Theorem smupval 16436

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 12868	. . . . 5 ⊢ ℕ0 = (ℤ≥‘0)
3	1, 2	eleqtrdi 2837	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
4		eluzfz2b 13516	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘0) ↔ 𝑁 ∈ (0...𝑁))
5	3, 4	sylib 217	. . 3 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
6		fveq2 6885	. . . . . 6 ⊢ (𝑥 = 0 → (𝑃‘𝑥) = (𝑃‘0))
7		fveq2 6885	. . . . . 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 2742	. . . . 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 6885	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝑃‘𝑥) = (𝑃‘𝑘))
11		fveq2 6885	. . . . . 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 2742	. . . . 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 6885	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝑃‘𝑥) = (𝑃‘(𝑘 + 1)))
15		fveq2 6885	. . . . . 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 2742	. . . . 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 6885	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑃‘𝑥) = (𝑃‘𝑁))
19		fveq2 6885	. . . . . 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 2742	. . . . 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 16427	. . . . . 6 ⊢ (𝜑 → (𝑃‘0) = ∅)
26		inss1 4223	. . . . . . . 8 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ 𝐴
27	26, 22	sstrid 3988	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ⊆ ℕ0)
28		eqid 2726	. . . . . . 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 16427	. . . . . 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 2769	. . . . 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 7412	. . . . . . 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 13642	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ (ℤ≥‘0))
36	35	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ (ℤ≥‘0))
37	36, 2	eleqtrrdi 2838	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℕ0)
38	33, 34, 24, 37	smupp1 16428	. . . . . . . 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 16428	. . . . . . . . 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 3959	. . . . . . . . . . . . . 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 629	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑘) ∈ 𝐵) ↔ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)))
45	44	rabbidv 3434	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛 − 𝑘) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ 𝐴 ∧ (𝑛 − 𝑘) ∈ 𝐵)})
46	45	oveq2d 7421	. . . . . . . . 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 2766	. . . . . . . 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 2742	. . . . . . 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 245	. . . . . 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 13756	. . 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 4224	. . . 4 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
55	54	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁))
56	1	nn0zd 12588	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
57		uzid 12841	. . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁))
58	56, 57	syl 17	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑁))
59	27, 23, 28, 1, 55, 58	smupvallem 16431	. 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 2766	1 ⊢ (𝜑 → (𝑃‘𝑁) = ((𝐴 ∩ (0..^𝑁)) smul 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3426   ∩ cin 3942   ⊆ wss 3943  ∅c0 4317  ifcif 4523  𝒫 cpw 4597   ↦ cmpt 5224  ‘cfv 6537  (class class class)co 7405   ∈ cmpo 7407  0cc0 11112  1c1 11113   + caddc 11115   − cmin 11448  ℕ₀cn0 12476  ℤcz 12562  ℤ_≥cuz 12826  ...cfz 13490  ..^cfzo 13633  seqcseq 13972   sadd csad 16368   smul csmu 16369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-xor 1505  df-tru 1536  df-fal 1546  df-had 1587  df-cad 1600  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-disj 5107  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-2o 8468  df-oadd 8471  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-rp 12981  df-fz 13491  df-fzo 13634  df-fl 13763  df-mod 13841  df-seq 13973  df-exp 14033  df-hash 14296  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15438  df-sum 15639  df-dvds 16205  df-bits 16370  df-sad 16399  df-smu 16424

This theorem is referenced by:  smup1  16437  smueqlem  16438