sadasslem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > sadasslem		Structured version Visualization version GIF version

Theorem sadasslem 16177

Description: Lemma for sadass 16178. (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 4162	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ 𝐴
2		sadasslem.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ ℕ₀)
3	1, 2	sstrid 3932	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀)
4		fzofi 13694	. . . . . . . . . . . 12 ⊢ (0..^𝑁) ∈ Fin
5	4	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (0..^𝑁) ∈ Fin)
6		inss2 4163	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
7		ssfi 8956	. . . . . . . . . . 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 9121	. . . . . . . . . 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 16152	. . . . . . . . . . 11 ⊢ (bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin)
12		f1ocnv 6728	. . . . . . . . . . 11 ⊢ ((bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin) → ^◡(bits ↾ ℕ₀):(𝒫 ℕ₀ ∩ Fin)–1-1-onto→ℕ₀)
13		f1of 6716	. . . . . . . . . . 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 6960	. . . . . . . . 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 12295	. . . . . . 7 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁))) ∈ ℂ)
18		inss1 4162	. . . . . . . . . . 11 ⊢ (𝐵 ∩ (0..^𝑁)) ⊆ 𝐵
19		sadasslem.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ⊆ ℕ₀)
20	18, 19	sstrid 3932	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ₀)
21		inss2 4163	. . . . . . . . . . 11 ⊢ (𝐵 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
22		ssfi 8956	. . . . . . . . . . 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 9121	. . . . . . . . . 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	ffvelrni 6960	. . . . . . . . 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 12295	. . . . . . 7 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁))) ∈ ℂ)
29		inss1 4162	. . . . . . . . . . 11 ⊢ (𝐶 ∩ (0..^𝑁)) ⊆ 𝐶
30		sadasslem.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ⊆ ℕ₀)
31	29, 30	sstrid 3932	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∩ (0..^𝑁)) ⊆ ℕ₀)
32		inss2 4163	. . . . . . . . . . 11 ⊢ (𝐶 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
33		ssfi 8956	. . . . . . . . . . 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 9121	. . . . . . . . . 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	ffvelrni 6960	. . . . . . . . 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 12295	. . . . . . 7 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(𝐶 ∩ (0..^𝑁))) ∈ ℂ)
40	17, 28, 39	addassd 10997	. . . . . 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 7290	. . . . 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 4162	. . . . . . . . . 10 ⊢ ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ⊆ (𝐴 sadd 𝐵)
43		sadcl 16169	. . . . . . . . . . 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 3932	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ⊆ ℕ₀)
46		inss2 4163	. . . . . . . . . 10 ⊢ ((𝐴 sadd 𝐵) ∩ (0..^𝑁)) ⊆ (0..^𝑁)
47		ssfi 8956	. . . . . . . . . 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 9121	. . . . . . . . 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	ffvelrni 6960	. . . . . . . 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 12294	. . . . . 6 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) ∈ ℝ)
54	16	nn0red 12294	. . . . . . 7 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁))) ∈ ℝ)
55	27	nn0red 12294	. . . . . . 7 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁))) ∈ ℝ)
56	54, 55	readdcld 11004	. . . . . 6 ⊢ (𝜑 → ((^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁))) + (^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ)
57	38	nn0red 12294	. . . . . 6 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(𝐶 ∩ (0..^𝑁))) ∈ ℝ)
58		2rp 12735	. . . . . . . 8 ⊢ 2 ∈ ℝ⁺
59	58	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℝ⁺)
60		sadasslem.4	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
61	60	nn0zd 12424	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
62	59, 61	rpexpcld 13962	. . . . . 6 ⊢ (𝜑 → (2↑𝑁) ∈ ℝ⁺)
63		eqid 2738	. . . . . . 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 2738	. . . . . . 7 ⊢ ^◡(bits ↾ ℕ₀) = ^◡(bits ↾ ℕ₀)
65	2, 19, 63, 60, 64	sadadd3 16168	. . . . . 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 2739	. . . . . 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 13647	. . . . 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 4162	. . . . . . . . . 10 ⊢ ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ⊆ (𝐵 sadd 𝐶)
69		sadcl 16169	. . . . . . . . . . 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 3932	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ⊆ ℕ₀)
72		inss2 4163	. . . . . . . . . 10 ⊢ ((𝐵 sadd 𝐶) ∩ (0..^𝑁)) ⊆ (0..^𝑁)
73		ssfi 8956	. . . . . . . . . 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 9121	. . . . . . . . 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	ffvelrni 6960	. . . . . . . 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 12294	. . . . . 6 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘((𝐵 sadd 𝐶) ∩ (0..^𝑁))) ∈ ℝ)
80	55, 57	readdcld 11004	. . . . . 6 ⊢ (𝜑 → ((^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁))) + (^◡(bits ↾ ℕ₀)‘(𝐶 ∩ (0..^𝑁)))) ∈ ℝ)
81		eqidd 2739	. . . . . 6 ⊢ (𝜑 → ((^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁))) mod (2↑𝑁)) = ((^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁))) mod (2↑𝑁)))
82		eqid 2738	. . . . . . 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 16168	. . . . . 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 13647	. . . . 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 2788	. . . 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 2738	. . . . 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 16168	. . . 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 2738	. . . . 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 16168	. . . 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 2788	. . 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 4162	. . . . . . . 8 ⊢ (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ⊆ ((𝐴 sadd 𝐵) sadd 𝐶)
92		sadcl 16169	. . . . . . . . 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 3932	. . . . . . 7 ⊢ (𝜑 → (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ⊆ ℕ₀)
95		inss2 4163	. . . . . . . 8 ⊢ (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)) ⊆ (0..^𝑁)
96		ssfi 8956	. . . . . . . 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 9121	. . . . . . 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	ffvelrni 6960	. . . . . 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 12294	. . . 4 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) ∈ ℝ)
103	101	nn0ge0d 12296	. . . 4 ⊢ (𝜑 → 0 ≤ (^◡(bits ↾ ℕ₀)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))))
104	101	fvresd 6794	. . . . . . . 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 7149	. . . . . . . . 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 2780	. . . . . . 7 ⊢ (𝜑 → (bits‘(^◡(bits ↾ ℕ₀)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))) = (((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))
108	107, 95	eqsstrdi 3975	. . . . . 6 ⊢ (𝜑 → (bits‘(^◡(bits ↾ ℕ₀)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁)))) ⊆ (0..^𝑁))
109	101	nn0zd 12424	. . . . . . 7 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) ∈ ℤ)
110		bitsfzo 16142	. . . . . . 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 256	. . . . 5 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)))
113		elfzolt2 13396	. . . . 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 13616	. . . 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 4162	. . . . . . . 8 ⊢ ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ⊆ (𝐴 sadd (𝐵 sadd 𝐶))
118		sadcl 16169	. . . . . . . . 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 3932	. . . . . . 7 ⊢ (𝜑 → ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ⊆ ℕ₀)
121		inss2 4163	. . . . . . . 8 ⊢ ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)) ⊆ (0..^𝑁)
122		ssfi 8956	. . . . . . . 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 9121	. . . . . . 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	ffvelrni 6960	. . . . . 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 12294	. . . 4 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ∈ ℝ)
129		2nn 12046	. . . . . . 7 ⊢ 2 ∈ ℕ
130	129	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℕ)
131	130, 60	nnexpcld 13960	. . . . 5 ⊢ (𝜑 → (2↑𝑁) ∈ ℕ)
132	131	nnrpd 12770	. . . 4 ⊢ (𝜑 → (2↑𝑁) ∈ ℝ⁺)
133	127	nn0ge0d 12296	. . . 4 ⊢ (𝜑 → 0 ≤ (^◡(bits ↾ ℕ₀)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))))
134	127	fvresd 6794	. . . . . . . 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 7149	. . . . . . . . 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 2780	. . . . . . 7 ⊢ (𝜑 → (bits‘(^◡(bits ↾ ℕ₀)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) = ((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))
138	137, 121	eqsstrdi 3975	. . . . . 6 ⊢ (𝜑 → (bits‘(^◡(bits ↾ ℕ₀)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁)))) ⊆ (0..^𝑁))
139	127	nn0zd 12424	. . . . . . 7 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ∈ ℤ)
140		bitsfzo 16142	. . . . . . 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 256	. . . . 5 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)))
143		elfzolt2 13396	. . . . 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 13616	. . . 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 2786	. 2 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘(((𝐴 sadd 𝐵) sadd 𝐶) ∩ (0..^𝑁))) = (^◡(bits ↾ ℕ₀)‘((𝐴 sadd (𝐵 sadd 𝐶)) ∩ (0..^𝑁))))
148		f1of1 6715	. . . . 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 7135	. . . 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 687	. . 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 231	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 205 ∧ wa 396 = wceq 1539 caddwcad 1608 ∈ wcel 2106 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 ifcif 4459 𝒫 cpw 4533 class class class wbr 5074 ↦ cmpt 5157 ^◡ccnv 5588 ↾ cres 5591 ⟶wf 6429 –1-1→wf1 6430 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 1_oc1o 8290 2_oc2o 8291 Fincfn 8733 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 < clt 11009 ≤ cle 11010 − cmin 11205 ℕcn 11973 2c2 12028 ℕ₀cn0 12233 ℤcz 12319 ℝ⁺crp 12730 ..^cfzo 13382 mod cmo 13589 seqcseq 13721 ↑cexp 13782 bitscbits 16126 sadd csad 16127

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-xor 1507 df-tru 1542 df-fal 1552 df-had 1595 df-cad 1609 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 df-dvds 15964 df-bits 16129 df-sad 16158

This theorem is referenced by: sadass 16178

W3C validator