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Theorem smupval 16525
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 12920 . . . . 5 0 = (ℤ‘0)
31, 2eleqtrdi 2851 . . . 4 (𝜑𝑁 ∈ (ℤ‘0))
4 eluzfz2b 13573 . . . 4 (𝑁 ∈ (ℤ‘0) ↔ 𝑁 ∈ (0...𝑁))
53, 4sylib 218 . . 3 (𝜑𝑁 ∈ (0...𝑁))
6 fveq2 6906 . . . . . 6 (𝑥 = 0 → (𝑃𝑥) = (𝑃‘0))
7 fveq2 6906 . . . . . 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 2753 . . . . 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 6906 . . . . . 6 (𝑥 = 𝑘 → (𝑃𝑥) = (𝑃𝑘))
11 fveq2 6906 . . . . . 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 2753 . . . . 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 6906 . . . . . 6 (𝑥 = (𝑘 + 1) → (𝑃𝑥) = (𝑃‘(𝑘 + 1)))
15 fveq2 6906 . . . . . 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 2753 . . . . 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 6906 . . . . . 6 (𝑥 = 𝑁 → (𝑃𝑥) = (𝑃𝑁))
19 fveq2 6906 . . . . . 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 2753 . . . . 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 16516 . . . . . 6 (𝜑 → (𝑃‘0) = ∅)
26 inss1 4237 . . . . . . . 8 (𝐴 ∩ (0..^𝑁)) ⊆ 𝐴
2726, 22sstrid 3995 . . . . . . 7 (𝜑 → (𝐴 ∩ (0..^𝑁)) ⊆ ℕ0)
28 eqid 2737 . . . . . . 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 16516 . . . . . 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 2780 . . . . 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 7438 . . . . . . 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 480 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝐴 ⊆ ℕ0)
3423adantr 480 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ0)
35 elfzouz 13703 . . . . . . . . . . 11 (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ (ℤ‘0))
3635adantl 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ (ℤ‘0))
3736, 2eleqtrrdi 2852 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℕ0)
3833, 34, 24, 37smupp1 16517 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑃‘(𝑘 + 1)) = ((𝑃𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)}))
3927adantr 480 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝐴 ∩ (0..^𝑁)) ⊆ ℕ0)
4039, 34, 28, 37smupp1 16517 . . . . . . . . 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 3967 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ↔ (𝑘𝐴𝑘 ∈ (0..^𝑁)))
4241rbaib 538 . . . . . . . . . . . . 13 (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ↔ 𝑘𝐴))
4342adantl 481 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ↔ 𝑘𝐴))
4443anbi1d 631 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (0..^𝑁)) → ((𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑘) ∈ 𝐵) ↔ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)))
4544rabbidv 3444 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0..^𝑁)) → {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (𝐴 ∩ (0..^𝑁)) ∧ (𝑛𝑘) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)})
4645oveq2d 7447 . . . . . . . . 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 2777 . . . . . . . 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 2753 . . . . . . 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, 48imbitrrid 246 . . . . . 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 413 . . . . 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 13824 . . 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 4238 . . . 4 (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
5554a1i 11 . . 3 (𝜑 → (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁))
561nn0zd 12639 . . . 4 (𝜑𝑁 ∈ ℤ)
57 uzid 12893 . . . 4 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
5856, 57syl 17 . . 3 (𝜑𝑁 ∈ (ℤ𝑁))
5927, 23, 28, 1, 55, 58smupvallem 16520 . 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 2777 1 (𝜑 → (𝑃𝑁) = ((𝐴 ∩ (0..^𝑁)) smul 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {crab 3436  cin 3950  wss 3951  c0 4333  ifcif 4525  𝒫 cpw 4600  cmpt 5225  cfv 6561  (class class class)co 7431  cmpo 7433  0cc0 11155  1c1 11156   + caddc 11158  cmin 11492  0cn0 12526  cz 12613  cuz 12878  ...cfz 13547  ..^cfzo 13694  seqcseq 14042   sadd csad 16457   smul csmu 16458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-xor 1512  df-tru 1543  df-fal 1553  df-had 1594  df-cad 1607  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-dvds 16291  df-bits 16459  df-sad 16488  df-smu 16513
This theorem is referenced by:  smup1  16526  smueqlem  16527
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