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Theorem smupp1 15133
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 11673 . . . . 5 0 = (ℤ‘0)
31, 2syl6eleq 2708 . . . 4 (𝜑𝑁 ∈ (ℤ‘0))
4 seqp1 12763 . . . 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 6154 . . 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 6154 . . . 4 (𝑃𝑁) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)
98oveq1i 6620 . . 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 2680 . 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 11259 . . . . . . 7 1 ∈ ℕ0
1211a1i 11 . . . . . 6 (𝜑 → 1 ∈ ℕ0)
131, 12nn0addcld 11306 . . . . 5 (𝜑 → (𝑁 + 1) ∈ ℕ0)
14 eqeq1 2625 . . . . . . 7 (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
15 oveq1 6617 . . . . . . 7 (𝑛 = (𝑁 + 1) → (𝑛 − 1) = ((𝑁 + 1) − 1))
1614, 15ifbieq2d 4088 . . . . . 6 (𝑛 = (𝑁 + 1) → if(𝑛 = 0, ∅, (𝑛 − 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
17 eqid 2621 . . . . . 6 (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))
18 0ex 4755 . . . . . . 7 ∅ ∈ V
19 ovex 6638 . . . . . . 7 ((𝑁 + 1) − 1) ∈ V
2018, 19ifex 4133 . . . . . 6 if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) ∈ V
2116, 17, 20fvmpt 6244 . . . . 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 11283 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
241, 23syl 17 . . . . . 6 (𝜑 → (𝑁 + 1) ∈ ℕ)
2524nnne0d 11016 . . . . 5 (𝜑 → (𝑁 + 1) ≠ 0)
26 ifnefalse 4075 . . . . 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 11304 . . . . 5 (𝜑𝑁 ∈ ℂ)
2912nn0cnd 11304 . . . . 5 (𝜑 → 1 ∈ ℂ)
3028, 29pncand 10344 . . . 4 (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
3122, 27, 303eqtrd 2659 . . 3 (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = 𝑁)
3231oveq2d 6626 . 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 15131 . . . 4 (𝜑𝑃:ℕ0⟶𝒫 ℕ0)
3635, 1ffvelrnd 6321 . . 3 (𝜑 → (𝑃𝑁) ∈ 𝒫 ℕ0)
37 simpl 473 . . . . 5 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → 𝑥 = (𝑃𝑁))
38 simpr 477 . . . . . . . . 9 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁)
3938eleq1d 2683 . . . . . . . 8 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → (𝑦𝐴𝑁𝐴))
4038oveq2d 6626 . . . . . . . . 9 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → (𝑘𝑦) = (𝑘𝑁))
4140eleq1d 2683 . . . . . . . 8 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → ((𝑘𝑦) ∈ 𝐵 ↔ (𝑘𝑁) ∈ 𝐵))
4239, 41anbi12d 746 . . . . . . 7 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → ((𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵) ↔ (𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵)))
4342rabbidv 3180 . . . . . 6 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)} = {𝑘 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵)})
44 oveq1 6617 . . . . . . . . 9 (𝑘 = 𝑛 → (𝑘𝑁) = (𝑛𝑁))
4544eleq1d 2683 . . . . . . . 8 (𝑘 = 𝑛 → ((𝑘𝑁) ∈ 𝐵 ↔ (𝑛𝑁) ∈ 𝐵))
4645anbi2d 739 . . . . . . 7 (𝑘 = 𝑛 → ((𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵) ↔ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)))
4746cbvrabv 3188 . . . . . 6 {𝑘 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}
4843, 47syl6eq 2671 . . . . 5 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)})
4937, 48oveq12d 6628 . . . 4 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → (𝑥 sadd {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)}) = ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}))
50 oveq1 6617 . . . . 5 (𝑝 = 𝑥 → (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}) = (𝑥 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))
51 eleq1 2686 . . . . . . . . 9 (𝑚 = 𝑦 → (𝑚𝐴𝑦𝐴))
52 oveq2 6618 . . . . . . . . . 10 (𝑚 = 𝑦 → (𝑛𝑚) = (𝑛𝑦))
5352eleq1d 2683 . . . . . . . . 9 (𝑚 = 𝑦 → ((𝑛𝑚) ∈ 𝐵 ↔ (𝑛𝑦) ∈ 𝐵))
5451, 53anbi12d 746 . . . . . . . 8 (𝑚 = 𝑦 → ((𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵) ↔ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)))
5554rabbidv 3180 . . . . . . 7 (𝑚 = 𝑦 → {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)})
56 oveq1 6617 . . . . . . . . . 10 (𝑘 = 𝑛 → (𝑘𝑦) = (𝑛𝑦))
5756eleq1d 2683 . . . . . . . . 9 (𝑘 = 𝑛 → ((𝑘𝑦) ∈ 𝐵 ↔ (𝑛𝑦) ∈ 𝐵))
5857anbi2d 739 . . . . . . . 8 (𝑘 = 𝑛 → ((𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵) ↔ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)))
5958cbvrabv 3188 . . . . . . 7 {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)}
6055, 59syl6eqr 2673 . . . . . 6 (𝑚 = 𝑦 → {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)} = {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)})
6160oveq2d 6626 . . . . 5 (𝑚 = 𝑦 → (𝑥 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}) = (𝑥 sadd {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)}))
6250, 61cbvmpt2v 6695 . . . 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 6638 . . . 4 ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}) ∈ V
6449, 62, 63ovmpt2a 6751 . . 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 692 . 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 2659 1 (𝜑 → (𝑃‘(𝑁 + 1)) = ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  {crab 2911  wss 3559  c0 3896  ifcif 4063  𝒫 cpw 4135  cmpt 4678  cfv 5852  (class class class)co 6610  cmpt2 6612  0cc0 9887  1c1 9888   + caddc 9890  cmin 10217  cn 10971  0cn0 11243  cuz 11638  seqcseq 12748   sadd csad 15073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-xor 1462  df-tru 1483  df-had 1530  df-cad 1543  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-n0 11244  df-z 11329  df-uz 11639  df-fz 12276  df-seq 12749  df-sad 15104
This theorem is referenced by:  smuval2  15135  smupvallem  15136  smu01lem  15138  smupval  15141  smup1  15142  smueqlem  15143
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