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Theorem sadaddlem 16405

Description: Lemma for sadadd 16406. (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 12230	. . . . . . . . . 10 ⊢ 2 ∈ ℕ
2	1	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℕ)
3		sadaddlem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
4	2, 3	nnexpcld 14180	. . . . . . . 8 ⊢ (𝜑 → (2↑𝑁) ∈ ℕ)
5	4	nnzd 12526	. . . . . . 7 ⊢ (𝜑 → (2↑𝑁) ∈ ℤ)
6		sadaddlem.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℤ)
7		inss1 4191	. . . . . . . . . . . 12 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ (bits‘𝐴)
8		bitsss 16365	. . . . . . . . . . . 12 ⊢ (bits‘𝐴) ⊆ ℕ₀
9	7, 8	sstri 3945	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ ℕ₀
10		fzofi 13909	. . . . . . . . . . . 12 ⊢ (0..^𝑁) ∈ Fin
11		inss2 4192	. . . . . . . . . . . 12 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ (0..^𝑁)
12		ssfi 9109	. . . . . . . . . . . 12 ⊢ (((0..^𝑁) ∈ Fin ∧ ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → ((bits‘𝐴) ∩ (0..^𝑁)) ∈ Fin)
13	10, 11, 12	mp2an 693	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ∈ Fin
14		elfpw 9266	. . . . . . . . . . 11 ⊢ (((bits‘𝐴) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ (((bits‘𝐴) ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ ((bits‘𝐴) ∩ (0..^𝑁)) ∈ Fin))
15	9, 13, 14	mpbir2an 712	. . . . . . . . . 10 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin)
16		bitsf1o 16384	. . . . . . . . . . . . 13 ⊢ (bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin)
17		f1ocnv 6794	. . . . . . . . . . . . 13 ⊢ ((bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin) → ^◡(bits ↾ ℕ₀):(𝒫 ℕ₀ ∩ Fin)–1-1-onto→ℕ₀)
18		f1of 6782	. . . . . . . . . . . . 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 6661	. . . . . . . . . . . 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 7037	. . . . . . . . . 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 12525	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) ∈ ℤ)
26	6, 25	zsubcld 12613	. . . . . . 7 ⊢ (𝜑 → (𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) ∈ ℤ)
27		sadaddlem.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℤ)
28		inss1 4191	. . . . . . . . . . . 12 ⊢ ((bits‘𝐵) ∩ (0..^𝑁)) ⊆ (bits‘𝐵)
29		bitsss 16365	. . . . . . . . . . . 12 ⊢ (bits‘𝐵) ⊆ ℕ₀
30	28, 29	sstri 3945	. . . . . . . . . . 11 ⊢ ((bits‘𝐵) ∩ (0..^𝑁)) ⊆ ℕ₀
31		inss2 4192	. . . . . . . . . . . 12 ⊢ ((bits‘𝐵) ∩ (0..^𝑁)) ⊆ (0..^𝑁)
32		ssfi 9109	. . . . . . . . . . . 12 ⊢ (((0..^𝑁) ∈ Fin ∧ ((bits‘𝐵) ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → ((bits‘𝐵) ∩ (0..^𝑁)) ∈ Fin)
33	10, 31, 32	mp2an 693	. . . . . . . . . . 11 ⊢ ((bits‘𝐵) ∩ (0..^𝑁)) ∈ Fin
34		elfpw 9266	. . . . . . . . . . 11 ⊢ (((bits‘𝐵) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ (((bits‘𝐵) ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ ((bits‘𝐵) ∩ (0..^𝑁)) ∈ Fin))
35	30, 33, 34	mpbir2an 712	. . . . . . . . . 10 ⊢ ((bits‘𝐵) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin)
36	22	ffvelcdmi 7037	. . . . . . . . . 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 12525	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))) ∈ ℤ)
39	27, 38	zsubcld 12613	. . . . . . 7 ⊢ (𝜑 → (𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) ∈ ℤ)
40	20	fveq1i 6843	. . . . . . . . . . . 12 ⊢ (𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) = (^◡(bits ↾ ℕ₀)‘((bits‘𝐴) ∩ (0..^𝑁)))
41	6, 4	zmodcld 13824	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 mod (2↑𝑁)) ∈ ℕ₀)
42	41	fvresd 6862	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐴 mod (2↑𝑁))) = (bits‘(𝐴 mod (2↑𝑁))))
43		bitsmod 16375	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(𝐴 mod (2↑𝑁))) = ((bits‘𝐴) ∩ (0..^𝑁)))
44	6, 3, 43	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (bits‘(𝐴 mod (2↑𝑁))) = ((bits‘𝐴) ∩ (0..^𝑁)))
45	42, 44	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐴 mod (2↑𝑁))) = ((bits‘𝐴) ∩ (0..^𝑁)))
46		f1ocnvfv 7234	. . . . . . . . . . . . . 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 588	. . . . . . . . . . . . 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 2784	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) = (𝐴 mod (2↑𝑁)))
50	49	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) = (𝐴 − (𝐴 mod (2↑𝑁))))
51	50	oveq1d 7383	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) / (2↑𝑁)) = ((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)))
52	6	zred 12608	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
53	4	nnrpd 12959	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝑁) ∈ ℝ⁺)
54		moddifz 13815	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (2↑𝑁) ∈ ℝ⁺) → ((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) ∈ ℤ)
55	52, 53, 54	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) ∈ ℤ)
56	51, 55	eqeltrd 2837	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) / (2↑𝑁)) ∈ ℤ)
57	4	nnne0d 12207	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝑁) ≠ 0)
58		dvdsval2 16194	. . . . . . . . 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 1374	. . . . . . . 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 6843	. . . . . . . . . . . 12 ⊢ (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))) = (^◡(bits ↾ ℕ₀)‘((bits‘𝐵) ∩ (0..^𝑁)))
62	27, 4	zmodcld 13824	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 mod (2↑𝑁)) ∈ ℕ₀)
63	62	fvresd 6862	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐵 mod (2↑𝑁))) = (bits‘(𝐵 mod (2↑𝑁))))
64		bitsmod 16375	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(𝐵 mod (2↑𝑁))) = ((bits‘𝐵) ∩ (0..^𝑁)))
65	27, 3, 64	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (bits‘(𝐵 mod (2↑𝑁))) = ((bits‘𝐵) ∩ (0..^𝑁)))
66	63, 65	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐵 mod (2↑𝑁))) = ((bits‘𝐵) ∩ (0..^𝑁)))
67		f1ocnvfv 7234	. . . . . . . . . . . . . 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 588	. . . . . . . . . . . . 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 2784	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))) = (𝐵 mod (2↑𝑁)))
71	70	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) = (𝐵 − (𝐵 mod (2↑𝑁))))
72	71	oveq1d 7383	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) / (2↑𝑁)) = ((𝐵 − (𝐵 mod (2↑𝑁))) / (2↑𝑁)))
73	27	zred 12608	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
74		moddifz 13815	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ (2↑𝑁) ∈ ℝ⁺) → ((𝐵 − (𝐵 mod (2↑𝑁))) / (2↑𝑁)) ∈ ℤ)
75	73, 53, 74	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − (𝐵 mod (2↑𝑁))) / (2↑𝑁)) ∈ ℤ)
76	72, 75	eqeltrd 2837	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) / (2↑𝑁)) ∈ ℤ)
77		dvdsval2 16194	. . . . . . . . 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 1374	. . . . . . . 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 16231	. . . . . 6 ⊢ (𝜑 → (2↑𝑁) ∥ ((𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) + (𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))))))
81	6	zcnd 12609	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
82	27	zcnd 12609	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
83	24	nn0cnd 12476	. . . . . . 7 ⊢ (𝜑 → (𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) ∈ ℂ)
84	37	nn0cnd 12476	. . . . . . 7 ⊢ (𝜑 → (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))) ∈ ℂ)
85	81, 82, 83, 84	addsub4d 11551	. . . . . 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 5128	. . . . 5 ⊢ (𝜑 → (2↑𝑁) ∥ ((𝐴 + 𝐵) − ((𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) + (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))))))
87	6, 27	zaddcld 12612	. . . . . 6 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ)
88	25, 38	zaddcld 12612	. . . . . 6 ⊢ (𝜑 → ((𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) + (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) ∈ ℤ)
89		moddvds 16202	. . . . . 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 1374	. . . . 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 16400	. . . 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 4191	. . . . . . . . 9 ⊢ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ⊆ ((bits‘𝐴) sadd (bits‘𝐵))
97		sadcl 16401	. . . . . . . . . 10 ⊢ (((bits‘𝐴) ⊆ ℕ₀ ∧ (bits‘𝐵) ⊆ ℕ₀) → ((bits‘𝐴) sadd (bits‘𝐵)) ⊆ ℕ₀)
98	8, 29, 97	mp2an 693	. . . . . . . . 9 ⊢ ((bits‘𝐴) sadd (bits‘𝐵)) ⊆ ℕ₀
99	96, 98	sstri 3945	. . . . . . . 8 ⊢ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ⊆ ℕ₀
100		inss2 4192	. . . . . . . . 9 ⊢ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ⊆ (0..^𝑁)
101		ssfi 9109	. . . . . . . . 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 693	. . . . . . . 8 ⊢ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ∈ Fin
103		elfpw 9266	. . . . . . . 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 712	. . . . . . 7 ⊢ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin)
105	22	ffvelcdmi 7037	. . . . . . 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 12475	. . . . 5 ⊢ (𝜑 → (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∈ ℝ)
108	106	nn0ge0d 12477	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))))
109	20	fveq1i 6843	. . . . . . . . . 10 ⊢ (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) = (^◡(bits ↾ ℕ₀)‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))
110	109	fveq2i 6845	. . . . . . . . 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 6862	. . . . . . . . 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 7233	. . . . . . . . . 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 588	. . . . . . . . 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 2796	. . . . . . . 8 ⊢ (𝜑 → (bits‘(𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))) = (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))
116	115, 100	eqsstrdi 3980	. . . . . . 7 ⊢ (𝜑 → (bits‘(𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))) ⊆ (0..^𝑁))
117	106	nn0zd 12525	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∈ ℤ)
118		bitsfzo 16374	. . . . . . . 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 585	. . . . . . 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 13596	. . . . . 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 13828	. . . . 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 839	. . . 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 2778	. . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) mod (2↑𝑁)) = (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))))
126	125	fveq2d 6846	. 2 ⊢ (𝜑 → (bits‘((𝐴 + 𝐵) mod (2↑𝑁))) = (bits‘(𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))))
127	126, 115	eqtr2d 2773	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 1542 caddwcad 1608 ∈ wcel 2114 ≠ wne 2933 ∩ cin 3902 ⊆ wss 3903 ∅c0 4287 ifcif 4481 𝒫 cpw 4556 class class class wbr 5100 ↦ cmpt 5181 ^◡ccnv 5631 ↾ cres 5634 ⟶wf 6496 –1-1-onto→wf1o 6499 ‘cfv 6500 (class class class)co 7368 ∈ cmpo 7370 1_oc1o 8400 2_oc2o 8401 Fincfn 8895 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 < clt 11178 ≤ cle 11179 − cmin 11376 / cdiv 11806 ℕcn 12157 2c2 12212 ℕ₀cn0 12413 ℤcz 12500 ℝ⁺crp 12917 ..^cfzo 13582 mod cmo 13801 seqcseq 13936 ↑cexp 13996 ∥ cdvds 16191 bitscbits 16358 sadd csad 16359

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-xor 1514 df-tru 1545 df-fal 1555 df-had 1596 df-cad 1609 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-disj 5068 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-map 8777 df-pm 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-oi 9427 df-dju 9825 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-xnn0 12487 df-z 12501 df-uz 12764 df-rp 12918 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-sum 15622 df-dvds 16192 df-bits 16361 df-sad 16390

This theorem is referenced by: sadadd 16406

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