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Theorem sadaddlem 16377

Description: Lemma for sadadd 16378. (Contributed by Mario Carneiro, 9-Sep-2016.)

**Hypotheses**
Ref	Expression
sadaddlem.c	⊢ 𝐶 = seq0((𝑐 ∈ 2_o, 𝑚 ∈ ℕ₀ ↦ if(cadd(𝑚 ∈ (bits‘𝐴), 𝑚 ∈ (bits‘𝐵), ∅ ∈ 𝑐), 1_o, ∅)), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
sadaddlem.k	⊢ 𝐾 = ^◡(bits ↾ ℕ₀)
sadaddlem.1	⊢ (𝜑 → 𝐴 ∈ ℤ)
sadaddlem.2	⊢ (𝜑 → 𝐵 ∈ ℤ)
sadaddlem.3	⊢ (𝜑 → 𝑁 ∈ ℕ₀)

**Assertion**
Ref	Expression
sadaddlem	⊢ (𝜑 → (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) = (bits‘((𝐴 + 𝐵) mod (2↑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)

**Proof of Theorem sadaddlem**
Step	Hyp	Ref	Expression
1		2nn 12198	. . . . . . . . . 10 ⊢ 2 ∈ ℕ
2	1	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℕ)
3		sadaddlem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
4	2, 3	nnexpcld 14152	. . . . . . . 8 ⊢ (𝜑 → (2↑𝑁) ∈ ℕ)
5	4	nnzd 12495	. . . . . . 7 ⊢ (𝜑 → (2↑𝑁) ∈ ℤ)
6		sadaddlem.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℤ)
7		inss1 4184	. . . . . . . . . . . 12 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ (bits‘𝐴)
8		bitsss 16337	. . . . . . . . . . . 12 ⊢ (bits‘𝐴) ⊆ ℕ₀
9	7, 8	sstri 3939	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ ℕ₀
10		fzofi 13881	. . . . . . . . . . . 12 ⊢ (0..^𝑁) ∈ Fin
11		inss2 4185	. . . . . . . . . . . 12 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ (0..^𝑁)
12		ssfi 9082	. . . . . . . . . . . 12 ⊢ (((0..^𝑁) ∈ Fin ∧ ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → ((bits‘𝐴) ∩ (0..^𝑁)) ∈ Fin)
13	10, 11, 12	mp2an 692	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ∈ Fin
14		elfpw 9238	. . . . . . . . . . 11 ⊢ (((bits‘𝐴) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ (((bits‘𝐴) ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ ((bits‘𝐴) ∩ (0..^𝑁)) ∈ Fin))
15	9, 13, 14	mpbir2an 711	. . . . . . . . . 10 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin)
16		bitsf1o 16356	. . . . . . . . . . . . 13 ⊢ (bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin)
17		f1ocnv 6775	. . . . . . . . . . . . 13 ⊢ ((bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin) → ^◡(bits ↾ ℕ₀):(𝒫 ℕ₀ ∩ Fin)–1-1-onto→ℕ₀)
18		f1of 6763	. . . . . . . . . . . . 13 ⊢ (^◡(bits ↾ ℕ₀):(𝒫 ℕ₀ ∩ Fin)–1-1-onto→ℕ₀ → ^◡(bits ↾ ℕ₀):(𝒫 ℕ₀ ∩ Fin)⟶ℕ₀)
19	16, 17, 18	mp2b 10	. . . . . . . . . . . 12 ⊢ ^◡(bits ↾ ℕ₀):(𝒫 ℕ₀ ∩ Fin)⟶ℕ₀
20		sadaddlem.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = ^◡(bits ↾ ℕ₀)
21	20	feq1i 6642	. . . . . . . . . . . 12 ⊢ (𝐾:(𝒫 ℕ₀ ∩ Fin)⟶ℕ₀ ↔ ^◡(bits ↾ ℕ₀):(𝒫 ℕ₀ ∩ Fin)⟶ℕ₀)
22	19, 21	mpbir 231	. . . . . . . . . . 11 ⊢ 𝐾:(𝒫 ℕ₀ ∩ Fin)⟶ℕ₀
23	22	ffvelcdmi 7016	. . . . . . . . . 10 ⊢ (((bits‘𝐴) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) → (𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) ∈ ℕ₀)
24	15, 23	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) ∈ ℕ₀)
25	24	nn0zd 12494	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) ∈ ℤ)
26	6, 25	zsubcld 12582	. . . . . . 7 ⊢ (𝜑 → (𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) ∈ ℤ)
27		sadaddlem.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℤ)
28		inss1 4184	. . . . . . . . . . . 12 ⊢ ((bits‘𝐵) ∩ (0..^𝑁)) ⊆ (bits‘𝐵)
29		bitsss 16337	. . . . . . . . . . . 12 ⊢ (bits‘𝐵) ⊆ ℕ₀
30	28, 29	sstri 3939	. . . . . . . . . . 11 ⊢ ((bits‘𝐵) ∩ (0..^𝑁)) ⊆ ℕ₀
31		inss2 4185	. . . . . . . . . . . 12 ⊢ ((bits‘𝐵) ∩ (0..^𝑁)) ⊆ (0..^𝑁)
32		ssfi 9082	. . . . . . . . . . . 12 ⊢ (((0..^𝑁) ∈ Fin ∧ ((bits‘𝐵) ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → ((bits‘𝐵) ∩ (0..^𝑁)) ∈ Fin)
33	10, 31, 32	mp2an 692	. . . . . . . . . . 11 ⊢ ((bits‘𝐵) ∩ (0..^𝑁)) ∈ Fin
34		elfpw 9238	. . . . . . . . . . 11 ⊢ (((bits‘𝐵) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ (((bits‘𝐵) ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ ((bits‘𝐵) ∩ (0..^𝑁)) ∈ Fin))
35	30, 33, 34	mpbir2an 711	. . . . . . . . . 10 ⊢ ((bits‘𝐵) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin)
36	22	ffvelcdmi 7016	. . . . . . . . . 10 ⊢ (((bits‘𝐵) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) → (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))) ∈ ℕ₀)
37	35, 36	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))) ∈ ℕ₀)
38	37	nn0zd 12494	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))) ∈ ℤ)
39	27, 38	zsubcld 12582	. . . . . . 7 ⊢ (𝜑 → (𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) ∈ ℤ)
40	20	fveq1i 6823	. . . . . . . . . . . 12 ⊢ (𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) = (^◡(bits ↾ ℕ₀)‘((bits‘𝐴) ∩ (0..^𝑁)))
41	6, 4	zmodcld 13796	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 mod (2↑𝑁)) ∈ ℕ₀)
42	41	fvresd 6842	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐴 mod (2↑𝑁))) = (bits‘(𝐴 mod (2↑𝑁))))
43		bitsmod 16347	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(𝐴 mod (2↑𝑁))) = ((bits‘𝐴) ∩ (0..^𝑁)))
44	6, 3, 43	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (bits‘(𝐴 mod (2↑𝑁))) = ((bits‘𝐴) ∩ (0..^𝑁)))
45	42, 44	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐴 mod (2↑𝑁))) = ((bits‘𝐴) ∩ (0..^𝑁)))
46		f1ocnvfv 7212	. . . . . . . . . . . . . 14 ⊢ (((bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin) ∧ (𝐴 mod (2↑𝑁)) ∈ ℕ₀) → (((bits ↾ ℕ₀)‘(𝐴 mod (2↑𝑁))) = ((bits‘𝐴) ∩ (0..^𝑁)) → (^◡(bits ↾ ℕ₀)‘((bits‘𝐴) ∩ (0..^𝑁))) = (𝐴 mod (2↑𝑁))))
47	16, 41, 46	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((bits ↾ ℕ₀)‘(𝐴 mod (2↑𝑁))) = ((bits‘𝐴) ∩ (0..^𝑁)) → (^◡(bits ↾ ℕ₀)‘((bits‘𝐴) ∩ (0..^𝑁))) = (𝐴 mod (2↑𝑁))))
48	45, 47	mpd 15	. . . . . . . . . . . 12 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘((bits‘𝐴) ∩ (0..^𝑁))) = (𝐴 mod (2↑𝑁)))
49	40, 48	eqtrid 2778	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) = (𝐴 mod (2↑𝑁)))
50	49	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) = (𝐴 − (𝐴 mod (2↑𝑁))))
51	50	oveq1d 7361	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) / (2↑𝑁)) = ((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)))
52	6	zred 12577	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
53	4	nnrpd 12932	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝑁) ∈ ℝ⁺)
54		moddifz 13787	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (2↑𝑁) ∈ ℝ⁺) → ((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) ∈ ℤ)
55	52, 53, 54	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) ∈ ℤ)
56	51, 55	eqeltrd 2831	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) / (2↑𝑁)) ∈ ℤ)
57	4	nnne0d 12175	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝑁) ≠ 0)
58		dvdsval2 16166	. . . . . . . . 9 ⊢ (((2↑𝑁) ∈ ℤ ∧ (2↑𝑁) ≠ 0 ∧ (𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) ∈ ℤ) → ((2↑𝑁) ∥ (𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) ↔ ((𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) / (2↑𝑁)) ∈ ℤ))
59	5, 57, 26, 58	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → ((2↑𝑁) ∥ (𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) ↔ ((𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) / (2↑𝑁)) ∈ ℤ))
60	56, 59	mpbird 257	. . . . . . 7 ⊢ (𝜑 → (2↑𝑁) ∥ (𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))))
61	20	fveq1i 6823	. . . . . . . . . . . 12 ⊢ (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))) = (^◡(bits ↾ ℕ₀)‘((bits‘𝐵) ∩ (0..^𝑁)))
62	27, 4	zmodcld 13796	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 mod (2↑𝑁)) ∈ ℕ₀)
63	62	fvresd 6842	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐵 mod (2↑𝑁))) = (bits‘(𝐵 mod (2↑𝑁))))
64		bitsmod 16347	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(𝐵 mod (2↑𝑁))) = ((bits‘𝐵) ∩ (0..^𝑁)))
65	27, 3, 64	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (bits‘(𝐵 mod (2↑𝑁))) = ((bits‘𝐵) ∩ (0..^𝑁)))
66	63, 65	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐵 mod (2↑𝑁))) = ((bits‘𝐵) ∩ (0..^𝑁)))
67		f1ocnvfv 7212	. . . . . . . . . . . . . 14 ⊢ (((bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin) ∧ (𝐵 mod (2↑𝑁)) ∈ ℕ₀) → (((bits ↾ ℕ₀)‘(𝐵 mod (2↑𝑁))) = ((bits‘𝐵) ∩ (0..^𝑁)) → (^◡(bits ↾ ℕ₀)‘((bits‘𝐵) ∩ (0..^𝑁))) = (𝐵 mod (2↑𝑁))))
68	16, 62, 67	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((bits ↾ ℕ₀)‘(𝐵 mod (2↑𝑁))) = ((bits‘𝐵) ∩ (0..^𝑁)) → (^◡(bits ↾ ℕ₀)‘((bits‘𝐵) ∩ (0..^𝑁))) = (𝐵 mod (2↑𝑁))))
69	66, 68	mpd 15	. . . . . . . . . . . 12 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘((bits‘𝐵) ∩ (0..^𝑁))) = (𝐵 mod (2↑𝑁)))
70	61, 69	eqtrid 2778	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))) = (𝐵 mod (2↑𝑁)))
71	70	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) = (𝐵 − (𝐵 mod (2↑𝑁))))
72	71	oveq1d 7361	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) / (2↑𝑁)) = ((𝐵 − (𝐵 mod (2↑𝑁))) / (2↑𝑁)))
73	27	zred 12577	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
74		moddifz 13787	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ (2↑𝑁) ∈ ℝ⁺) → ((𝐵 − (𝐵 mod (2↑𝑁))) / (2↑𝑁)) ∈ ℤ)
75	73, 53, 74	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − (𝐵 mod (2↑𝑁))) / (2↑𝑁)) ∈ ℤ)
76	72, 75	eqeltrd 2831	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) / (2↑𝑁)) ∈ ℤ)
77		dvdsval2 16166	. . . . . . . . 9 ⊢ (((2↑𝑁) ∈ ℤ ∧ (2↑𝑁) ≠ 0 ∧ (𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) ∈ ℤ) → ((2↑𝑁) ∥ (𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) ↔ ((𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) / (2↑𝑁)) ∈ ℤ))
78	5, 57, 39, 77	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → ((2↑𝑁) ∥ (𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) ↔ ((𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) / (2↑𝑁)) ∈ ℤ))
79	76, 78	mpbird 257	. . . . . . 7 ⊢ (𝜑 → (2↑𝑁) ∥ (𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))))
80	5, 26, 39, 60, 79	dvds2addd 16203	. . . . . 6 ⊢ (𝜑 → (2↑𝑁) ∥ ((𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) + (𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))))))
81	6	zcnd 12578	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
82	27	zcnd 12578	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
83	24	nn0cnd 12444	. . . . . . 7 ⊢ (𝜑 → (𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) ∈ ℂ)
84	37	nn0cnd 12444	. . . . . . 7 ⊢ (𝜑 → (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))) ∈ ℂ)
85	81, 82, 83, 84	addsub4d 11519	. . . . . 6 ⊢ (𝜑 → ((𝐴 + 𝐵) − ((𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) + (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))))) = ((𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) + (𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))))))
86	80, 85	breqtrrd 5117	. . . . 5 ⊢ (𝜑 → (2↑𝑁) ∥ ((𝐴 + 𝐵) − ((𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) + (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))))))
87	6, 27	zaddcld 12581	. . . . . 6 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ)
88	25, 38	zaddcld 12581	. . . . . 6 ⊢ (𝜑 → ((𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) + (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) ∈ ℤ)
89		moddvds 16174	. . . . . 6 ⊢ (((2↑𝑁) ∈ ℕ ∧ (𝐴 + 𝐵) ∈ ℤ ∧ ((𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) + (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) ∈ ℤ) → (((𝐴 + 𝐵) mod (2↑𝑁)) = (((𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) + (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) mod (2↑𝑁)) ↔ (2↑𝑁) ∥ ((𝐴 + 𝐵) − ((𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) + (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))))))
90	4, 87, 88, 89	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (((𝐴 + 𝐵) mod (2↑𝑁)) = (((𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) + (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) mod (2↑𝑁)) ↔ (2↑𝑁) ∥ ((𝐴 + 𝐵) − ((𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) + (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))))))
91	86, 90	mpbird 257	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) mod (2↑𝑁)) = (((𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) + (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) mod (2↑𝑁)))
92	8	a1i 11	. . . . 5 ⊢ (𝜑 → (bits‘𝐴) ⊆ ℕ₀)
93	29	a1i 11	. . . . 5 ⊢ (𝜑 → (bits‘𝐵) ⊆ ℕ₀)
94		sadaddlem.c	. . . . 5 ⊢ 𝐶 = seq0((𝑐 ∈ 2_o, 𝑚 ∈ ℕ₀ ↦ if(cadd(𝑚 ∈ (bits‘𝐴), 𝑚 ∈ (bits‘𝐵), ∅ ∈ 𝑐), 1_o, ∅)), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
95	92, 93, 94, 3, 20	sadadd3 16372	. . . 4 ⊢ (𝜑 → ((𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) mod (2↑𝑁)) = (((𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) + (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) mod (2↑𝑁)))
96		inss1 4184	. . . . . . . . 9 ⊢ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ⊆ ((bits‘𝐴) sadd (bits‘𝐵))
97		sadcl 16373	. . . . . . . . . 10 ⊢ (((bits‘𝐴) ⊆ ℕ₀ ∧ (bits‘𝐵) ⊆ ℕ₀) → ((bits‘𝐴) sadd (bits‘𝐵)) ⊆ ℕ₀)
98	8, 29, 97	mp2an 692	. . . . . . . . 9 ⊢ ((bits‘𝐴) sadd (bits‘𝐵)) ⊆ ℕ₀
99	96, 98	sstri 3939	. . . . . . . 8 ⊢ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ⊆ ℕ₀
100		inss2 4185	. . . . . . . . 9 ⊢ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ⊆ (0..^𝑁)
101		ssfi 9082	. . . . . . . . 9 ⊢ (((0..^𝑁) ∈ Fin ∧ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ∈ Fin)
102	10, 100, 101	mp2an 692	. . . . . . . 8 ⊢ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ∈ Fin
103		elfpw 9238	. . . . . . . 8 ⊢ ((((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ∈ Fin))
104	99, 102, 103	mpbir2an 711	. . . . . . 7 ⊢ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin)
105	22	ffvelcdmi 7016	. . . . . . 7 ⊢ ((((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) → (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∈ ℕ₀)
106	104, 105	mp1i 13	. . . . . 6 ⊢ (𝜑 → (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∈ ℕ₀)
107	106	nn0red 12443	. . . . 5 ⊢ (𝜑 → (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∈ ℝ)
108	106	nn0ge0d 12445	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))))
109	20	fveq1i 6823	. . . . . . . . . 10 ⊢ (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) = (^◡(bits ↾ ℕ₀)‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))
110	109	fveq2i 6825	. . . . . . . . 9 ⊢ ((bits ↾ ℕ₀)‘(𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))) = ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))))
111	106	fvresd 6842	. . . . . . . . 9 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))) = (bits‘(𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))))
112	104	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
113		f1ocnvfv2 7211	. . . . . . . . . 10 ⊢ (((bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin) ∧ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin)) → ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))) = (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))
114	16, 112, 113	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))) = (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))
115	110, 111, 114	3eqtr3a 2790	. . . . . . . 8 ⊢ (𝜑 → (bits‘(𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))) = (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))
116	115, 100	eqsstrdi 3974	. . . . . . 7 ⊢ (𝜑 → (bits‘(𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))) ⊆ (0..^𝑁))
117	106	nn0zd 12494	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∈ ℤ)
118		bitsfzo 16346	. . . . . . . 8 ⊢ (((𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) ↔ (bits‘(𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))) ⊆ (0..^𝑁)))
119	117, 3, 118	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) ↔ (bits‘(𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))) ⊆ (0..^𝑁)))
120	116, 119	mpbird 257	. . . . . 6 ⊢ (𝜑 → (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)))
121		elfzolt2 13568	. . . . . 6 ⊢ ((𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) → (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) < (2↑𝑁))
122	120, 121	syl 17	. . . . 5 ⊢ (𝜑 → (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) < (2↑𝑁))
123		modid 13800	. . . . 5 ⊢ ((((𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∈ ℝ ∧ (2↑𝑁) ∈ ℝ⁺) ∧ (0 ≤ (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∧ (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) < (2↑𝑁))) → ((𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) mod (2↑𝑁)) = (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))))
124	107, 53, 108, 122, 123	syl22anc 838	. . . 4 ⊢ (𝜑 → ((𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) mod (2↑𝑁)) = (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))))
125	91, 95, 124	3eqtr2d 2772	. . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) mod (2↑𝑁)) = (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))))
126	125	fveq2d 6826	. 2 ⊢ (𝜑 → (bits‘((𝐴 + 𝐵) mod (2↑𝑁))) = (bits‘(𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))))
127	126, 115	eqtr2d 2767	1 ⊢ (𝜑 → (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) = (bits‘((𝐴 + 𝐵) mod (2↑𝑁))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 = wceq 1541 caddwcad 1607 ∈ wcel 2111 ≠ wne 2928 ∩ cin 3896 ⊆ wss 3897 ∅c0 4280 ifcif 4472 𝒫 cpw 4547 class class class wbr 5089 ↦ cmpt 5170 ^◡ccnv 5613 ↾ cres 5616 ⟶wf 6477 –1-1-onto→wf1o 6480 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 1_oc1o 8378 2_oc2o 8379 Fincfn 8869 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 < clt 11146 ≤ cle 11147 − cmin 11344 / cdiv 11774 ℕcn 12125 2c2 12180 ℕ₀cn0 12381 ℤcz 12468 ℝ⁺crp 12890 ..^cfzo 13554 mod cmo 13773 seqcseq 13908 ↑cexp 13968 ∥ cdvds 16163 bitscbits 16330 sadd csad 16331

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1513 df-tru 1544 df-fal 1554 df-had 1595 df-cad 1608 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-disj 5057 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-er 8622 df-map 8752 df-pm 8753 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-n0 12382 df-xnn0 12455 df-z 12469 df-uz 12733 df-rp 12891 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-dvds 16164 df-bits 16333 df-sad 16362

This theorem is referenced by: sadadd 16378

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