Theorem smupval 16534

Description: Rewrite the elements of the partial sum sequence in terms of sequence multiplication. (Contributed by Mario Carneiro, 20-Sep-2016.)

Hypotheses

Ref	Expression
smupval.a	⊢ (𝜑 → 𝐴 ⊆ ℕ0)
smupval.b	⊢ (𝜑 → 𝐵 ⊆ ℕ0)
smupval.p	⊢ 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
smupval.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)

Assertion

Ref	Expression
smupval	⊢ (𝜑 → (𝑃‘𝑁) = ((𝐴 ∩ (0..^𝑁)) smul 𝐵))

Distinct variable groups:   𝑚,𝑛,𝑝,𝐴   𝐵,𝑚,𝑛,𝑝   𝑚,𝑁,𝑛,𝑝   𝜑,𝑛

Allowed substitution hints:   𝜑(𝑚,𝑝)   𝑃(𝑚,𝑛,𝑝)

Proof of Theorem smupval

Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smupval.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
2		nn0uz 12945	. . . . 5 ⊢ ℕ0 = (ℤ≥‘0)
3	1, 2	eleqtrdi 2854	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
4		eluzfz2b 13593	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘0) ↔ 𝑁 ∈ (0...𝑁))
5	3, 4	sylib 218	. . 3 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
6		fveq2 6920	. . . . . 6 ⊢ (𝑥 = 0 → (𝑃‘𝑥) = (𝑃‘0))
7		fveq2 6920	. . . . . 6 ⊢ (𝑥 = 0 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0))
8	6, 7	eqeq12d 2756	. . . . 5 ⊢ (𝑥 = 0 → ((𝑃‘𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) ↔ (𝑃‘0) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0)))
9	8	imbi2d 340	. . . 4 ⊢ (𝑥 = 0 → ((𝜑 → (𝑃‘𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥)) ↔ (𝜑 → (𝑃‘0) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0))))
10		fveq2 6920	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝑃‘𝑥) = (𝑃‘𝑘))
11		fveq2 6920	. . . . . 6 ⊢ (𝑥 = 𝑘 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘))
12	10, 11	eqeq12d 2756	. . . . 5 ⊢ (𝑥 = 𝑘 → ((𝑃‘𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) ↔ (𝑃‘𝑘) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘)))
13	12	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑘 → ((𝜑 → (𝑃‘𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥)) ↔ (𝜑 → (𝑃‘𝑘) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘))))
14		fveq2 6920	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝑃‘𝑥) = (𝑃‘(𝑘 + 1)))
15		fveq2 6920	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)))
16	14, 15	eqeq12d 2756	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → ((𝑃‘𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) ↔ (𝑃‘(𝑘 + 1)) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1))))
17	16	imbi2d 340	. . . 4 ⊢ (𝑥 = (𝑘 + 1) → ((𝜑 → (𝑃‘𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥)) ↔ (𝜑 → (𝑃‘(𝑘 + 1)) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)))))
18		fveq2 6920	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑃‘𝑥) = (𝑃‘𝑁))
19		fveq2 6920	. . . . . 6 ⊢ (𝑥 = 𝑁 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁))
20	18, 19	eqeq12d 2756	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑃‘𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) ↔ (𝑃‘𝑁) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)))
21	20	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑃‘𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥)) ↔ (𝜑 → (𝑃‘𝑁) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁))))
22		smupval.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℕ0)
23		smupval.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ ℕ0)
24		smupval.p	. . . . . . 7 ⊢ 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
25	22, 23, 24	smup0 16525	. . . . . 6 ⊢ (𝜑 → (𝑃‘0) = ∅)
26		inss1 4258	. . . . . . . 8 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ 𝐴
27	26, 22	sstrid 4020	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ⊆ ℕ0)
28		eqid 2740	. . . . . . 7 ⊢ seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
29	27, 23, 28	smup0 16525	. . . . . 6 ⊢ (𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0) = ∅)
30	25, 29	eqtr4d 2783	. . . . 5 ⊢ (𝜑 → (𝑃‘0) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0))
31	30	a1i 11	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘0) → (𝜑 → (𝑃‘0) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0)))
32		oveq1 7455	. . . . . . 7 ⊢ ((𝑃‘𝑘) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) → ((𝑃‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)}) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)}))
33	22	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝐴 ⊆ ℕ0)
34	23	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ0)
35		elfzouz 13720	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ (ℤ≥‘0))
36	35	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ (ℤ≥‘0))
37	36, 2	eleqtrrdi 2855	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℕ0)
38	33, 34, 24, 37	smupp1 16526	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑃‘(𝑘 + 1)) = ((𝑃‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)}))
39	27	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝐴 ∩ (0..^𝑁)) ⊆ ℕ0)
40	39, 34, 28, 37	smupp1 16526	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑘) ∈ 𝐵)}))
41		elin 3992	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ↔ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ (0..^𝑁)))
42	41	rbaib 538	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ↔ 𝑘 ∈ 𝐴))
43	42	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ↔ 𝑘 ∈ 𝐴))
44	43	anbi1d 630	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑘) ∈ 𝐵) ↔ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)))
45	44	rabbidv 3451	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑘) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)})
46	45	oveq2d 7464	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑘) ∈ 𝐵)}) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)}))
47	40, 46	eqtrd 2780	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)}))
48	38, 47	eqeq12d 2756	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((𝑃‘(𝑘 + 1)) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) ↔ ((𝑃‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)}) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)})))
49	32, 48	imbitrrid 246	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((𝑃‘𝑘) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) → (𝑃‘(𝑘 + 1)) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1))))
50	49	expcom 413	. . . . 5 ⊢ (𝑘 ∈ (0..^𝑁) → (𝜑 → ((𝑃‘𝑘) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) → (𝑃‘(𝑘 + 1)) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)))))
51	50	a2d 29	. . . 4 ⊢ (𝑘 ∈ (0..^𝑁) → ((𝜑 → (𝑃‘𝑘) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘)) → (𝜑 → (𝑃‘(𝑘 + 1)) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)))))
52	9, 13, 17, 21, 31, 51	fzind2 13835	. . 3 ⊢ (𝑁 ∈ (0...𝑁) → (𝜑 → (𝑃‘𝑁) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)))
53	5, 52	mpcom 38	. 2 ⊢ (𝜑 → (𝑃‘𝑁) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁))
54		inss2 4259	. . . 4 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
55	54	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁))
56	1	nn0zd 12665	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
57		uzid 12918	. . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁))
58	56, 57	syl 17	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑁))
59	27, 23, 28, 1, 55, 58	smupvallem 16529	. 2 ⊢ (𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁) = ((𝐴 ∩ (0..^𝑁)) smul 𝐵))
60	53, 59	eqtrd 2780	1 ⊢ (𝜑 → (𝑃‘𝑁) = ((𝐴 ∩ (0..^𝑁)) smul 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {crab 3443   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ifcif 4548  𝒫 cpw 4622   ↦ cmpt 5249  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  0cc0 11184  1c1 11185   + caddc 11187   − cmin 11520  ℕ₀cn0 12553  ℤcz 12639  ℤ_≥cuz 12903  ...cfz 13567  ..^cfzo 13711  seqcseq 14052   sadd csad 16466   smul csmu 16467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-xor 1509  df-tru 1540  df-fal 1550  df-had 1591  df-cad 1604  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-disj 5134  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-oi 9579  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-rp 13058  df-fz 13568  df-fzo 13712  df-fl 13843  df-mod 13921  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735  df-dvds 16303  df-bits 16468  df-sad 16497  df-smu 16522

This theorem is referenced by:  smup1  16535  smueqlem  16536