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 Description: Sum of initial segments of the sadd sequence. (Contributed by Mario Carneiro, 8-Sep-2016.)
Hypotheses
Ref Expression
sadval.a (𝜑𝐴 ⊆ ℕ0)
sadval.b (𝜑𝐵 ⊆ ℕ0)
sadval.c 𝐶 = seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚𝐴, 𝑚𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
sadcp1.n (𝜑𝑁 ∈ ℕ0)
Assertion
Ref Expression
sadadd2 (𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) + if(∅ ∈ (𝐶𝑁), (2↑𝑁), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
Distinct variable groups:   𝑚,𝑐,𝑛   𝐴,𝑐,𝑚   𝐵,𝑐,𝑚   𝑛,𝑁
Allowed substitution hints:   𝜑(𝑚,𝑛,𝑐)   𝐴(𝑛)   𝐵(𝑛)   𝐶(𝑚,𝑛,𝑐)   𝐾(𝑚,𝑛,𝑐)   𝑁(𝑚,𝑐)

Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sadcp1.n . 2 (𝜑𝑁 ∈ ℕ0)
2 oveq2 7144 . . . . . . . . . . 11 (𝑥 = 0 → (0..^𝑥) = (0..^0))
3 fzo0 13059 . . . . . . . . . . 11 (0..^0) = ∅
42, 3eqtrdi 2849 . . . . . . . . . 10 (𝑥 = 0 → (0..^𝑥) = ∅)
54ineq2d 4139 . . . . . . . . 9 (𝑥 = 0 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ ∅))
6 in0 4299 . . . . . . . . 9 ((𝐴 sadd 𝐵) ∩ ∅) = ∅
75, 6eqtrdi 2849 . . . . . . . 8 (𝑥 = 0 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ∅)
87fveq2d 6650 . . . . . . 7 (𝑥 = 0 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘∅))
9 sadcadd.k . . . . . . . . 9 𝐾 = (bits ↾ ℕ0)
10 0nn0 11903 . . . . . . . . . . 11 0 ∈ ℕ0
11 fvres 6665 . . . . . . . . . . 11 (0 ∈ ℕ0 → ((bits ↾ ℕ0)‘0) = (bits‘0))
1210, 11ax-mp 5 . . . . . . . . . 10 ((bits ↾ ℕ0)‘0) = (bits‘0)
13 0bits 15781 . . . . . . . . . 10 (bits‘0) = ∅
1412, 13eqtr2i 2822 . . . . . . . . 9 ∅ = ((bits ↾ ℕ0)‘0)
159, 14fveq12i 6652 . . . . . . . 8 (𝐾‘∅) = ((bits ↾ ℕ0)‘((bits ↾ ℕ0)‘0))
16 bitsf1o 15787 . . . . . . . . 9 (bits ↾ ℕ0):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin)
17 f1ocnvfv1 7012 . . . . . . . . 9 (((bits ↾ ℕ0):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin) ∧ 0 ∈ ℕ0) → ((bits ↾ ℕ0)‘((bits ↾ ℕ0)‘0)) = 0)
1816, 10, 17mp2an 691 . . . . . . . 8 ((bits ↾ ℕ0)‘((bits ↾ ℕ0)‘0)) = 0
1915, 18eqtri 2821 . . . . . . 7 (𝐾‘∅) = 0
208, 19eqtrdi 2849 . . . . . 6 (𝑥 = 0 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = 0)
21 fveq2 6646 . . . . . . . 8 (𝑥 = 0 → (𝐶𝑥) = (𝐶‘0))
2221eleq2d 2875 . . . . . . 7 (𝑥 = 0 → (∅ ∈ (𝐶𝑥) ↔ ∅ ∈ (𝐶‘0)))
23 oveq2 7144 . . . . . . 7 (𝑥 = 0 → (2↑𝑥) = (2↑0))
2422, 23ifbieq1d 4448 . . . . . 6 (𝑥 = 0 → if(∅ ∈ (𝐶𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘0), (2↑0), 0))
2520, 24oveq12d 7154 . . . . 5 (𝑥 = 0 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶𝑥), (2↑𝑥), 0)) = (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)))
264ineq2d 4139 . . . . . . . . . 10 (𝑥 = 0 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ ∅))
27 in0 4299 . . . . . . . . . 10 (𝐴 ∩ ∅) = ∅
2826, 27eqtrdi 2849 . . . . . . . . 9 (𝑥 = 0 → (𝐴 ∩ (0..^𝑥)) = ∅)
2928fveq2d 6650 . . . . . . . 8 (𝑥 = 0 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘∅))
3029, 19eqtrdi 2849 . . . . . . 7 (𝑥 = 0 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = 0)
314ineq2d 4139 . . . . . . . . . 10 (𝑥 = 0 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ ∅))
32 in0 4299 . . . . . . . . . 10 (𝐵 ∩ ∅) = ∅
3331, 32eqtrdi 2849 . . . . . . . . 9 (𝑥 = 0 → (𝐵 ∩ (0..^𝑥)) = ∅)
3433fveq2d 6650 . . . . . . . 8 (𝑥 = 0 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘∅))
3534, 19eqtrdi 2849 . . . . . . 7 (𝑥 = 0 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = 0)
3630, 35oveq12d 7154 . . . . . 6 (𝑥 = 0 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = (0 + 0))
37 00id 10807 . . . . . 6 (0 + 0) = 0
3836, 37eqtrdi 2849 . . . . 5 (𝑥 = 0 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = 0)
3925, 38eqeq12d 2814 . . . 4 (𝑥 = 0 → (((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶𝑥), (2↑𝑥), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) ↔ (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)) = 0))
4039imbi2d 344 . . 3 (𝑥 = 0 → ((𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶𝑥), (2↑𝑥), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥))))) ↔ (𝜑 → (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)) = 0)))
41 oveq2 7144 . . . . . . . 8 (𝑥 = 𝑘 → (0..^𝑥) = (0..^𝑘))
4241ineq2d 4139 . . . . . . 7 (𝑥 = 𝑘 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^𝑘)))
4342fveq2d 6650 . . . . . 6 (𝑥 = 𝑘 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))))
44 fveq2 6646 . . . . . . . 8 (𝑥 = 𝑘 → (𝐶𝑥) = (𝐶𝑘))
4544eleq2d 2875 . . . . . . 7 (𝑥 = 𝑘 → (∅ ∈ (𝐶𝑥) ↔ ∅ ∈ (𝐶𝑘)))
46 oveq2 7144 . . . . . . 7 (𝑥 = 𝑘 → (2↑𝑥) = (2↑𝑘))
4745, 46ifbieq1d 4448 . . . . . 6 (𝑥 = 𝑘 → if(∅ ∈ (𝐶𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶𝑘), (2↑𝑘), 0))
4843, 47oveq12d 7154 . . . . 5 (𝑥 = 𝑘 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶𝑥), (2↑𝑥), 0)) = ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶𝑘), (2↑𝑘), 0)))
4941ineq2d 4139 . . . . . . 7 (𝑥 = 𝑘 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^𝑘)))
5049fveq2d 6650 . . . . . 6 (𝑥 = 𝑘 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^𝑘))))
5141ineq2d 4139 . . . . . . 7 (𝑥 = 𝑘 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^𝑘)))
5251fveq2d 6650 . . . . . 6 (𝑥 = 𝑘 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^𝑘))))
5350, 52oveq12d 7154 . . . . 5 (𝑥 = 𝑘 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))
5448, 53eqeq12d 2814 . . . 4 (𝑥 = 𝑘 → (((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶𝑥), (2↑𝑥), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) ↔ ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))))
5554imbi2d 344 . . 3 (𝑥 = 𝑘 → ((𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶𝑥), (2↑𝑥), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥))))) ↔ (𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))))
56 oveq2 7144 . . . . . . . 8 (𝑥 = (𝑘 + 1) → (0..^𝑥) = (0..^(𝑘 + 1)))
5756ineq2d 4139 . . . . . . 7 (𝑥 = (𝑘 + 1) → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1))))
5857fveq2d 6650 . . . . . 6 (𝑥 = (𝑘 + 1) → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))))
59 fveq2 6646 . . . . . . . 8 (𝑥 = (𝑘 + 1) → (𝐶𝑥) = (𝐶‘(𝑘 + 1)))
6059eleq2d 2875 . . . . . . 7 (𝑥 = (𝑘 + 1) → (∅ ∈ (𝐶𝑥) ↔ ∅ ∈ (𝐶‘(𝑘 + 1))))
61 oveq2 7144 . . . . . . 7 (𝑥 = (𝑘 + 1) → (2↑𝑥) = (2↑(𝑘 + 1)))
6260, 61ifbieq1d 4448 . . . . . 6 (𝑥 = (𝑘 + 1) → if(∅ ∈ (𝐶𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0))
6358, 62oveq12d 7154 . . . . 5 (𝑥 = (𝑘 + 1) → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶𝑥), (2↑𝑥), 0)) = ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))) + if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0)))
6456ineq2d 4139 . . . . . . 7 (𝑥 = (𝑘 + 1) → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^(𝑘 + 1))))
6564fveq2d 6650 . . . . . 6 (𝑥 = (𝑘 + 1) → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))))
6656ineq2d 4139 . . . . . . 7 (𝑥 = (𝑘 + 1) → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^(𝑘 + 1))))
6766fveq2d 6650 . . . . . 6 (𝑥 = (𝑘 + 1) → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1)))))
6865, 67oveq12d 7154 . . . . 5 (𝑥 = (𝑘 + 1) → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1))))))
6963, 68eqeq12d 2814 . . . 4 (𝑥 = (𝑘 + 1) → (((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶𝑥), (2↑𝑥), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) ↔ ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))) + if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0)) = ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1)))))))
7069imbi2d 344 . . 3 (𝑥 = (𝑘 + 1) → ((𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶𝑥), (2↑𝑥), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥))))) ↔ (𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))) + if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0)) = ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1))))))))
71 oveq2 7144 . . . . . . . 8 (𝑥 = 𝑁 → (0..^𝑥) = (0..^𝑁))
7271ineq2d 4139 . . . . . . 7 (𝑥 = 𝑁 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^𝑁)))
7372fveq2d 6650 . . . . . 6 (𝑥 = 𝑁 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))))
74 fveq2 6646 . . . . . . . 8 (𝑥 = 𝑁 → (𝐶𝑥) = (𝐶𝑁))
7574eleq2d 2875 . . . . . . 7 (𝑥 = 𝑁 → (∅ ∈ (𝐶𝑥) ↔ ∅ ∈ (𝐶𝑁)))
76 oveq2 7144 . . . . . . 7 (𝑥 = 𝑁 → (2↑𝑥) = (2↑𝑁))
7775, 76ifbieq1d 4448 . . . . . 6 (𝑥 = 𝑁 → if(∅ ∈ (𝐶𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶𝑁), (2↑𝑁), 0))
7873, 77oveq12d 7154 . . . . 5 (𝑥 = 𝑁 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶𝑥), (2↑𝑥), 0)) = ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) + if(∅ ∈ (𝐶𝑁), (2↑𝑁), 0)))
7971ineq2d 4139 . . . . . . 7 (𝑥 = 𝑁 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^𝑁)))
8079fveq2d 6650 . . . . . 6 (𝑥 = 𝑁 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^𝑁))))
8171ineq2d 4139 . . . . . . 7 (𝑥 = 𝑁 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^𝑁)))
8281fveq2d 6650 . . . . . 6 (𝑥 = 𝑁 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^𝑁))))
8380, 82oveq12d 7154 . . . . 5 (𝑥 = 𝑁 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
8478, 83eqeq12d 2814 . . . 4 (𝑥 = 𝑁 → (((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶𝑥), (2↑𝑥), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) ↔ ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) + if(∅ ∈ (𝐶𝑁), (2↑𝑁), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁))))))
8584imbi2d 344 . . 3 (𝑥 = 𝑁 → ((𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶𝑥), (2↑𝑥), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥))))) ↔ (𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) + if(∅ ∈ (𝐶𝑁), (2↑𝑁), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))))
86 sadval.a . . . . . . 7 (𝜑𝐴 ⊆ ℕ0)
87 sadval.b . . . . . . 7 (𝜑𝐵 ⊆ ℕ0)
88 sadval.c . . . . . . 7 𝐶 = seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚𝐴, 𝑚𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
8986, 87, 88sadc0 15796 . . . . . 6 (𝜑 → ¬ ∅ ∈ (𝐶‘0))
9089iffalsed 4436 . . . . 5 (𝜑 → if(∅ ∈ (𝐶‘0), (2↑0), 0) = 0)
9190oveq2d 7152 . . . 4 (𝜑 → (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)) = (0 + 0))
9291, 37eqtrdi 2849 . . 3 (𝜑 → (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)) = 0)
9386ad2antrr 725 . . . . . . 7 (((𝜑𝑘 ∈ ℕ0) ∧ ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))) → 𝐴 ⊆ ℕ0)
9487ad2antrr 725 . . . . . . 7 (((𝜑𝑘 ∈ ℕ0) ∧ ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))) → 𝐵 ⊆ ℕ0)
95 simplr 768 . . . . . . 7 (((𝜑𝑘 ∈ ℕ0) ∧ ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))) → 𝑘 ∈ ℕ0)
96 simpr 488 . . . . . . 7 (((𝜑𝑘 ∈ ℕ0) ∧ ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))) → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))
9793, 94, 88, 95, 9, 96sadadd2lem 15801 . . . . . 6 (((𝜑𝑘 ∈ ℕ0) ∧ ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))) → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))) + if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0)) = ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1))))))
9897ex 416 . . . . 5 ((𝜑𝑘 ∈ ℕ0) → (((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))) → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))) + if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0)) = ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1)))))))
9998expcom 417 . . . 4 (𝑘 ∈ ℕ0 → (𝜑 → (((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))) → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))) + if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0)) = ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1))))))))
10099a2d 29 . . 3 (𝑘 ∈ ℕ0 → ((𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))) → (𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))) + if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0)) = ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1))))))))
10140, 55, 70, 85, 92, 100nn0ind 12068 . 2 (𝑁 ∈ ℕ0 → (𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) + if(∅ ∈ (𝐶𝑁), (2↑𝑁), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁))))))
1021, 101mpcom 38 1 (𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) + if(∅ ∈ (𝐶𝑁), (2↑𝑁), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))