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Theorem sadasslem 15819

Description: Lemma for sadass 15820. (Contributed by Mario Carneiro, 9-Sep-2016.)

**Hypotheses**
Ref	Expression
sadasslem.1	⊢ (𝜑 → 𝐴 ⊆ ℕ₀)
sadasslem.2	⊢ (𝜑 → 𝐵 ⊆ ℕ₀)
sadasslem.3	⊢ (𝜑 → 𝐶 ⊆ ℕ₀)
sadasslem.4	⊢ (𝜑 → 𝑁 ∈ ℕ₀)

**Assertion**
Ref	Expression
sadasslem	⊢ (𝜑 → (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) = ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))

Proof of Theorem sadasslem Dummy variables 𝑐 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 4205	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ 𝐴
2		sadasslem.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ ℕ₀)
3	1, 2	sstrid 3978	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀)
4		fzofi 13343	. . . . . . . . . . . 12 ⊢ (0..^𝑁) ∈ Fin
5	4	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (0..^𝑁) ∈ Fin)
6		inss2 4206	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
7		ssfi 8738	. . . . . . . . . . 11 ⊢ (((0..^𝑁) ∈ Fin ∧ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (𝐴 ∩ (0..^𝑁)) ∈ Fin)
8	5, 6, 7	sylancl 588	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ∈ Fin)
9		elfpw 8826	. . . . . . . . . 10 ⊢ ((𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ (𝐴 ∩ (0..^𝑁)) ∈ Fin))
10	3, 8, 9	sylanbrc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
11		bitsf1o 15794	. . . . . . . . . . 11 ⊢ (bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin)
12		f1ocnv 6627	. . . . . . . . . . 11 ⊢ ((bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin) → ^◡(bits ↾ ℕ₀):(𝒫 ℕ₀ ∩ Fin)–1-1-onto→ℕ₀)
13		f1of 6615	. . . . . . . . . . 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	ffvelrni 6850	. . . . . . . . 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 11958	. . . . . . 7 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁))) ∈ ℂ)
18		inss1 4205	. . . . . . . . . . 11 ⊢ (𝐵 ∩ (0..^𝑁)) ⊆ 𝐵
19		sadasslem.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ⊆ ℕ₀)
20	18, 19	sstrid 3978	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ₀)
21		inss2 4206	. . . . . . . . . . 11 ⊢ (𝐵 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
22		ssfi 8738	. . . . . . . . . . 11 ⊢ (((0..^𝑁) ∈ Fin ∧ (𝐵 ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ∈ Fin)
23	5, 21, 22	sylancl 588	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ (0..^𝑁)) ∈ Fin)
24		elfpw 8826	. . . . . . . . . 10 ⊢ ((𝐵 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((𝐵 ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ (𝐵 ∩ (0..^𝑁)) ∈ Fin))
25	20, 23, 24	sylanbrc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
26	14	ffvelrni 6850	. . . . . . . . 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 11958	. . . . . . 7 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁))) ∈ ℂ)
29		inss1 4205	. . . . . . . . . . 11 ⊢ (𝐶 ∩ (0..^𝑁)) ⊆ 𝐶
30		sadasslem.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ⊆ ℕ₀)
31	29, 30	sstrid 3978	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∩ (0..^𝑁)) ⊆ ℕ₀)
32		inss2 4206	. . . . . . . . . . 11 ⊢ (𝐶 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
33		ssfi 8738	. . . . . . . . . . 11 ⊢ (((0..^𝑁) ∈ Fin ∧ (𝐶 ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (𝐶 ∩ (0..^𝑁)) ∈ Fin)
34	5, 32, 33	sylancl 588	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∩ (0..^𝑁)) ∈ Fin)
35		elfpw 8826	. . . . . . . . . 10 ⊢ ((𝐶 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((𝐶 ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ (𝐶 ∩ (0..^𝑁)) ∈ Fin))
36	31, 34, 35	sylanbrc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
37	14	ffvelrni 6850	. . . . . . . . 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 11958	. . . . . . 7 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(𝐶 ∩ (0..^𝑁))) ∈ ℂ)
40	17, 28, 39	addassd 10663	. . . . . 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 7171	. . . . 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 4205	. . . . . . . . . 10 ⊢ ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ⊆ (𝐴 sadd 𝐵)
43		sadcl 15811	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝐵 ⊆ ℕ₀) → (𝐴 sadd 𝐵) ⊆ ℕ₀)
44	2, 19, 43	syl2anc 586	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 sadd 𝐵) ⊆ ℕ₀)
45	42, 44	sstrid 3978	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ⊆ ℕ₀)
46		inss2 4206	. . . . . . . . . 10 ⊢ ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ⊆ (0..^𝑁)
47		ssfi 8738	. . . . . . . . . 10 ⊢ (((0..^𝑁) ∈ Fin ∧ ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ∈ Fin)
48	5, 46, 47	sylancl 588	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ∈ Fin)
49		elfpw 8826	. . . . . . . . 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 585	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
51	14	ffvelrni 6850	. . . . . . . 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 11957	. . . . . 6 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) ∈ ℝ)
54	16	nn0red 11957	. . . . . . 7 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁))) ∈ ℝ)
55	27	nn0red 11957	. . . . . . 7 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁))) ∈ ℝ)
56	54, 55	readdcld 10670	. . . . . 6 ⊢ (𝜑 → ((^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁))) + (^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ)
57	38	nn0red 11957	. . . . . 6 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(𝐶 ∩ (0..^𝑁))) ∈ ℝ)
58		2rp 12395	. . . . . . . 8 ⊢ 2 ∈ ℝ⁺
59	58	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℝ⁺)
60		sadasslem.4	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
61	60	nn0zd 12086	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
62	59, 61	rpexpcld 13609	. . . . . 6 ⊢ (𝜑 → (2↑𝑁) ∈ ℝ⁺)
63		eqid 2821	. . . . . . 7 ⊢ seq0((𝑐 ∈ 2_o, 𝑚 ∈ ℕ₀ ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1_o, ∅)), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = seq0((𝑐 ∈ 2_o, 𝑚 ∈ ℕ₀ ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1_o, ∅)), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
64		eqid 2821	. . . . . . 7 ⊢ ^◡(bits ↾ ℕ₀) = ^◡(bits ↾ ℕ₀)
65	2, 19, 63, 60, 64	sadadd3 15810	. . . . . 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 2822	. . . . . 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 13296	. . . . 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 4205	. . . . . . . . . 10 ⊢ ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ⊆ (𝐵 sadd 𝐶)
69		sadcl 15811	. . . . . . . . . . 11 ⊢ ((𝐵 ⊆ ℕ₀ ∧ 𝐶 ⊆ ℕ₀) → (𝐵 sadd 𝐶) ⊆ ℕ₀)
70	19, 30, 69	syl2anc 586	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 sadd 𝐶) ⊆ ℕ₀)
71	68, 70	sstrid 3978	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ⊆ ℕ₀)
72		inss2 4206	. . . . . . . . . 10 ⊢ ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ⊆ (0..^𝑁)
73		ssfi 8738	. . . . . . . . . 10 ⊢ (((0..^𝑁) ∈ Fin ∧ ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ∈ Fin)
74	5, 72, 73	sylancl 588	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ∈ Fin)
75		elfpw 8826	. . . . . . . . 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 585	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
77	14	ffvelrni 6850	. . . . . . . 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 11957	. . . . . 6 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘((𝐵 sadd 𝐶) ∩ (0..^𝑁))) ∈ ℝ)
80	55, 57	readdcld 10670	. . . . . 6 ⊢ (𝜑 → ((^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁))) + (^◡(bits ↾ ℕ₀)‘(𝐶 ∩ (0..^𝑁)))) ∈ ℝ)
81		eqidd 2822	. . . . . 6 ⊢ (𝜑 → ((^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁))) mod (2↑𝑁)) = ((^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁))) mod (2↑𝑁)))
82		eqid 2821	. . . . . . 7 ⊢ seq0((𝑐 ∈ 2_o, 𝑚 ∈ ℕ₀ ↦ if(cadd(𝑚 ∈ 𝐵, 𝑚 ∈ 𝐶, ∅ ∈ 𝑐), 1_o, ∅)), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = seq0((𝑐 ∈ 2_o, 𝑚 ∈ ℕ₀ ↦ if(cadd(𝑚 ∈ 𝐵, 𝑚 ∈ 𝐶, ∅ ∈ 𝑐), 1_o, ∅)), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
83	19, 30, 82, 60, 64	sadadd3 15810	. . . . . 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 13296	. . . . 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 2866	. . . 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 2821	. . . . 5 ⊢ seq0((𝑐 ∈ 2_o, 𝑚 ∈ ℕ₀ ↦ if(cadd(𝑚 ∈ (𝐴 sadd 𝐵), 𝑚 ∈ 𝐶, ∅ ∈ 𝑐), 1_o, ∅)), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = seq0((𝑐 ∈ 2_o, 𝑚 ∈ ℕ₀ ↦ if(cadd(𝑚 ∈ (𝐴 sadd 𝐵), 𝑚 ∈ 𝐶, ∅ ∈ 𝑐), 1_o, ∅)), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
87	44, 30, 86, 60, 64	sadadd3 15810	. . . 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 2821	. . . . 5 ⊢ seq0((𝑐 ∈ 2_o, 𝑚 ∈ ℕ₀ ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ (𝐵 sadd 𝐶), ∅ ∈ 𝑐), 1_o, ∅)), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = seq0((𝑐 ∈ 2_o, 𝑚 ∈ ℕ₀ ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ (𝐵 sadd 𝐶), ∅ ∈ 𝑐), 1_o, ∅)), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
89	2, 70, 88, 60, 64	sadadd3 15810	. . . 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 2866	. . 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 4205	. . . . . . . 8 ⊢ (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ⊆ ((𝐴 sadd 𝐵) sadd 𝐶)
92		sadcl 15811	. . . . . . . . 9 ⊢ (((𝐴 sadd 𝐵) ⊆ ℕ₀ ∧ 𝐶 ⊆ ℕ₀) → ((𝐴 sadd 𝐵) sadd 𝐶) ⊆ ℕ₀)
93	44, 30, 92	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 sadd 𝐵) sadd 𝐶) ⊆ ℕ₀)
94	91, 93	sstrid 3978	. . . . . . 7 ⊢ (𝜑 → (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ⊆ ℕ₀)
95		inss2 4206	. . . . . . . 8 ⊢ (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ⊆ (0..^𝑁)
96		ssfi 8738	. . . . . . . 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 588	. . . . . . 7 ⊢ (𝜑 → (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ∈ Fin)
98		elfpw 8826	. . . . . . 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 585	. . . . . 6 ⊢ (𝜑 → (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
100	14	ffvelrni 6850	. . . . . 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 11957	. . . 4 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) ∈ ℝ)
103	101	nn0ge0d 11959	. . . 4 ⊢ (𝜑 → 0 ≤ (^◡(bits ↾ ℕ₀)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))))
104	101	fvresd 6690	. . . . . . . 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 7034	. . . . . . . . 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 589	. . . . . . . 8 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))) = (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))
107	104, 106	eqtr3d 2858	. . . . . . 7 ⊢ (𝜑 → (bits‘(^◡(bits ↾ ℕ₀)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))) = (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))
108	107, 95	eqsstrdi 4021	. . . . . 6 ⊢ (𝜑 → (bits‘(^◡(bits ↾ ℕ₀)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))) ⊆ (0..^𝑁))
109	101	nn0zd 12086	. . . . . . 7 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) ∈ ℤ)
110		bitsfzo 15784	. . . . . . 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 586	. . . . . 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 259	. . . . 5 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)))
113		elfzolt2 13048	. . . . 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 13265	. . . 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 836	. . 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 4205	. . . . . . . 8 ⊢ ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ⊆ (𝐴 sadd (𝐵 sadd 𝐶))
118		sadcl 15811	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℕ₀ ∧ (𝐵 sadd 𝐶) ⊆ ℕ₀) → (𝐴 sadd (𝐵 sadd 𝐶)) ⊆ ℕ₀)
119	2, 70, 118	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → (𝐴 sadd (𝐵 sadd 𝐶)) ⊆ ℕ₀)
120	117, 119	sstrid 3978	. . . . . . 7 ⊢ (𝜑 → ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ⊆ ℕ₀)
121		inss2 4206	. . . . . . . 8 ⊢ ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ⊆ (0..^𝑁)
122		ssfi 8738	. . . . . . . 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 588	. . . . . . 7 ⊢ (𝜑 → ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ∈ Fin)
124		elfpw 8826	. . . . . . 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 585	. . . . . 6 ⊢ (𝜑 → ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
126	14	ffvelrni 6850	. . . . . 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 11957	. . . 4 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ∈ ℝ)
129		2nn 11711	. . . . . . 7 ⊢ 2 ∈ ℕ
130	129	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℕ)
131	130, 60	nnexpcld 13607	. . . . 5 ⊢ (𝜑 → (2↑𝑁) ∈ ℕ)
132	131	nnrpd 12430	. . . 4 ⊢ (𝜑 → (2↑𝑁) ∈ ℝ⁺)
133	127	nn0ge0d 11959	. . . 4 ⊢ (𝜑 → 0 ≤ (^◡(bits ↾ ℕ₀)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))))
134	127	fvresd 6690	. . . . . . . 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 7034	. . . . . . . . 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 589	. . . . . . . 8 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) = ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))
137	134, 136	eqtr3d 2858	. . . . . . 7 ⊢ (𝜑 → (bits‘(^◡(bits ↾ ℕ₀)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) = ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))
138	137, 121	eqsstrdi 4021	. . . . . 6 ⊢ (𝜑 → (bits‘(^◡(bits ↾ ℕ₀)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) ⊆ (0..^𝑁))
139	127	nn0zd 12086	. . . . . . 7 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ∈ ℤ)
140		bitsfzo 15784	. . . . . . 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 586	. . . . . 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 259	. . . . 5 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)))
143		elfzolt2 13048	. . . . 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 13265	. . . 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 836	. . 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 2864	. 2 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) = (^◡(bits ↾ ℕ₀)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))))
148		f1of1 6614	. . . . 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 7020	. . . 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 688	. . 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 586	. 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 234	1 ⊢ (𝜑 → (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) = ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 caddwcad 1607 ∈ wcel 2114 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 ifcif 4467 𝒫 cpw 4539 class class class wbr 5066 ↦ cmpt 5146 ^◡ccnv 5554 ↾ cres 5557 ⟶wf 6351 –1-1→wf1 6352 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 1_oc1o 8095 2_oc2o 8096 Fincfn 8509 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 < clt 10675 ≤ cle 10676 − cmin 10870 ℕcn 11638 2c2 11693 ℕ₀cn0 11898 ℤcz 11982 ℝ⁺crp 12390 ..^cfzo 13034 mod cmo 13238 seqcseq 13370 ↑cexp 13430 bitscbits 15768 sadd csad 15769

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1502 df-tru 1540 df-fal 1550 df-had 1594 df-cad 1608 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-disj 5032 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-dvds 15608 df-bits 15771 df-sad 15800

This theorem is referenced by: sadass 15820

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