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Theorem mplcoe3 19514
 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 4123 . . . . . . . . . . 11 (𝑥 = 0 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 0, 0))
3 ifid 4158 . . . . . . . . . . 11 if(𝑘 = 𝑋, 0, 0) = 0
42, 3syl6eq 2701 . . . . . . . . . 10 (𝑥 = 0 → if(𝑘 = 𝑋, 𝑥, 0) = 0)
54mpteq2dv 4778 . . . . . . . . 9 (𝑥 = 0 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ 0))
6 fconstmpt 5197 . . . . . . . . 9 (𝐼 × {0}) = (𝑘𝐼 ↦ 0)
75, 6syl6eqr 2703 . . . . . . . 8 (𝑥 = 0 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝐼 × {0}))
87eqeq2d 2661 . . . . . . 7 (𝑥 = 0 → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝐼 × {0})))
98ifbid 4141 . . . . . 6 (𝑥 = 0 → if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝐼 × {0}), 1 , 0 ))
109mpteq2dv 4778 . . . . 5 (𝑥 = 0 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
11 oveq1 6697 . . . . 5 (𝑥 = 0 → (𝑥 (𝑉𝑋)) = (0 (𝑉𝑋)))
1210, 11eqeq12d 2666 . . . 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 329 . . 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 4123 . . . . . . . . 9 (𝑥 = 𝑛 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑛, 0))
1514mpteq2dv 4778 . . . . . . . 8 (𝑥 = 𝑛 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
1615eqeq2d 2661 . . . . . . 7 (𝑥 = 𝑛 → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0))))
1716ifbid 4141 . . . . . 6 (𝑥 = 𝑛 → if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))
1817mpteq2dv 4778 . . . . 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 6697 . . . . 5 (𝑥 = 𝑛 → (𝑥 (𝑉𝑋)) = (𝑛 (𝑉𝑋)))
2018, 19eqeq12d 2666 . . . 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 329 . . 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 4123 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
2322mpteq2dv 4778 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)))
2423eqeq2d 2661 . . . . . . 7 (𝑥 = (𝑛 + 1) → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0))))
2524ifbid 4141 . . . . . 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 4778 . . . . 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 6697 . . . . 5 (𝑥 = (𝑛 + 1) → (𝑥 (𝑉𝑋)) = ((𝑛 + 1) (𝑉𝑋)))
2826, 27eqeq12d 2666 . . . 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 329 . . 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 4123 . . . . . . . . 9 (𝑥 = 𝑁 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑁, 0))
3130mpteq2dv 4778 . . . . . . . 8 (𝑥 = 𝑁 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)))
3231eqeq2d 2661 . . . . . . 7 (𝑥 = 𝑁 → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0))))
3332ifbid 4141 . . . . . 6 (𝑥 = 𝑁 → if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 ))
3433mpteq2dv 4778 . . . . 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 6697 . . . . 5 (𝑥 = 𝑁 → (𝑥 (𝑉𝑋)) = (𝑁 (𝑉𝑋)))
3634, 35eqeq12d 2666 . . . 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 329 . . 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 2651 . . . . . 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 19497 . . . . 5 (𝜑 → (𝑉𝑋) ∈ (Base‘𝑃))
45 mplcoe2.g . . . . . . 7 𝐺 = (mulGrp‘𝑃)
4645, 40mgpbas 18541 . . . . . 6 (Base‘𝑃) = (Base‘𝐺)
47 eqid 2651 . . . . . . 7 (1r𝑃) = (1r𝑃)
4845, 47ringidval 18549 . . . . . 6 (1r𝑃) = (0g𝐺)
49 mplcoe2.m . . . . . 6 = (.g𝐺)
5046, 48, 49mulg0 17593 . . . . 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 19492 . . . 4 (𝜑 → (1r𝑃) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
5651, 55eqtr2d 2686 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (0 (𝑉𝑋)))
57 oveq1 6697 . . . . . 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 480 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → 𝐼𝑊)
5942adantr 480 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
6052snifpsrbag 19414 . . . . . . . . . 10 ((𝐼𝑊𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
6141, 60sylan 487 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
62 eqid 2651 . . . . . . . . 9 (.r𝑃) = (.r𝑃)
63 1nn0 11346 . . . . . . . . . . 11 1 ∈ ℕ0
6463a1i 11 . . . . . . . . . 10 (𝑛 ∈ ℕ0 → 1 ∈ ℕ0)
6552snifpsrbag 19414 . . . . . . . . . 10 ((𝐼𝑊 ∧ 1 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
6641, 64, 65syl2an 493 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
6738, 40, 53, 54, 52, 58, 59, 61, 62, 66mplmonmul 19512 . . . . . . . 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 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → 𝑋𝐼)
6939, 52, 53, 54, 58, 59, 68mvrval 19469 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → (𝑉𝑋) = (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )))
7069eqcomd 2657 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )) = (𝑉𝑋))
7170oveq2d 6706 . . . . . . . 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 807 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ0) ∧ 𝑘𝐼) → 𝑛 ∈ ℕ0)
73 0nn0 11345 . . . . . . . . . . . . . 14 0 ∈ ℕ0
74 ifcl 4163 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑘 = 𝑋, 𝑛, 0) ∈ ℕ0)
7572, 73, 74sylancl 695 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ0) ∧ 𝑘𝐼) → if(𝑘 = 𝑋, 𝑛, 0) ∈ ℕ0)
7663, 73keepel 4188 . . . . . . . . . . . . . 14 if(𝑘 = 𝑋, 1, 0) ∈ ℕ0
7776a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ0) ∧ 𝑘𝐼) → if(𝑘 = 𝑋, 1, 0) ∈ ℕ0)
78 eqidd 2652 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
79 eqidd 2652 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)))
8058, 75, 77, 78, 79offval2 6956 . . . . . . . . . . . 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 4125 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 𝑛)
82 iftrue 4125 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 1)
8381, 82oveq12d 6708 . . . . . . . . . . . . . . 15 (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (𝑛 + 1))
84 iftrue 4125 . . . . . . . . . . . . . . 15 (𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = (𝑛 + 1))
8583, 84eqtr4d 2688 . . . . . . . . . . . . . 14 (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
86 00id 10249 . . . . . . . . . . . . . . 15 (0 + 0) = 0
87 iffalse 4128 . . . . . . . . . . . . . . . 16 𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 0)
88 iffalse 4128 . . . . . . . . . . . . . . . 16 𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 0)
8987, 88oveq12d 6708 . . . . . . . . . . . . . . 15 𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (0 + 0))
90 iffalse 4128 . . . . . . . . . . . . . . 15 𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = 0)
9186, 89, 903eqtr4a 2711 . . . . . . . . . . . . . 14 𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
9285, 91pm2.61i 176 . . . . . . . . . . . . 13 (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0)
9392mpteq2i 4774 . . . . . . . . . . . 12 (𝑘𝐼 ↦ (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0))) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0))
9480, 93syl6eq 2701 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → ((𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∘𝑓 + (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0))) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)))
9594eqeq2d 2661 . . . . . . . . . 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 4141 . . . . . . . . 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 4778 . . . . . . . 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 2694 . . . . . . 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 19500 . . . . . . . . . . 11 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ Ring)
10041, 42, 99syl2anc 694 . . . . . . . . . 10 (𝜑𝑃 ∈ Ring)
10145ringmgp 18599 . . . . . . . . . 10 (𝑃 ∈ Ring → 𝐺 ∈ Mnd)
102100, 101syl 17 . . . . . . . . 9 (𝜑𝐺 ∈ Mnd)
103102adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝐺 ∈ Mnd)
104 simpr 476 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
10544adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → (𝑉𝑋) ∈ (Base‘𝑃))
10645, 62mgpplusg 18539 . . . . . . . . 9 (.r𝑃) = (+g𝐺)
10746, 49, 106mulgnn0p1 17599 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑛 ∈ ℕ0 ∧ (𝑉𝑋) ∈ (Base‘𝑃)) → ((𝑛 + 1) (𝑉𝑋)) = ((𝑛 (𝑉𝑋))(.r𝑃)(𝑉𝑋)))
108103, 104, 105, 107syl3anc 1366 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 + 1) (𝑉𝑋)) = ((𝑛 (𝑉𝑋))(.r𝑃)(𝑉𝑋)))
10998, 108eqeq12d 2666 . . . . . 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 236 . . . . 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 450 . . . 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 11510 . 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 383   = wceq 1523   ∈ wcel 2030  {crab 2945  ifcif 4119  {csn 4210   ↦ cmpt 4762   × cxp 5141  ◡ccnv 5142   “ cima 5146  ‘cfv 5926  (class class class)co 6690   ∘𝑓 cof 6937   ↑𝑚 cmap 7899  Fincfn 7997  0cc0 9974  1c1 9975   + caddc 9977  ℕcn 11058  ℕ0cn0 11330  Basecbs 15904  .rcmulr 15989  0gc0g 16147  Mndcmnd 17341  .gcmg 17587  mulGrpcmgp 18535  1rcur 18547  Ringcrg 18593   mVar cmvr 19400   mPoly cmpl 19401 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-tset 16007  df-0g 16149  df-gsum 16150  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-mulg 17588  df-subg 17638  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-subrg 18826  df-psr 19404  df-mvr 19405  df-mpl 19406 This theorem is referenced by:  mplcoe5  19516  coe1tm  19691
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