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Theorem smupval 16286
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 (𝜑𝐴 ⊆ ℕ0)
smupval.b (𝜑𝐵 ⊆ ℕ0)
smupval.p 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
smupval.n (𝜑𝑁 ∈ ℕ0)
Assertion
Ref Expression
smupval (𝜑 → (𝑃𝑁) = ((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.
StepHypRef Expression
1 smupval.n . . . . 5 (𝜑𝑁 ∈ ℕ0)
2 nn0uz 12713 . . . . 5 0 = (ℤ‘0)
31, 2eleqtrdi 2847 . . . 4 (𝜑𝑁 ∈ (ℤ‘0))
4 eluzfz2b 13358 . . . 4 (𝑁 ∈ (ℤ‘0) ↔ 𝑁 ∈ (0...𝑁))
53, 4sylib 217 . . 3 (𝜑𝑁 ∈ (0...𝑁))
6 fveq2 6819 . . . . . 6 (𝑥 = 0 → (𝑃𝑥) = (𝑃‘0))
7 fveq2 6819 . . . . . 6 (𝑥 = 0 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0))
86, 7eqeq12d 2752 . . . . 5 (𝑥 = 0 → ((𝑃𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 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)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0)))
98imbi2d 340 . . . 4 (𝑥 = 0 → ((𝜑 → (𝑃𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 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)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0))))
10 fveq2 6819 . . . . . 6 (𝑥 = 𝑘 → (𝑃𝑥) = (𝑃𝑘))
11 fveq2 6819 . . . . . 6 (𝑥 = 𝑘 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘))
1210, 11eqeq12d 2752 . . . . 5 (𝑥 = 𝑘 → ((𝑃𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) ↔ (𝑃𝑘) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘)))
1312imbi2d 340 . . . 4 (𝑥 = 𝑘 → ((𝜑 → (𝑃𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥)) ↔ (𝜑 → (𝑃𝑘) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘))))
14 fveq2 6819 . . . . . 6 (𝑥 = (𝑘 + 1) → (𝑃𝑥) = (𝑃‘(𝑘 + 1)))
15 fveq2 6819 . . . . . 6 (𝑥 = (𝑘 + 1) → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)))
1614, 15eqeq12d 2752 . . . . 5 (𝑥 = (𝑘 + 1) → ((𝑃𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 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)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1))))
1716imbi2d 340 . . . 4 (𝑥 = (𝑘 + 1) → ((𝜑 → (𝑃𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 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)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)))))
18 fveq2 6819 . . . . . 6 (𝑥 = 𝑁 → (𝑃𝑥) = (𝑃𝑁))
19 fveq2 6819 . . . . . 6 (𝑥 = 𝑁 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁))
2018, 19eqeq12d 2752 . . . . 5 (𝑥 = 𝑁 → ((𝑃𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) ↔ (𝑃𝑁) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)))
2120imbi2d 340 . . . 4 (𝑥 = 𝑁 → ((𝜑 → (𝑃𝑥) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥)) ↔ (𝜑 → (𝑃𝑁) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁))))
22 smupval.a . . . . . . 7 (𝜑𝐴 ⊆ ℕ0)
23 smupval.b . . . . . . 7 (𝜑𝐵 ⊆ ℕ0)
24 smupval.p . . . . . . 7 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
2522, 23, 24smup0 16277 . . . . . 6 (𝜑 → (𝑃‘0) = ∅)
26 inss1 4174 . . . . . . . 8 (𝐴 ∩ (0..^𝑁)) ⊆ 𝐴
2726, 22sstrid 3942 . . . . . . 7 (𝜑 → (𝐴 ∩ (0..^𝑁)) ⊆ ℕ0)
28 eqid 2736 . . . . . . 7 seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
2927, 23, 28smup0 16277 . . . . . 6 (𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0) = ∅)
3025, 29eqtr4d 2779 . . . . 5 (𝜑 → (𝑃‘0) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0))
3130a1i 11 . . . 4 (𝑁 ∈ (ℤ‘0) → (𝜑 → (𝑃‘0) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0)))
32 oveq1 7336 . . . . . . 7 ((𝑃𝑘) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) → ((𝑃𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)}) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)}))
3322adantr 481 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝐴 ⊆ ℕ0)
3423adantr 481 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ0)
35 elfzouz 13484 . . . . . . . . . . 11 (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ (ℤ‘0))
3635adantl 482 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ (ℤ‘0))
3736, 2eleqtrrdi 2848 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℕ0)
3833, 34, 24, 37smupp1 16278 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑃‘(𝑘 + 1)) = ((𝑃𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)}))
3927adantr 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝐴 ∩ (0..^𝑁)) ⊆ ℕ0)
4039, 34, 28, 37smupp1 16278 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑘) ∈ 𝐵)}))
41 elin 3913 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ↔ (𝑘𝐴𝑘 ∈ (0..^𝑁)))
4241rbaib 539 . . . . . . . . . . . . 13 (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ↔ 𝑘𝐴))
4342adantl 482 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ↔ 𝑘𝐴))
4443anbi1d 630 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (0..^𝑁)) → ((𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑘) ∈ 𝐵) ↔ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)))
4544rabbidv 3411 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0..^𝑁)) → {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑘) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)})
4645oveq2d 7345 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑘) ∈ 𝐵)}) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)}))
4740, 46eqtrd 2776 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)}))
4838, 47eqeq12d 2752 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → ((𝑃‘(𝑘 + 1)) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) ↔ ((𝑃𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)}) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)})))
4932, 48syl5ibr 245 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → ((𝑃𝑘) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 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)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1))))
5049expcom 414 . . . . 5 (𝑘 ∈ (0..^𝑁) → (𝜑 → ((𝑃𝑘) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 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)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)))))
5150a2d 29 . . . 4 (𝑘 ∈ (0..^𝑁) → ((𝜑 → (𝑃𝑘) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 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)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)))))
529, 13, 17, 21, 31, 51fzind2 13598 . . 3 (𝑁 ∈ (0...𝑁) → (𝜑 → (𝑃𝑁) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)))
535, 52mpcom 38 . 2 (𝜑 → (𝑃𝑁) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁))
54 inss2 4175 . . . 4 (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
5554a1i 11 . . 3 (𝜑 → (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁))
561nn0zd 12517 . . . 4 (𝜑𝑁 ∈ ℤ)
57 uzid 12690 . . . 4 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
5856, 57syl 17 . . 3 (𝜑𝑁 ∈ (ℤ𝑁))
5927, 23, 28, 1, 55, 58smupvallem 16281 . 2 (𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁) = ((𝐴 ∩ (0..^𝑁)) smul 𝐵))
6053, 59eqtrd 2776 1 (𝜑 → (𝑃𝑁) = ((𝐴 ∩ (0..^𝑁)) smul 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  {crab 3403  cin 3896  wss 3897  c0 4268  ifcif 4472  𝒫 cpw 4546  cmpt 5172  cfv 6473  (class class class)co 7329  cmpo 7331  0cc0 10964  1c1 10965   + caddc 10967  cmin 11298  0cn0 12326  cz 12412  cuz 12675  ...cfz 13332  ..^cfzo 13475  seqcseq 13814   sadd csad 16218   smul csmu 16219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-inf2 9490  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041  ax-pre-sup 11042
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-xor 1509  df-tru 1543  df-fal 1553  df-had 1594  df-cad 1607  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-disj 5055  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-se 5570  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-isom 6482  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-1st 7891  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-1o 8359  df-2o 8360  df-oadd 8363  df-er 8561  df-map 8680  df-pm 8681  df-en 8797  df-dom 8798  df-sdom 8799  df-fin 8800  df-sup 9291  df-inf 9292  df-oi 9359  df-dju 9750  df-card 9788  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-div 11726  df-nn 12067  df-2 12129  df-3 12130  df-n0 12327  df-xnn0 12399  df-z 12413  df-uz 12676  df-rp 12824  df-fz 13333  df-fzo 13476  df-fl 13605  df-mod 13683  df-seq 13815  df-exp 13876  df-hash 14138  df-cj 14901  df-re 14902  df-im 14903  df-sqrt 15037  df-abs 15038  df-clim 15288  df-sum 15489  df-dvds 16055  df-bits 16220  df-sad 16249  df-smu 16274
This theorem is referenced by:  smup1  16287  smueqlem  16288
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