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Theorem smupp1 15040
 Description: The initial element of the partial sum sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)
Hypotheses
Ref Expression
smuval.a (𝜑𝐴 ⊆ ℕ0)
smuval.b (𝜑𝐵 ⊆ ℕ0)
smuval.p 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
smuval.n (𝜑𝑁 ∈ ℕ0)
Assertion
Ref Expression
smupp1 (𝜑 → (𝑃‘(𝑁 + 1)) = ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}))
Distinct variable groups:   𝑚,𝑛,𝑝,𝐴   𝑛,𝑁   𝜑,𝑛   𝐵,𝑚,𝑛,𝑝
Allowed substitution hints:   𝜑(𝑚,𝑝)   𝑃(𝑚,𝑛,𝑝)   𝑁(𝑚,𝑝)

Proof of Theorem smupp1
Dummy variables 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smuval.n . . . . 5 (𝜑𝑁 ∈ ℕ0)
2 nn0uz 11598 . . . . 5 0 = (ℤ‘0)
31, 2syl6eleq 2698 . . . 4 (𝜑𝑁 ∈ (ℤ‘0))
4 seqp1 12678 . . . 4 (𝑁 ∈ (ℤ‘0) → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑁 + 1)) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))))
53, 4syl 17 . . 3 (𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑁 + 1)) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))))
6 smuval.p . . . 4 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
76fveq1i 6104 . . 3 (𝑃‘(𝑁 + 1)) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑁 + 1))
86fveq1i 6104 . . . 4 (𝑃𝑁) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)
98oveq1i 6559 . . 3 ((𝑃𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)))
105, 7, 93eqtr4g 2669 . 2 (𝜑 → (𝑃‘(𝑁 + 1)) = ((𝑃𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))))
11 1nn0 11185 . . . . . . 7 1 ∈ ℕ0
1211a1i 11 . . . . . 6 (𝜑 → 1 ∈ ℕ0)
131, 12nn0addcld 11232 . . . . 5 (𝜑 → (𝑁 + 1) ∈ ℕ0)
14 eqeq1 2614 . . . . . . 7 (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
15 oveq1 6556 . . . . . . 7 (𝑛 = (𝑁 + 1) → (𝑛 − 1) = ((𝑁 + 1) − 1))
1614, 15ifbieq2d 4061 . . . . . 6 (𝑛 = (𝑁 + 1) → if(𝑛 = 0, ∅, (𝑛 − 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
17 eqid 2610 . . . . . 6 (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))
18 0ex 4718 . . . . . . 7 ∅ ∈ V
19 ovex 6577 . . . . . . 7 ((𝑁 + 1) − 1) ∈ V
2018, 19ifex 4106 . . . . . 6 if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) ∈ V
2116, 17, 20fvmpt 6191 . . . . 5 ((𝑁 + 1) ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
2213, 21syl 17 . . . 4 (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
23 nn0p1nn 11209 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
241, 23syl 17 . . . . . 6 (𝜑 → (𝑁 + 1) ∈ ℕ)
2524nnne0d 10942 . . . . 5 (𝜑 → (𝑁 + 1) ≠ 0)
26 ifnefalse 4048 . . . . 5 ((𝑁 + 1) ≠ 0 → if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) = ((𝑁 + 1) − 1))
2725, 26syl 17 . . . 4 (𝜑 → if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) = ((𝑁 + 1) − 1))
281nn0cnd 11230 . . . . 5 (𝜑𝑁 ∈ ℂ)
2912nn0cnd 11230 . . . . 5 (𝜑 → 1 ∈ ℂ)
3028, 29pncand 10272 . . . 4 (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
3122, 27, 303eqtrd 2648 . . 3 (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = 𝑁)
3231oveq2d 6565 . 2 (𝜑 → ((𝑃𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))) = ((𝑃𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))𝑁))
33 smuval.a . . . . 5 (𝜑𝐴 ⊆ ℕ0)
34 smuval.b . . . . 5 (𝜑𝐵 ⊆ ℕ0)
3533, 34, 6smupf 15038 . . . 4 (𝜑𝑃:ℕ0⟶𝒫 ℕ0)
3635, 1ffvelrnd 6268 . . 3 (𝜑 → (𝑃𝑁) ∈ 𝒫 ℕ0)
37 simpl 472 . . . . 5 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → 𝑥 = (𝑃𝑁))
38 simpr 476 . . . . . . . . 9 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁)
3938eleq1d 2672 . . . . . . . 8 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → (𝑦𝐴𝑁𝐴))
4038oveq2d 6565 . . . . . . . . 9 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → (𝑘𝑦) = (𝑘𝑁))
4140eleq1d 2672 . . . . . . . 8 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → ((𝑘𝑦) ∈ 𝐵 ↔ (𝑘𝑁) ∈ 𝐵))
4239, 41anbi12d 743 . . . . . . 7 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → ((𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵) ↔ (𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵)))
4342rabbidv 3164 . . . . . 6 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)} = {𝑘 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵)})
44 oveq1 6556 . . . . . . . . 9 (𝑘 = 𝑛 → (𝑘𝑁) = (𝑛𝑁))
4544eleq1d 2672 . . . . . . . 8 (𝑘 = 𝑛 → ((𝑘𝑁) ∈ 𝐵 ↔ (𝑛𝑁) ∈ 𝐵))
4645anbi2d 736 . . . . . . 7 (𝑘 = 𝑛 → ((𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵) ↔ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)))
4746cbvrabv 3172 . . . . . 6 {𝑘 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}
4843, 47syl6eq 2660 . . . . 5 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)})
4937, 48oveq12d 6567 . . . 4 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → (𝑥 sadd {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)}) = ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}))
50 oveq1 6556 . . . . 5 (𝑝 = 𝑥 → (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}) = (𝑥 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))
51 eleq1 2676 . . . . . . . . 9 (𝑚 = 𝑦 → (𝑚𝐴𝑦𝐴))
52 oveq2 6557 . . . . . . . . . 10 (𝑚 = 𝑦 → (𝑛𝑚) = (𝑛𝑦))
5352eleq1d 2672 . . . . . . . . 9 (𝑚 = 𝑦 → ((𝑛𝑚) ∈ 𝐵 ↔ (𝑛𝑦) ∈ 𝐵))
5451, 53anbi12d 743 . . . . . . . 8 (𝑚 = 𝑦 → ((𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵) ↔ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)))
5554rabbidv 3164 . . . . . . 7 (𝑚 = 𝑦 → {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)})
56 oveq1 6556 . . . . . . . . . 10 (𝑘 = 𝑛 → (𝑘𝑦) = (𝑛𝑦))
5756eleq1d 2672 . . . . . . . . 9 (𝑘 = 𝑛 → ((𝑘𝑦) ∈ 𝐵 ↔ (𝑛𝑦) ∈ 𝐵))
5857anbi2d 736 . . . . . . . 8 (𝑘 = 𝑛 → ((𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵) ↔ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)))
5958cbvrabv 3172 . . . . . . 7 {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)}
6055, 59syl6eqr 2662 . . . . . 6 (𝑚 = 𝑦 → {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)} = {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)})
6160oveq2d 6565 . . . . 5 (𝑚 = 𝑦 → (𝑥 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}) = (𝑥 sadd {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)}))
6250, 61cbvmpt2v 6633 . . . 4 (𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})) = (𝑥 ∈ 𝒫 ℕ0, 𝑦 ∈ ℕ0 ↦ (𝑥 sadd {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)}))
63 ovex 6577 . . . 4 ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}) ∈ V
6449, 62, 63ovmpt2a 6689 . . 3 (((𝑃𝑁) ∈ 𝒫 ℕ0𝑁 ∈ ℕ0) → ((𝑃𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))𝑁) = ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}))
6536, 1, 64syl2anc 691 . 2 (𝜑 → ((𝑃𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))𝑁) = ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}))
6610, 32, 653eqtrd 2648 1 (𝜑 → (𝑃‘(𝑁 + 1)) = ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900   ⊆ wss 3540  ∅c0 3874  ifcif 4036  𝒫 cpw 4108   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ℤ≥cuz 11563  seqcseq 12663   sadd csad 14980 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-xor 1457  df-tru 1478  df-had 1524  df-cad 1537  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664  df-sad 15011 This theorem is referenced by:  smuval2  15042  smupvallem  15043  smu01lem  15045  smupval  15048  smup1  15049  smueqlem  15050
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