| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | inss1 4236 | . . . . . . . . . . 11
⊢ (𝐴 ∩ (0..^𝑁)) ⊆ 𝐴 | 
| 2 |  | sadasslem.1 | . . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ⊆
ℕ0) | 
| 3 | 1, 2 | sstrid 3994 | . . . . . . . . . 10
⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ⊆
ℕ0) | 
| 4 |  | fzofi 14016 | . . . . . . . . . . . 12
⊢
(0..^𝑁) ∈
Fin | 
| 5 | 4 | a1i 11 | . . . . . . . . . . 11
⊢ (𝜑 → (0..^𝑁) ∈ Fin) | 
| 6 |  | inss2 4237 | . . . . . . . . . . 11
⊢ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁) | 
| 7 |  | ssfi 9214 | . . . . . . . . . . 11
⊢
(((0..^𝑁) ∈ Fin
∧ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (𝐴 ∩ (0..^𝑁)) ∈ Fin) | 
| 8 | 5, 6, 7 | sylancl 586 | . . . . . . . . . 10
⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ∈ Fin) | 
| 9 |  | elfpw 9395 | . . . . . . . . . 10
⊢ ((𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin) ↔ ((𝐴 ∩
(0..^𝑁)) ⊆
ℕ0 ∧ (𝐴 ∩ (0..^𝑁)) ∈ Fin)) | 
| 10 | 3, 8, 9 | sylanbrc 583 | . . . . . . . . 9
⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin)) | 
| 11 |  | bitsf1o 16483 | . . . . . . . . . . 11
⊢ (bits
↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩
Fin) | 
| 12 |  | f1ocnv 6859 | . . . . . . . . . . 11
⊢ ((bits
↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin)
→ ◡(bits ↾
ℕ0):(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0) | 
| 13 |  | f1of 6847 | . . . . . . . . . . 11
⊢ (◡(bits ↾
ℕ0):(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0 → ◡(bits ↾
ℕ0):(𝒫 ℕ0 ∩
Fin)⟶ℕ0) | 
| 14 | 11, 12, 13 | mp2b 10 | . . . . . . . . . 10
⊢ ◡(bits ↾
ℕ0):(𝒫 ℕ0 ∩
Fin)⟶ℕ0 | 
| 15 | 14 | ffvelcdmi 7102 | . . . . . . . . 9
⊢ ((𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin) → (◡(bits ↾
ℕ0)‘(𝐴 ∩ (0..^𝑁))) ∈
ℕ0) | 
| 16 | 10, 15 | syl 17 | . . . . . . . 8
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘(𝐴 ∩ (0..^𝑁))) ∈
ℕ0) | 
| 17 | 16 | nn0cnd 12591 | . . . . . . 7
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘(𝐴 ∩ (0..^𝑁))) ∈ ℂ) | 
| 18 |  | inss1 4236 | . . . . . . . . . . 11
⊢ (𝐵 ∩ (0..^𝑁)) ⊆ 𝐵 | 
| 19 |  | sadasslem.2 | . . . . . . . . . . 11
⊢ (𝜑 → 𝐵 ⊆
ℕ0) | 
| 20 | 18, 19 | sstrid 3994 | . . . . . . . . . 10
⊢ (𝜑 → (𝐵 ∩ (0..^𝑁)) ⊆
ℕ0) | 
| 21 |  | inss2 4237 | . . . . . . . . . . 11
⊢ (𝐵 ∩ (0..^𝑁)) ⊆ (0..^𝑁) | 
| 22 |  | ssfi 9214 | . . . . . . . . . . 11
⊢
(((0..^𝑁) ∈ Fin
∧ (𝐵 ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ∈ Fin) | 
| 23 | 5, 21, 22 | sylancl 586 | . . . . . . . . . 10
⊢ (𝜑 → (𝐵 ∩ (0..^𝑁)) ∈ Fin) | 
| 24 |  | elfpw 9395 | . . . . . . . . . 10
⊢ ((𝐵 ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin) ↔ ((𝐵 ∩
(0..^𝑁)) ⊆
ℕ0 ∧ (𝐵 ∩ (0..^𝑁)) ∈ Fin)) | 
| 25 | 20, 23, 24 | sylanbrc 583 | . . . . . . . . 9
⊢ (𝜑 → (𝐵 ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin)) | 
| 26 | 14 | ffvelcdmi 7102 | . . . . . . . . 9
⊢ ((𝐵 ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin) → (◡(bits ↾
ℕ0)‘(𝐵 ∩ (0..^𝑁))) ∈
ℕ0) | 
| 27 | 25, 26 | syl 17 | . . . . . . . 8
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘(𝐵 ∩ (0..^𝑁))) ∈
ℕ0) | 
| 28 | 27 | nn0cnd 12591 | . . . . . . 7
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘(𝐵 ∩ (0..^𝑁))) ∈ ℂ) | 
| 29 |  | inss1 4236 | . . . . . . . . . . 11
⊢ (𝐶 ∩ (0..^𝑁)) ⊆ 𝐶 | 
| 30 |  | sadasslem.3 | . . . . . . . . . . 11
⊢ (𝜑 → 𝐶 ⊆
ℕ0) | 
| 31 | 29, 30 | sstrid 3994 | . . . . . . . . . 10
⊢ (𝜑 → (𝐶 ∩ (0..^𝑁)) ⊆
ℕ0) | 
| 32 |  | inss2 4237 | . . . . . . . . . . 11
⊢ (𝐶 ∩ (0..^𝑁)) ⊆ (0..^𝑁) | 
| 33 |  | ssfi 9214 | . . . . . . . . . . 11
⊢
(((0..^𝑁) ∈ Fin
∧ (𝐶 ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (𝐶 ∩ (0..^𝑁)) ∈ Fin) | 
| 34 | 5, 32, 33 | sylancl 586 | . . . . . . . . . 10
⊢ (𝜑 → (𝐶 ∩ (0..^𝑁)) ∈ Fin) | 
| 35 |  | elfpw 9395 | . . . . . . . . . 10
⊢ ((𝐶 ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin) ↔ ((𝐶 ∩
(0..^𝑁)) ⊆
ℕ0 ∧ (𝐶 ∩ (0..^𝑁)) ∈ Fin)) | 
| 36 | 31, 34, 35 | sylanbrc 583 | . . . . . . . . 9
⊢ (𝜑 → (𝐶 ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin)) | 
| 37 | 14 | ffvelcdmi 7102 | . . . . . . . . 9
⊢ ((𝐶 ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin) → (◡(bits ↾
ℕ0)‘(𝐶 ∩ (0..^𝑁))) ∈
ℕ0) | 
| 38 | 36, 37 | syl 17 | . . . . . . . 8
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘(𝐶 ∩ (0..^𝑁))) ∈
ℕ0) | 
| 39 | 38 | nn0cnd 12591 | . . . . . . 7
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘(𝐶 ∩ (0..^𝑁))) ∈ ℂ) | 
| 40 | 17, 28, 39 | addassd 11284 | . . . . . 6
⊢ (𝜑 → (((◡(bits ↾
ℕ0)‘(𝐴 ∩ (0..^𝑁))) + (◡(bits ↾
ℕ0)‘(𝐵 ∩ (0..^𝑁)))) + (◡(bits ↾
ℕ0)‘(𝐶 ∩ (0..^𝑁)))) = ((◡(bits ↾
ℕ0)‘(𝐴 ∩ (0..^𝑁))) + ((◡(bits ↾
ℕ0)‘(𝐵 ∩ (0..^𝑁))) + (◡(bits ↾
ℕ0)‘(𝐶 ∩ (0..^𝑁)))))) | 
| 41 | 40 | oveq1d 7447 | . . . . 5
⊢ (𝜑 → ((((◡(bits ↾
ℕ0)‘(𝐴 ∩ (0..^𝑁))) + (◡(bits ↾
ℕ0)‘(𝐵 ∩ (0..^𝑁)))) + (◡(bits ↾
ℕ0)‘(𝐶 ∩ (0..^𝑁)))) mod (2↑𝑁)) = (((◡(bits ↾
ℕ0)‘(𝐴 ∩ (0..^𝑁))) + ((◡(bits ↾
ℕ0)‘(𝐵 ∩ (0..^𝑁))) + (◡(bits ↾
ℕ0)‘(𝐶 ∩ (0..^𝑁))))) mod (2↑𝑁))) | 
| 42 |  | inss1 4236 | . . . . . . . . . 10
⊢ ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ⊆ (𝐴 sadd 𝐵) | 
| 43 |  | sadcl 16500 | . . . . . . . . . . 11
⊢ ((𝐴 ⊆ ℕ0
∧ 𝐵 ⊆
ℕ0) → (𝐴 sadd 𝐵) ⊆
ℕ0) | 
| 44 | 2, 19, 43 | syl2anc 584 | . . . . . . . . . 10
⊢ (𝜑 → (𝐴 sadd 𝐵) ⊆
ℕ0) | 
| 45 | 42, 44 | sstrid 3994 | . . . . . . . . 9
⊢ (𝜑 → ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ⊆
ℕ0) | 
| 46 |  | inss2 4237 | . . . . . . . . . 10
⊢ ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ⊆ (0..^𝑁) | 
| 47 |  | ssfi 9214 | . . . . . . . . . 10
⊢
(((0..^𝑁) ∈ Fin
∧ ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ∈ Fin) | 
| 48 | 5, 46, 47 | sylancl 586 | . . . . . . . . 9
⊢ (𝜑 → ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ∈ Fin) | 
| 49 |  | elfpw 9395 | . . . . . . . . 9
⊢ (((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin) ↔ (((𝐴 sadd
𝐵) ∩ (0..^𝑁)) ⊆ ℕ0
∧ ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ∈ Fin)) | 
| 50 | 45, 48, 49 | sylanbrc 583 | . . . . . . . 8
⊢ (𝜑 → ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin)) | 
| 51 | 14 | ffvelcdmi 7102 | . . . . . . . 8
⊢ (((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin) → (◡(bits ↾
ℕ0)‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) ∈
ℕ0) | 
| 52 | 50, 51 | syl 17 | . . . . . . 7
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) ∈
ℕ0) | 
| 53 | 52 | nn0red 12590 | . . . . . 6
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) ∈ ℝ) | 
| 54 | 16 | nn0red 12590 | . . . . . . 7
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘(𝐴 ∩ (0..^𝑁))) ∈ ℝ) | 
| 55 | 27 | nn0red 12590 | . . . . . . 7
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘(𝐵 ∩ (0..^𝑁))) ∈ ℝ) | 
| 56 | 54, 55 | readdcld 11291 | . . . . . 6
⊢ (𝜑 → ((◡(bits ↾
ℕ0)‘(𝐴 ∩ (0..^𝑁))) + (◡(bits ↾
ℕ0)‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ) | 
| 57 | 38 | nn0red 12590 | . . . . . 6
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘(𝐶 ∩ (0..^𝑁))) ∈ ℝ) | 
| 58 |  | 2rp 13040 | . . . . . . . 8
⊢ 2 ∈
ℝ+ | 
| 59 | 58 | a1i 11 | . . . . . . 7
⊢ (𝜑 → 2 ∈
ℝ+) | 
| 60 |  | sadasslem.4 | . . . . . . . 8
⊢ (𝜑 → 𝑁 ∈
ℕ0) | 
| 61 | 60 | nn0zd 12641 | . . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℤ) | 
| 62 | 59, 61 | rpexpcld 14287 | . . . . . 6
⊢ (𝜑 → (2↑𝑁) ∈
ℝ+) | 
| 63 |  | eqid 2736 | . . . . . . 7
⊢
seq0((𝑐 ∈
2o, 𝑚 ∈
ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 − 1)))) =
seq0((𝑐 ∈
2o, 𝑚 ∈
ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 −
1)))) | 
| 64 |  | eqid 2736 | . . . . . . 7
⊢ ◡(bits ↾ ℕ0) = ◡(bits ↾
ℕ0) | 
| 65 | 2, 19, 63, 60, 64 | sadadd3 16499 | . . . . . 6
⊢ (𝜑 → ((◡(bits ↾
ℕ0)‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) mod (2↑𝑁)) = (((◡(bits ↾
ℕ0)‘(𝐴 ∩ (0..^𝑁))) + (◡(bits ↾
ℕ0)‘(𝐵 ∩ (0..^𝑁)))) mod (2↑𝑁))) | 
| 66 |  | eqidd 2737 | . . . . . 6
⊢ (𝜑 → ((◡(bits ↾
ℕ0)‘(𝐶 ∩ (0..^𝑁))) mod (2↑𝑁)) = ((◡(bits ↾
ℕ0)‘(𝐶 ∩ (0..^𝑁))) mod (2↑𝑁))) | 
| 67 | 53, 56, 57, 57, 62, 65, 66 | modadd12d 13969 | . . . . 5
⊢ (𝜑 → (((◡(bits ↾
ℕ0)‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) + (◡(bits ↾
ℕ0)‘(𝐶 ∩ (0..^𝑁)))) mod (2↑𝑁)) = ((((◡(bits ↾
ℕ0)‘(𝐴 ∩ (0..^𝑁))) + (◡(bits ↾
ℕ0)‘(𝐵 ∩ (0..^𝑁)))) + (◡(bits ↾
ℕ0)‘(𝐶 ∩ (0..^𝑁)))) mod (2↑𝑁))) | 
| 68 |  | inss1 4236 | . . . . . . . . . 10
⊢ ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ⊆ (𝐵 sadd 𝐶) | 
| 69 |  | sadcl 16500 | . . . . . . . . . . 11
⊢ ((𝐵 ⊆ ℕ0
∧ 𝐶 ⊆
ℕ0) → (𝐵 sadd 𝐶) ⊆
ℕ0) | 
| 70 | 19, 30, 69 | syl2anc 584 | . . . . . . . . . 10
⊢ (𝜑 → (𝐵 sadd 𝐶) ⊆
ℕ0) | 
| 71 | 68, 70 | sstrid 3994 | . . . . . . . . 9
⊢ (𝜑 → ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ⊆
ℕ0) | 
| 72 |  | inss2 4237 | . . . . . . . . . 10
⊢ ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ⊆ (0..^𝑁) | 
| 73 |  | ssfi 9214 | . . . . . . . . . 10
⊢
(((0..^𝑁) ∈ Fin
∧ ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ∈ Fin) | 
| 74 | 5, 72, 73 | sylancl 586 | . . . . . . . . 9
⊢ (𝜑 → ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ∈ Fin) | 
| 75 |  | elfpw 9395 | . . . . . . . . 9
⊢ (((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin) ↔ (((𝐵 sadd
𝐶) ∩ (0..^𝑁)) ⊆ ℕ0
∧ ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ∈ Fin)) | 
| 76 | 71, 74, 75 | sylanbrc 583 | . . . . . . . 8
⊢ (𝜑 → ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin)) | 
| 77 | 14 | ffvelcdmi 7102 | . . . . . . . 8
⊢ (((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin) → (◡(bits ↾
ℕ0)‘((𝐵 sadd 𝐶) ∩ (0..^𝑁))) ∈
ℕ0) | 
| 78 | 76, 77 | syl 17 | . . . . . . 7
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘((𝐵 sadd 𝐶) ∩ (0..^𝑁))) ∈
ℕ0) | 
| 79 | 78 | nn0red 12590 | . . . . . 6
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘((𝐵 sadd 𝐶) ∩ (0..^𝑁))) ∈ ℝ) | 
| 80 | 55, 57 | readdcld 11291 | . . . . . 6
⊢ (𝜑 → ((◡(bits ↾
ℕ0)‘(𝐵 ∩ (0..^𝑁))) + (◡(bits ↾
ℕ0)‘(𝐶 ∩ (0..^𝑁)))) ∈ ℝ) | 
| 81 |  | eqidd 2737 | . . . . . 6
⊢ (𝜑 → ((◡(bits ↾
ℕ0)‘(𝐴 ∩ (0..^𝑁))) mod (2↑𝑁)) = ((◡(bits ↾
ℕ0)‘(𝐴 ∩ (0..^𝑁))) mod (2↑𝑁))) | 
| 82 |  | eqid 2736 | . . . . . . 7
⊢
seq0((𝑐 ∈
2o, 𝑚 ∈
ℕ0 ↦ if(cadd(𝑚 ∈ 𝐵, 𝑚 ∈ 𝐶, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 − 1)))) =
seq0((𝑐 ∈
2o, 𝑚 ∈
ℕ0 ↦ if(cadd(𝑚 ∈ 𝐵, 𝑚 ∈ 𝐶, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 −
1)))) | 
| 83 | 19, 30, 82, 60, 64 | sadadd3 16499 | . . . . . 6
⊢ (𝜑 → ((◡(bits ↾
ℕ0)‘((𝐵 sadd 𝐶) ∩ (0..^𝑁))) mod (2↑𝑁)) = (((◡(bits ↾
ℕ0)‘(𝐵 ∩ (0..^𝑁))) + (◡(bits ↾
ℕ0)‘(𝐶 ∩ (0..^𝑁)))) mod (2↑𝑁))) | 
| 84 | 54, 54, 79, 80, 62, 81, 83 | modadd12d 13969 | . . . . 5
⊢ (𝜑 → (((◡(bits ↾
ℕ0)‘(𝐴 ∩ (0..^𝑁))) + (◡(bits ↾
ℕ0)‘((𝐵 sadd 𝐶) ∩ (0..^𝑁)))) mod (2↑𝑁)) = (((◡(bits ↾
ℕ0)‘(𝐴 ∩ (0..^𝑁))) + ((◡(bits ↾
ℕ0)‘(𝐵 ∩ (0..^𝑁))) + (◡(bits ↾
ℕ0)‘(𝐶 ∩ (0..^𝑁))))) mod (2↑𝑁))) | 
| 85 | 41, 67, 84 | 3eqtr4d 2786 | . . . 4
⊢ (𝜑 → (((◡(bits ↾
ℕ0)‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) + (◡(bits ↾
ℕ0)‘(𝐶 ∩ (0..^𝑁)))) mod (2↑𝑁)) = (((◡(bits ↾
ℕ0)‘(𝐴 ∩ (0..^𝑁))) + (◡(bits ↾
ℕ0)‘((𝐵 sadd 𝐶) ∩ (0..^𝑁)))) mod (2↑𝑁))) | 
| 86 |  | eqid 2736 | . . . . 5
⊢
seq0((𝑐 ∈
2o, 𝑚 ∈
ℕ0 ↦ if(cadd(𝑚 ∈ (𝐴 sadd 𝐵), 𝑚 ∈ 𝐶, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 − 1)))) =
seq0((𝑐 ∈
2o, 𝑚 ∈
ℕ0 ↦ if(cadd(𝑚 ∈ (𝐴 sadd 𝐵), 𝑚 ∈ 𝐶, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 −
1)))) | 
| 87 | 44, 30, 86, 60, 64 | sadadd3 16499 | . . . 4
⊢ (𝜑 → ((◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) mod (2↑𝑁)) = (((◡(bits ↾
ℕ0)‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) + (◡(bits ↾
ℕ0)‘(𝐶 ∩ (0..^𝑁)))) mod (2↑𝑁))) | 
| 88 |  | eqid 2736 | . . . . 5
⊢
seq0((𝑐 ∈
2o, 𝑚 ∈
ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ (𝐵 sadd 𝐶), ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 − 1)))) =
seq0((𝑐 ∈
2o, 𝑚 ∈
ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ (𝐵 sadd 𝐶), ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 −
1)))) | 
| 89 | 2, 70, 88, 60, 64 | sadadd3 16499 | . . . 4
⊢ (𝜑 → ((◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) mod (2↑𝑁)) = (((◡(bits ↾
ℕ0)‘(𝐴 ∩ (0..^𝑁))) + (◡(bits ↾
ℕ0)‘((𝐵 sadd 𝐶) ∩ (0..^𝑁)))) mod (2↑𝑁))) | 
| 90 | 85, 87, 89 | 3eqtr4d 2786 | . . 3
⊢ (𝜑 → ((◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) mod (2↑𝑁)) = ((◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) mod (2↑𝑁))) | 
| 91 |  | inss1 4236 | . . . . . . . 8
⊢ (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ⊆ ((𝐴 sadd 𝐵) sadd 𝐶) | 
| 92 |  | sadcl 16500 | . . . . . . . . 9
⊢ (((𝐴 sadd 𝐵) ⊆ ℕ0 ∧ 𝐶 ⊆ ℕ0)
→ ((𝐴 sadd 𝐵) sadd 𝐶) ⊆
ℕ0) | 
| 93 | 44, 30, 92 | syl2anc 584 | . . . . . . . 8
⊢ (𝜑 → ((𝐴 sadd 𝐵) sadd 𝐶) ⊆
ℕ0) | 
| 94 | 91, 93 | sstrid 3994 | . . . . . . 7
⊢ (𝜑 → (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ⊆
ℕ0) | 
| 95 |  | inss2 4237 | . . . . . . . 8
⊢ (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ⊆ (0..^𝑁) | 
| 96 |  | ssfi 9214 | . . . . . . . 8
⊢
(((0..^𝑁) ∈ Fin
∧ (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ∈ Fin) | 
| 97 | 5, 95, 96 | sylancl 586 | . . . . . . 7
⊢ (𝜑 → (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ∈ Fin) | 
| 98 |  | elfpw 9395 | . . . . . . 7
⊢ ((((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin) ↔ ((((𝐴
sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ⊆ ℕ0 ∧
(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ∈ Fin)) | 
| 99 | 94, 97, 98 | sylanbrc 583 | . . . . . 6
⊢ (𝜑 → (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin)) | 
| 100 | 14 | ffvelcdmi 7102 | . . . . . 6
⊢ ((((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin) → (◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) ∈
ℕ0) | 
| 101 | 99, 100 | syl 17 | . . . . 5
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) ∈
ℕ0) | 
| 102 | 101 | nn0red 12590 | . . . 4
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) ∈ ℝ) | 
| 103 | 101 | nn0ge0d 12592 | . . . 4
⊢ (𝜑 → 0 ≤ (◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))) | 
| 104 | 101 | fvresd 6925 | . . . . . . . 8
⊢ (𝜑 → ((bits ↾
ℕ0)‘(◡(bits
↾ ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))) = (bits‘(◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))))) | 
| 105 |  | f1ocnvfv2 7298 | . . . . . . . . 9
⊢ (((bits
↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin)
∧ (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin)) → ((bits ↾ ℕ0)‘(◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))) = (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) | 
| 106 | 11, 99, 105 | sylancr 587 | . . . . . . . 8
⊢ (𝜑 → ((bits ↾
ℕ0)‘(◡(bits
↾ ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))) = (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) | 
| 107 | 104, 106 | eqtr3d 2778 | . . . . . . 7
⊢ (𝜑 → (bits‘(◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))) = (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) | 
| 108 | 107, 95 | eqsstrdi 4027 | . . . . . 6
⊢ (𝜑 → (bits‘(◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))) ⊆ (0..^𝑁)) | 
| 109 | 101 | nn0zd 12641 | . . . . . . 7
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) ∈ ℤ) | 
| 110 |  | bitsfzo 16473 | . . . . . . 7
⊢ (((◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) ↔ (bits‘(◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))) ⊆ (0..^𝑁))) | 
| 111 | 109, 60, 110 | syl2anc 584 | . . . . . 6
⊢ (𝜑 → ((◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) ↔ (bits‘(◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))) ⊆ (0..^𝑁))) | 
| 112 | 108, 111 | mpbird 257 | . . . . 5
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁))) | 
| 113 |  | elfzolt2 13709 | . . . . 5
⊢ ((◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) → (◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) < (2↑𝑁)) | 
| 114 | 112, 113 | syl 17 | . . . 4
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) < (2↑𝑁)) | 
| 115 |  | modid 13937 | . . . 4
⊢ ((((◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) ∈ ℝ ∧ (2↑𝑁) ∈ ℝ+)
∧ (0 ≤ (◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) ∧ (◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) < (2↑𝑁))) → ((◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) mod (2↑𝑁)) = (◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))) | 
| 116 | 102, 62, 103, 114, 115 | syl22anc 838 | . . 3
⊢ (𝜑 → ((◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) mod (2↑𝑁)) = (◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))) | 
| 117 |  | inss1 4236 | . . . . . . . 8
⊢ ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ⊆ (𝐴 sadd (𝐵 sadd 𝐶)) | 
| 118 |  | sadcl 16500 | . . . . . . . . 9
⊢ ((𝐴 ⊆ ℕ0
∧ (𝐵 sadd 𝐶) ⊆ ℕ0)
→ (𝐴 sadd (𝐵 sadd 𝐶)) ⊆
ℕ0) | 
| 119 | 2, 70, 118 | syl2anc 584 | . . . . . . . 8
⊢ (𝜑 → (𝐴 sadd (𝐵 sadd 𝐶)) ⊆
ℕ0) | 
| 120 | 117, 119 | sstrid 3994 | . . . . . . 7
⊢ (𝜑 → ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ⊆
ℕ0) | 
| 121 |  | inss2 4237 | . . . . . . . 8
⊢ ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ⊆ (0..^𝑁) | 
| 122 |  | ssfi 9214 | . . . . . . . 8
⊢
(((0..^𝑁) ∈ Fin
∧ ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ∈ Fin) | 
| 123 | 5, 121, 122 | sylancl 586 | . . . . . . 7
⊢ (𝜑 → ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ∈ Fin) | 
| 124 |  | elfpw 9395 | . . . . . . 7
⊢ (((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin) ↔ (((𝐴 sadd
(𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ⊆ ℕ0 ∧ ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ∈ Fin)) | 
| 125 | 120, 123,
124 | sylanbrc 583 | . . . . . 6
⊢ (𝜑 → ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin)) | 
| 126 | 14 | ffvelcdmi 7102 | . . . . . 6
⊢ (((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin) → (◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ∈
ℕ0) | 
| 127 | 125, 126 | syl 17 | . . . . 5
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ∈
ℕ0) | 
| 128 | 127 | nn0red 12590 | . . . 4
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ∈ ℝ) | 
| 129 |  | 2nn 12340 | . . . . . . 7
⊢ 2 ∈
ℕ | 
| 130 | 129 | a1i 11 | . . . . . 6
⊢ (𝜑 → 2 ∈
ℕ) | 
| 131 | 130, 60 | nnexpcld 14285 | . . . . 5
⊢ (𝜑 → (2↑𝑁) ∈ ℕ) | 
| 132 | 131 | nnrpd 13076 | . . . 4
⊢ (𝜑 → (2↑𝑁) ∈
ℝ+) | 
| 133 | 127 | nn0ge0d 12592 | . . . 4
⊢ (𝜑 → 0 ≤ (◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) | 
| 134 | 127 | fvresd 6925 | . . . . . . . 8
⊢ (𝜑 → ((bits ↾
ℕ0)‘(◡(bits
↾ ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) = (bits‘(◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))))) | 
| 135 |  | f1ocnvfv2 7298 | . . . . . . . . 9
⊢ (((bits
↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin)
∧ ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin)) → ((bits ↾ ℕ0)‘(◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) = ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) | 
| 136 | 11, 125, 135 | sylancr 587 | . . . . . . . 8
⊢ (𝜑 → ((bits ↾
ℕ0)‘(◡(bits
↾ ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) = ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) | 
| 137 | 134, 136 | eqtr3d 2778 | . . . . . . 7
⊢ (𝜑 → (bits‘(◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) = ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) | 
| 138 | 137, 121 | eqsstrdi 4027 | . . . . . 6
⊢ (𝜑 → (bits‘(◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) ⊆ (0..^𝑁)) | 
| 139 | 127 | nn0zd 12641 | . . . . . . 7
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ∈ ℤ) | 
| 140 |  | bitsfzo 16473 | . . . . . . 7
⊢ (((◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) ↔ (bits‘(◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) ⊆ (0..^𝑁))) | 
| 141 | 139, 60, 140 | syl2anc 584 | . . . . . 6
⊢ (𝜑 → ((◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) ↔ (bits‘(◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) ⊆ (0..^𝑁))) | 
| 142 | 138, 141 | mpbird 257 | . . . . 5
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁))) | 
| 143 |  | elfzolt2 13709 | . . . . 5
⊢ ((◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) → (◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) < (2↑𝑁)) | 
| 144 | 142, 143 | syl 17 | . . . 4
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) < (2↑𝑁)) | 
| 145 |  | modid 13937 | . . . 4
⊢ ((((◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ∈ ℝ ∧ (2↑𝑁) ∈ ℝ+)
∧ (0 ≤ (◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ∧ (◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) < (2↑𝑁))) → ((◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) mod (2↑𝑁)) = (◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) | 
| 146 | 128, 132,
133, 144, 145 | syl22anc 838 | . . 3
⊢ (𝜑 → ((◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) mod (2↑𝑁)) = (◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) | 
| 147 | 90, 116, 146 | 3eqtr3d 2784 | . 2
⊢ (𝜑 → (◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) = (◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) | 
| 148 |  | f1of1 6846 | . . . . 5
⊢ (◡(bits ↾
ℕ0):(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0 → ◡(bits ↾
ℕ0):(𝒫 ℕ0 ∩ Fin)–1-1→ℕ0) | 
| 149 | 11, 12, 148 | mp2b 10 | . . . 4
⊢ ◡(bits ↾
ℕ0):(𝒫 ℕ0 ∩ Fin)–1-1→ℕ0 | 
| 150 |  | f1fveq 7283 | . . . 4
⊢ ((◡(bits ↾
ℕ0):(𝒫 ℕ0 ∩ Fin)–1-1→ℕ0 ∧ ((((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin) ∧ ((𝐴 sadd
(𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin))) → ((◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) = (◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ↔ (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) = ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) | 
| 151 | 149, 150 | mpan 690 | . . 3
⊢
(((((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin) ∧ ((𝐴 sadd
(𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ∈ (𝒫 ℕ0
∩ Fin)) → ((◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) = (◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ↔ (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) = ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) | 
| 152 | 99, 125, 151 | syl2anc 584 | . 2
⊢ (𝜑 → ((◡(bits ↾
ℕ0)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) = (◡(bits ↾
ℕ0)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ↔ (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) = ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) | 
| 153 | 147, 152 | mpbid 232 | 1
⊢ (𝜑 → (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) = ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) |