 0 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 0, 0))
3 ifid 4518 . . . . . . . . . . 11 if(𝑘 = 𝑋, 0, 0) = 0
42, 3eqtrdi 2793 . . . . . . . . . 10 (𝑥 = 0 → if(𝑘 = 𝑋, 𝑥, 0) = 0)
54mpteq2dv 5199 . . . . . . . . 9 (𝑥 = 0 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ 0))
6 fconstmpt 5685 . . . . . . . . 9 (𝐼 × {0}) = (𝑘𝐼 ↦ 0)
75, 6eqtr4di 2795 . . . . . . . 8 (𝑥 = 0 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝐼 × {0}))
87eqeq2d 2748 . . . . . . 7 (𝑥 = 0 → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝐼 × {0})))
98ifbid 4501 . . . . . 6 (𝑥 = 0 → if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝐼 × {0}), 1 , 0 ))
109mpteq2dv 5199 . . . . 5 (𝑥 = 0 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
11 oveq1 7349 . . . . 5 (𝑥 = 0 → (𝑥 (𝑉𝑋)) = (0 (𝑉𝑋)))
1210, 11eqeq12d 2753 . . . 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 341 . . 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 4482 . . . . . . . . 9 (𝑥 = 𝑛 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑛, 0))
1514mpteq2dv 5199 . . . . . . . 8 (𝑥 = 𝑛 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
1615eqeq2d 2748 . . . . . . 7 (𝑥 = 𝑛 → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0))))
1716ifbid 4501 . . . . . 6 (𝑥 = 𝑛 → if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))
1817mpteq2dv 5199 . . . . 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 7349 . . . . 5 (𝑥 = 𝑛 → (𝑥 (𝑉𝑋)) = (𝑛 (𝑉𝑋)))
2018, 19eqeq12d 2753 . . . 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 341 . . 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 4482 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
2322mpteq2dv 5199 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)))
2423eqeq2d 2748 . . . . . . 7 (𝑥 = (𝑛 + 1) → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0))))
2524ifbid 4501 . . . . . 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 5199 . . . . 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 7349 . . . . 5 (𝑥 = (𝑛 + 1) → (𝑥 (𝑉𝑋)) = ((𝑛 + 1) (𝑉𝑋)))
2826, 27eqeq12d 2753 . . . 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 341 . . 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 4482 . . . . . . . . 9 (𝑥 = 𝑁 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑁, 0))
3130mpteq2dv 5199 . . . . . . . 8 (𝑥 = 𝑁 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)))
3231eqeq2d 2748 . . . . . . 7 (𝑥 = 𝑁 → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0))))
3332ifbid 4501 . . . . . 6 (𝑥 = 𝑁 → if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 ))
3433mpteq2dv 5199 . . . . 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 7349 . . . . 5 (𝑥 = 𝑁 → (𝑥 (𝑉𝑋)) = (𝑁 (𝑉𝑋)))
3634, 35eqeq12d 2753 . . . 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 341 . . 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 2737 . . . . . 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 21327 . . . . 5 (𝜑 → (𝑉𝑋) ∈ (Base‘𝑃))
45 mplcoe2.g . . . . . . 7 𝐺 = (mulGrp‘𝑃)
4645, 40mgpbas 19821 . . . . . 6 (Base‘𝑃) = (Base‘𝐺)
47 eqid 2737 . . . . . . 7 (1r𝑃) = (1r𝑃)
4845, 47ringidval 19834 . . . . . 6 (1r𝑃) = (0g𝐺)
49 mplcoe2.m . . . . . 6 = (.g𝐺)
5046, 48, 49mulg0 18804 . . . . 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 21322 . . . 4 (𝜑 → (1r𝑃) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
5651, 55eqtr2d 2778 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (0 (𝑉𝑋)))
57 oveq1 7349 . . . . . 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 482 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → 𝐼𝑊)
5942adantr 482 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
6052snifpsrbag 21231 . . . . . . . . . 10 ((𝐼𝑊𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
6141, 60sylan 581 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
62 eqid 2737 . . . . . . . . 9 (.r𝑃) = (.r𝑃)
63 1nn0 12355 . . . . . . . . . . 11 1 ∈ ℕ0
6463a1i 11 . . . . . . . . . 10 (𝑛 ∈ ℕ0 → 1 ∈ ℕ0)
6552snifpsrbag 21231 . . . . . . . . . 10 ((𝐼𝑊 ∧ 1 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
6641, 64, 65syl2an 597 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
6738, 40, 53, 54, 52, 58, 59, 61, 62, 66mplmonmul 21343 . . . . . . . 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 482 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → 𝑋𝐼)
6939, 52, 53, 54, 58, 59, 68mvrval 21296 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → (𝑉𝑋) = (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )))
7069eqcomd 2743 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )) = (𝑉𝑋))
7170oveq2d 7358 . . . . . . . 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 12354 . . . . . . . . . . . . . 14 0 ∈ ℕ0
74 ifcl 4523 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑘 = 𝑋, 𝑛, 0) ∈ ℕ0)
7572, 73, 74sylancl 587 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ0) ∧ 𝑘𝐼) → if(𝑘 = 𝑋, 𝑛, 0) ∈ ℕ0)
7663, 73ifcli 4525 . . . . . . . . . . . . . 14 if(𝑘 = 𝑋, 1, 0) ∈ ℕ0
7776a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ0) ∧ 𝑘𝐼) → if(𝑘 = 𝑋, 1, 0) ∈ ℕ0)
78 eqidd 2738 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
79 eqidd 2738 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)))
8058, 75, 77, 78, 79offval2 7620 . . . . . . . . . . . 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 4484 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 𝑛)
82 iftrue 4484 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 1)
8381, 82oveq12d 7360 . . . . . . . . . . . . . . 15 (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (𝑛 + 1))
84 iftrue 4484 . . . . . . . . . . . . . . 15 (𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = (𝑛 + 1))
8583, 84eqtr4d 2780 . . . . . . . . . . . . . 14 (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
86 00id 11256 . . . . . . . . . . . . . . 15 (0 + 0) = 0
87 iffalse 4487 . . . . . . . . . . . . . . . 16 𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 0)
88 iffalse 4487 . . . . . . . . . . . . . . . 16 𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 0)
8987, 88oveq12d 7360 . . . . . . . . . . . . . . 15 𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (0 + 0))
90 iffalse 4487 . . . . . . . . . . . . . . 15 𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = 0)
9186, 89, 903eqtr4a 2803 . . . . . . . . . . . . . 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 5202 . . . . . . . . . . . 12 (𝑘𝐼 ↦ (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0))) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0))
9480, 93eqtrdi 2793 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → ((𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∘f + (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0))) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)))
9594eqeq2d 2748 . . . . . . . . . 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 4501 . . . . . . . . 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 5199 . . . . . . . 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 2786 . . . . . . 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 21330 . . . . . . . . . . 11 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ Ring)
10041, 42, 99syl2anc 585 . . . . . . . . . 10 (𝜑𝑃 ∈ Ring)
10145ringmgp 19884 . . . . . . . . . 10 (𝑃 ∈ Ring → 𝐺 ∈ Mnd)
102100, 101syl 17 . . . . . . . . 9 (𝜑𝐺 ∈ Mnd)
103102adantr 482 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝐺 ∈ Mnd)
104 simpr 486 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
10544adantr 482 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → (𝑉𝑋) ∈ (Base‘𝑃))
10645, 62mgpplusg 19819 . . . . . . . . 9 (.r𝑃) = (+g𝐺)
10746, 49, 106mulgnn0p1 18812 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑛 ∈ ℕ0 ∧ (𝑉𝑋) ∈ (Base‘𝑃)) → ((𝑛 + 1) (𝑉𝑋)) = ((𝑛 (𝑉𝑋))(.r𝑃)(𝑉𝑋)))
108103, 104, 105, 107syl3anc 1371 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 + 1) (𝑉𝑋)) = ((𝑛 (𝑉𝑋))(.r𝑃)(𝑉𝑋)))
10998, 108eqeq12d 2753 . . . . . 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 246 . . . . 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 415 . . . 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 12521 . 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 397   = wceq 1541  wcel 2106  {crab 3404  ifcif 4478  {csn 4578  cmpt 5180   × cxp 5623  ccnv 5624  cima 5628  cfv 6484  (class class class)co 7342  f cof 7598  m cmap 8691  Fincfn 8809  0cc0 10977  1c1 10978   + caddc 10980  cn 12079  0cn0 12339  Basecbs 17010  .rcmulr 17061  0gc0g 17248  Mndcmnd 18483  .gcmg 18797  mulGrpcmgp 19815  1rcur 19832  Ringcrg 19878   mVar cmvr 21214   mPoly cmpl 21215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655  ax-cnex 11033  ax-resscn 11034  ax-1cn 11035  ax-icn 11036  ax-addcl 11037  ax-addrcl 11038  ax-mulcl 11039  ax-mulrcl 11040  ax-mulcom 11041  ax-addass 11042  ax-mulass 11043  ax-distr 11044  ax-i2m1 11045  ax-1ne0 11046  ax-1rid 11047  ax-rnegex 11048  ax-rrecex 11049  ax-cnre 11050  ax-pre-lttri 11051  ax-pre-lttrn 11052  ax-pre-ltadd 11053  ax-pre-mulgt0 11054
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3921  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4858  df-int 4900  df-iun 4948  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5181  df-tr 5215  df-id 5523  df-eprel 5529  df-po 5537  df-so 5538  df-fr 5580  df-se 5581  df-we 5582  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6243  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-isom 6493  df-riota 7298  df-ov 7345  df-oprab 7346  df-mpo 7347  df-of 7600  df-ofr 7601  df-om 7786  df-1st 7904  df-2nd 7905  df-supp 8053  df-frecs 8172  df-wrecs 8203  df-recs 8277  df-rdg 8316  df-1o 8372  df-er 8574  df-map 8693  df-pm 8694  df-ixp 8762  df-en 8810  df-dom 8811  df-sdom 8812  df-fin 8813  df-fsupp 9232  df-oi 9372  df-card 9801  df-pnf 11117  df-mnf 11118  df-xr 11119  df-ltxr 11120  df-le 11121  df-sub 11313  df-neg 11314  df-nn 12080  df-2 12142  df-3 12143  df-4 12144  df-5 12145  df-6 12146  df-7 12147  df-8 12148  df-9 12149  df-n0 12340  df-z 12426  df-uz 12689  df-fz 13346  df-fzo 13489  df-seq 13828  df-hash 14151  df-struct 16946  df-sets 16963  df-slot 16981  df-ndx 16993  df-base 17011  df-ress 17040  df-plusg 17073  df-mulr 17074  df-sca 17076  df-vsca 17077  df-tset 17079  df-0g 17250  df-gsum 17251  df-mre 17393  df-mrc 17394  df-acs 17396  df-mgm 18424  df-sgrp 18473  df-mnd 18484  df-mhm 18528  df-submnd 18529  df-grp 18677  df-minusg 18678  df-mulg 18798  df-subg 18849  df-ghm 18929  df-cntz 19020  df-cmn 19484  df-abl 19485  df-mgp 19816  df-ur 19833  df-ring 19880  df-subrg 20127  df-psr 21218  df-mvr 21219  df-mpl 21220
This theorem is referenced by:  mplcoe5  21347  coe1tm  21550
  Copyright terms: Public domain W3C validator