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Theorem mplcoe3 19973
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 4349 . . . . . . . . . . 11 (𝑥 = 0 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 0, 0))
3 ifid 4384 . . . . . . . . . . 11 if(𝑘 = 𝑋, 0, 0) = 0
42, 3syl6eq 2825 . . . . . . . . . 10 (𝑥 = 0 → if(𝑘 = 𝑋, 𝑥, 0) = 0)
54mpteq2dv 5020 . . . . . . . . 9 (𝑥 = 0 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ 0))
6 fconstmpt 5461 . . . . . . . . 9 (𝐼 × {0}) = (𝑘𝐼 ↦ 0)
75, 6syl6eqr 2827 . . . . . . . 8 (𝑥 = 0 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝐼 × {0}))
87eqeq2d 2783 . . . . . . 7 (𝑥 = 0 → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝐼 × {0})))
98ifbid 4367 . . . . . 6 (𝑥 = 0 → if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝐼 × {0}), 1 , 0 ))
109mpteq2dv 5020 . . . . 5 (𝑥 = 0 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
11 oveq1 6982 . . . . 5 (𝑥 = 0 → (𝑥 (𝑉𝑋)) = (0 (𝑉𝑋)))
1210, 11eqeq12d 2788 . . . 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 333 . . 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 4349 . . . . . . . . 9 (𝑥 = 𝑛 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑛, 0))
1514mpteq2dv 5020 . . . . . . . 8 (𝑥 = 𝑛 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
1615eqeq2d 2783 . . . . . . 7 (𝑥 = 𝑛 → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0))))
1716ifbid 4367 . . . . . 6 (𝑥 = 𝑛 → if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))
1817mpteq2dv 5020 . . . . 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 6982 . . . . 5 (𝑥 = 𝑛 → (𝑥 (𝑉𝑋)) = (𝑛 (𝑉𝑋)))
2018, 19eqeq12d 2788 . . . 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 333 . . 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 4349 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
2322mpteq2dv 5020 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)))
2423eqeq2d 2783 . . . . . . 7 (𝑥 = (𝑛 + 1) → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0))))
2524ifbid 4367 . . . . . 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 5020 . . . . 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 6982 . . . . 5 (𝑥 = (𝑛 + 1) → (𝑥 (𝑉𝑋)) = ((𝑛 + 1) (𝑉𝑋)))
2826, 27eqeq12d 2788 . . . 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 333 . . 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 4349 . . . . . . . . 9 (𝑥 = 𝑁 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑁, 0))
3130mpteq2dv 5020 . . . . . . . 8 (𝑥 = 𝑁 → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)))
3231eqeq2d 2783 . . . . . . 7 (𝑥 = 𝑁 → (𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0))))
3332ifbid 4367 . . . . . 6 (𝑥 = 𝑁 → if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 ))
3433mpteq2dv 5020 . . . . 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 6982 . . . . 5 (𝑥 = 𝑁 → (𝑥 (𝑉𝑋)) = (𝑁 (𝑉𝑋)))
3634, 35eqeq12d 2788 . . . 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 333 . . 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 2773 . . . . . 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 19956 . . . . 5 (𝜑 → (𝑉𝑋) ∈ (Base‘𝑃))
45 mplcoe2.g . . . . . . 7 𝐺 = (mulGrp‘𝑃)
4645, 40mgpbas 18981 . . . . . 6 (Base‘𝑃) = (Base‘𝐺)
47 eqid 2773 . . . . . . 7 (1r𝑃) = (1r𝑃)
4845, 47ringidval 18989 . . . . . 6 (1r𝑃) = (0g𝐺)
49 mplcoe2.m . . . . . 6 = (.g𝐺)
5046, 48, 49mulg0 18031 . . . . 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 19951 . . . 4 (𝜑 → (1r𝑃) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
5651, 55eqtr2d 2810 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (0 (𝑉𝑋)))
57 oveq1 6982 . . . . . 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 473 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → 𝐼𝑊)
5942adantr 473 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
6052snifpsrbag 19873 . . . . . . . . . 10 ((𝐼𝑊𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
6141, 60sylan 572 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
62 eqid 2773 . . . . . . . . 9 (.r𝑃) = (.r𝑃)
63 1nn0 11724 . . . . . . . . . . 11 1 ∈ ℕ0
6463a1i 11 . . . . . . . . . 10 (𝑛 ∈ ℕ0 → 1 ∈ ℕ0)
6552snifpsrbag 19873 . . . . . . . . . 10 ((𝐼𝑊 ∧ 1 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
6641, 64, 65syl2an 587 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
6738, 40, 53, 54, 52, 58, 59, 61, 62, 66mplmonmul 19971 . . . . . . . 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 473 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → 𝑋𝐼)
6939, 52, 53, 54, 58, 59, 68mvrval 19928 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → (𝑉𝑋) = (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )))
7069eqcomd 2779 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )) = (𝑉𝑋))
7170oveq2d 6991 . . . . . . . 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 757 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ0) ∧ 𝑘𝐼) → 𝑛 ∈ ℕ0)
73 0nn0 11723 . . . . . . . . . . . . . 14 0 ∈ ℕ0
74 ifcl 4389 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑘 = 𝑋, 𝑛, 0) ∈ ℕ0)
7572, 73, 74sylancl 578 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ0) ∧ 𝑘𝐼) → if(𝑘 = 𝑋, 𝑛, 0) ∈ ℕ0)
7663, 73ifcli 4391 . . . . . . . . . . . . . 14 if(𝑘 = 𝑋, 1, 0) ∈ ℕ0
7776a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ0) ∧ 𝑘𝐼) → if(𝑘 = 𝑋, 1, 0) ∈ ℕ0)
78 eqidd 2774 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
79 eqidd 2774 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0)))
8058, 75, 77, 78, 79offval2 7243 . . . . . . . . . . . 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 4351 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 𝑛)
82 iftrue 4351 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 1)
8381, 82oveq12d 6993 . . . . . . . . . . . . . . 15 (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (𝑛 + 1))
84 iftrue 4351 . . . . . . . . . . . . . . 15 (𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = (𝑛 + 1))
8583, 84eqtr4d 2812 . . . . . . . . . . . . . 14 (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
86 00id 10614 . . . . . . . . . . . . . . 15 (0 + 0) = 0
87 iffalse 4354 . . . . . . . . . . . . . . . 16 𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 0)
88 iffalse 4354 . . . . . . . . . . . . . . . 16 𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 0)
8987, 88oveq12d 6993 . . . . . . . . . . . . . . 15 𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (0 + 0))
90 iffalse 4354 . . . . . . . . . . . . . . 15 𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = 0)
9186, 89, 903eqtr4a 2835 . . . . . . . . . . . . . 14 𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
9285, 91pm2.61i 177 . . . . . . . . . . . . 13 (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0)
9392mpteq2i 5016 . . . . . . . . . . . 12 (𝑘𝐼 ↦ (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0))) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0))
9480, 93syl6eq 2825 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → ((𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∘𝑓 + (𝑘𝐼 ↦ if(𝑘 = 𝑋, 1, 0))) = (𝑘𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)))
9594eqeq2d 2783 . . . . . . . . . 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 4367 . . . . . . . . 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 5020 . . . . . . . 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 2818 . . . . . . 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 19959 . . . . . . . . . . 11 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ Ring)
10041, 42, 99syl2anc 576 . . . . . . . . . 10 (𝜑𝑃 ∈ Ring)
10145ringmgp 19039 . . . . . . . . . 10 (𝑃 ∈ Ring → 𝐺 ∈ Mnd)
102100, 101syl 17 . . . . . . . . 9 (𝜑𝐺 ∈ Mnd)
103102adantr 473 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝐺 ∈ Mnd)
104 simpr 477 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
10544adantr 473 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → (𝑉𝑋) ∈ (Base‘𝑃))
10645, 62mgpplusg 18979 . . . . . . . . 9 (.r𝑃) = (+g𝐺)
10746, 49, 106mulgnn0p1 18037 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑛 ∈ ℕ0 ∧ (𝑉𝑋) ∈ (Base‘𝑃)) → ((𝑛 + 1) (𝑉𝑋)) = ((𝑛 (𝑉𝑋))(.r𝑃)(𝑉𝑋)))
108103, 104, 105, 107syl3anc 1352 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 + 1) (𝑉𝑋)) = ((𝑛 (𝑉𝑋))(.r𝑃)(𝑉𝑋)))
10998, 108eqeq12d 2788 . . . . . 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 238 . . . . 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 406 . . . 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 11889 . 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 387   = wceq 1508  wcel 2051  {crab 3087  ifcif 4345  {csn 4436  cmpt 5005   × cxp 5402  ccnv 5403  cima 5407  cfv 6186  (class class class)co 6975  𝑓 cof 7224  𝑚 cmap 8205  Fincfn 8305  0cc0 10334  1c1 10335   + caddc 10337  cn 11438  0cn0 11706  Basecbs 16338  .rcmulr 16421  0gc0g 16568  Mndcmnd 17775  .gcmg 18024  mulGrpcmgp 18975  1rcur 18987  Ringcrg 19033   mVar cmvr 19859   mPoly cmpl 19860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-cnex 10390  ax-resscn 10391  ax-1cn 10392  ax-icn 10393  ax-addcl 10394  ax-addrcl 10395  ax-mulcl 10396  ax-mulrcl 10397  ax-mulcom 10398  ax-addass 10399  ax-mulass 10400  ax-distr 10401  ax-i2m1 10402  ax-1ne0 10403  ax-1rid 10404  ax-rnegex 10405  ax-rrecex 10406  ax-cnre 10407  ax-pre-lttri 10408  ax-pre-lttrn 10409  ax-pre-ltadd 10410  ax-pre-mulgt0 10411
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-reu 3090  df-rmo 3091  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-int 4747  df-iun 4791  df-iin 4792  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-se 5364  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-isom 6195  df-riota 6936  df-ov 6978  df-oprab 6979  df-mpo 6980  df-of 7226  df-ofr 7227  df-om 7396  df-1st 7500  df-2nd 7501  df-supp 7633  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-1o 7904  df-2o 7905  df-oadd 7908  df-er 8088  df-map 8207  df-pm 8208  df-ixp 8259  df-en 8306  df-dom 8307  df-sdom 8308  df-fin 8309  df-fsupp 8628  df-oi 8768  df-card 9161  df-pnf 10475  df-mnf 10476  df-xr 10477  df-ltxr 10478  df-le 10479  df-sub 10671  df-neg 10672  df-nn 11439  df-2 11502  df-3 11503  df-4 11504  df-5 11505  df-6 11506  df-7 11507  df-8 11508  df-9 11509  df-n0 11707  df-z 11793  df-uz 12058  df-fz 12708  df-fzo 12849  df-seq 13184  df-hash 13505  df-struct 16340  df-ndx 16341  df-slot 16342  df-base 16344  df-sets 16345  df-ress 16346  df-plusg 16433  df-mulr 16434  df-sca 16436  df-vsca 16437  df-tset 16439  df-0g 16570  df-gsum 16571  df-mre 16728  df-mrc 16729  df-acs 16731  df-mgm 17723  df-sgrp 17765  df-mnd 17776  df-mhm 17816  df-submnd 17817  df-grp 17907  df-minusg 17908  df-mulg 18025  df-subg 18073  df-ghm 18140  df-cntz 18231  df-cmn 18681  df-abl 18682  df-mgp 18976  df-ur 18988  df-ring 19035  df-subrg 19269  df-psr 19863  df-mvr 19864  df-mpl 19865
This theorem is referenced by:  mplcoe5  19975  coe1tm  20160
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