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Theorem sadaddlem 16648

Description: Lemma for sadadd 16649. (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 12426	. . . . . . . . . 10 ⊢ 2 ∈ ℕ
2	1	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℕ)
3		sadaddlem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
4	2, 3	nnexpcld 14399	. . . . . . . 8 ⊢ (𝜑 → (2↑𝑁) ∈ ℕ)
5	4	nnzd 12729	. . . . . . 7 ⊢ (𝜑 → (2↑𝑁) ∈ ℤ)
6		sadaddlem.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℤ)
7		inss1 4135	. . . . . . . . . . . 12 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ (bits‘𝐴)
8		bitsss 16608	. . . . . . . . . . . 12 ⊢ (bits‘𝐴) ⊆ ℕ₀
9	7, 8	sstri 3921	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ ℕ₀
10		fzofi 14127	. . . . . . . . . . . 12 ⊢ (0..^𝑁) ∈ Fin
11		inss2 4136	. . . . . . . . . . . 12 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ (0..^𝑁)
12		ssfi 9199	. . . . . . . . . . . 12 ⊢ (((0..^𝑁) ∈ Fin ∧ ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → ((bits‘𝐴) ∩ (0..^𝑁)) ∈ Fin)
13	10, 11, 12	mp2an 705	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ∈ Fin
14		elfpw 9354	. . . . . . . . . . 11 ⊢ (((bits‘𝐴) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ (((bits‘𝐴) ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ ((bits‘𝐴) ∩ (0..^𝑁)) ∈ Fin))
15	9, 13, 14	mpbir2an 724	. . . . . . . . . 10 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin)
16		bitsf1o 16627	. . . . . . . . . . . . 13 ⊢ (bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin)
17		f1ocnv 6845	. . . . . . . . . . . . 13 ⊢ ((bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin) → ^◡(bits ↾ ℕ₀):(𝒫 ℕ₀ ∩ Fin)–1-1-onto→ℕ₀)
18		f1of 6832	. . . . . . . . . . . . 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 6708	. . . . . . . . . . . 12 ⊢ (𝐾:(𝒫 ℕ₀ ∩ Fin)⟶ℕ₀ ↔ ^◡(bits ↾ ℕ₀):(𝒫 ℕ₀ ∩ Fin)⟶ℕ₀)
22	19, 21	mpbir 234	. . . . . . . . . . 11 ⊢ 𝐾:(𝒫 ℕ₀ ∩ Fin)⟶ℕ₀
23	22	ffvelcdmi 7091	. . . . . . . . . 10 ⊢ (((bits‘𝐴) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) → (𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) ∈ ℕ₀)
24	15, 23	mp1i 14	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) ∈ ℕ₀)
25	24	nn0zd 12728	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) ∈ ℤ)
26	6, 25	zsubcld 12818	. . . . . . 7 ⊢ (𝜑 → (𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) ∈ ℤ)
27		sadaddlem.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℤ)
28		inss1 4135	. . . . . . . . . . . 12 ⊢ ((bits‘𝐵) ∩ (0..^𝑁)) ⊆ (bits‘𝐵)
29		bitsss 16608	. . . . . . . . . . . 12 ⊢ (bits‘𝐵) ⊆ ℕ₀
30	28, 29	sstri 3921	. . . . . . . . . . 11 ⊢ ((bits‘𝐵) ∩ (0..^𝑁)) ⊆ ℕ₀
31		inss2 4136	. . . . . . . . . . . 12 ⊢ ((bits‘𝐵) ∩ (0..^𝑁)) ⊆ (0..^𝑁)
32		ssfi 9199	. . . . . . . . . . . 12 ⊢ (((0..^𝑁) ∈ Fin ∧ ((bits‘𝐵) ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → ((bits‘𝐵) ∩ (0..^𝑁)) ∈ Fin)
33	10, 31, 32	mp2an 705	. . . . . . . . . . 11 ⊢ ((bits‘𝐵) ∩ (0..^𝑁)) ∈ Fin
34		elfpw 9354	. . . . . . . . . . 11 ⊢ (((bits‘𝐵) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ (((bits‘𝐵) ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ ((bits‘𝐵) ∩ (0..^𝑁)) ∈ Fin))
35	30, 33, 34	mpbir2an 724	. . . . . . . . . 10 ⊢ ((bits‘𝐵) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin)
36	22	ffvelcdmi 7091	. . . . . . . . . 10 ⊢ (((bits‘𝐵) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) → (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))) ∈ ℕ₀)
37	35, 36	mp1i 14	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))) ∈ ℕ₀)
38	37	nn0zd 12728	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))) ∈ ℤ)
39	27, 38	zsubcld 12818	. . . . . . 7 ⊢ (𝜑 → (𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) ∈ ℤ)
40	20	fveq1i 6894	. . . . . . . . . . . 12 ⊢ (𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) = (^◡(bits ↾ ℕ₀)‘((bits‘𝐴) ∩ (0..^𝑁)))
41	6, 4	zmodcld 14042	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 mod (2↑𝑁)) ∈ ℕ₀)
42	41	fvresd 6913	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐴 mod (2↑𝑁))) = (bits‘(𝐴 mod (2↑𝑁))))
43		bitsmod 16618	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(𝐴 mod (2↑𝑁))) = ((bits‘𝐴) ∩ (0..^𝑁)))
44	6, 3, 43	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (bits‘(𝐴 mod (2↑𝑁))) = ((bits‘𝐴) ∩ (0..^𝑁)))
45	42, 44	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐴 mod (2↑𝑁))) = ((bits‘𝐴) ∩ (0..^𝑁)))
46		f1ocnvfv 7293	. . . . . . . . . . . . . 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 599	. . . . . . . . . . . . 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 16	. . . . . . . . . . . 12 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘((bits‘𝐴) ∩ (0..^𝑁))) = (𝐴 mod (2↑𝑁)))
49	40, 48	eqtrid 2810	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) = (𝐴 mod (2↑𝑁)))
50	49	oveq2d 7443	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) = (𝐴 − (𝐴 mod (2↑𝑁))))
51	50	oveq1d 7442	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) / (2↑𝑁)) = ((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)))
52	6	zred 12813	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
53	4	nnrpd 13172	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝑁) ∈ ℝ⁺)
54		moddifz 14033	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (2↑𝑁) ∈ ℝ⁺) → ((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) ∈ ℤ)
55	52, 53, 54	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) ∈ ℤ)
56	51, 55	eqeltrd 2863	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) / (2↑𝑁)) ∈ ℤ)
57	4	nnne0d 12398	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝑁) ≠ 0)
58		dvdsval2 16437	. . . . . . . . 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 1398	. . . . . . . 8 ⊢ (𝜑 → ((2↑𝑁) ∥ (𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) ↔ ((𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) / (2↑𝑁)) ∈ ℤ))
60	56, 59	mpbird 260	. . . . . . 7 ⊢ (𝜑 → (2↑𝑁) ∥ (𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))))
61	20	fveq1i 6894	. . . . . . . . . . . 12 ⊢ (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))) = (^◡(bits ↾ ℕ₀)‘((bits‘𝐵) ∩ (0..^𝑁)))
62	27, 4	zmodcld 14042	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 mod (2↑𝑁)) ∈ ℕ₀)
63	62	fvresd 6913	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐵 mod (2↑𝑁))) = (bits‘(𝐵 mod (2↑𝑁))))
64		bitsmod 16618	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(𝐵 mod (2↑𝑁))) = ((bits‘𝐵) ∩ (0..^𝑁)))
65	27, 3, 64	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (bits‘(𝐵 mod (2↑𝑁))) = ((bits‘𝐵) ∩ (0..^𝑁)))
66	63, 65	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐵 mod (2↑𝑁))) = ((bits‘𝐵) ∩ (0..^𝑁)))
67		f1ocnvfv 7293	. . . . . . . . . . . . . 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 599	. . . . . . . . . . . . 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 16	. . . . . . . . . . . 12 ⊢ (𝜑 → (^◡(bits ↾ ℕ₀)‘((bits‘𝐵) ∩ (0..^𝑁))) = (𝐵 mod (2↑𝑁)))
70	61, 69	eqtrid 2810	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))) = (𝐵 mod (2↑𝑁)))
71	70	oveq2d 7443	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) = (𝐵 − (𝐵 mod (2↑𝑁))))
72	71	oveq1d 7442	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) / (2↑𝑁)) = ((𝐵 − (𝐵 mod (2↑𝑁))) / (2↑𝑁)))
73	27	zred 12813	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
74		moddifz 14033	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ (2↑𝑁) ∈ ℝ⁺) → ((𝐵 − (𝐵 mod (2↑𝑁))) / (2↑𝑁)) ∈ ℤ)
75	73, 53, 74	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − (𝐵 mod (2↑𝑁))) / (2↑𝑁)) ∈ ℤ)
76	72, 75	eqeltrd 2863	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) / (2↑𝑁)) ∈ ℤ)
77		dvdsval2 16437	. . . . . . . . 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 1398	. . . . . . . 8 ⊢ (𝜑 → ((2↑𝑁) ∥ (𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) ↔ ((𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) / (2↑𝑁)) ∈ ℤ))
79	76, 78	mpbird 260	. . . . . . 7 ⊢ (𝜑 → (2↑𝑁) ∥ (𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))))
80	5, 26, 39, 60, 79	dvds2addd 16474	. . . . . 6 ⊢ (𝜑 → (2↑𝑁) ∥ ((𝐴 − (𝐾‘((bits‘𝐴) ∩ (0..^𝑁)))) + (𝐵 − (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))))))
81	6	zcnd 12814	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
82	27	zcnd 12814	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
83	24	nn0cnd 12679	. . . . . . 7 ⊢ (𝜑 → (𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) ∈ ℂ)
84	37	nn0cnd 12679	. . . . . . 7 ⊢ (𝜑 → (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))) ∈ ℂ)
85	81, 82, 83, 84	addsub4d 11726	. . . . . 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 5135	. . . . 5 ⊢ (𝜑 → (2↑𝑁) ∥ ((𝐴 + 𝐵) − ((𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) + (𝐾‘((bits‘𝐵) ∩ (0..^𝑁))))))
87	6, 27	zaddcld 12817	. . . . . 6 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ)
88	25, 38	zaddcld 12817	. . . . . 6 ⊢ (𝜑 → ((𝐾‘((bits‘𝐴) ∩ (0..^𝑁))) + (𝐾‘((bits‘𝐵) ∩ (0..^𝑁)))) ∈ ℤ)
89		moddvds 16445	. . . . . 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 1398	. . . . 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 260	. . . 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 16643	. . . 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 4135	. . . . . . . . 9 ⊢ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ⊆ ((bits‘𝐴) sadd (bits‘𝐵))
97		sadcl 16644	. . . . . . . . . 10 ⊢ (((bits‘𝐴) ⊆ ℕ₀ ∧ (bits‘𝐵) ⊆ ℕ₀) → ((bits‘𝐴) sadd (bits‘𝐵)) ⊆ ℕ₀)
98	8, 29, 97	mp2an 705	. . . . . . . . 9 ⊢ ((bits‘𝐴) sadd (bits‘𝐵)) ⊆ ℕ₀
99	96, 98	sstri 3921	. . . . . . . 8 ⊢ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ⊆ ℕ₀
100		inss2 4136	. . . . . . . . 9 ⊢ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ⊆ (0..^𝑁)
101		ssfi 9199	. . . . . . . . 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 705	. . . . . . . 8 ⊢ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ∈ Fin
103		elfpw 9354	. . . . . . . 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 724	. . . . . . 7 ⊢ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin)
105	22	ffvelcdmi 7091	. . . . . . 7 ⊢ ((((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) → (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∈ ℕ₀)
106	104, 105	mp1i 14	. . . . . 6 ⊢ (𝜑 → (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∈ ℕ₀)
107	106	nn0red 12678	. . . . 5 ⊢ (𝜑 → (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∈ ℝ)
108	106	nn0ge0d 12680	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))))
109	20	fveq1i 6894	. . . . . . . . . 10 ⊢ (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) = (^◡(bits ↾ ℕ₀)‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))
110	109	fveq2i 6896	. . . . . . . . 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 6913	. . . . . . . . 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 7292	. . . . . . . . . 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 599	. . . . . . . . 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 2822	. . . . . . . 8 ⊢ (𝜑 → (bits‘(𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))) = (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))
116	115, 100	eqsstrdi 3956	. . . . . . 7 ⊢ (𝜑 → (bits‘(𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))) ⊆ (0..^𝑁))
117	106	nn0zd 12728	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∈ ℤ)
118		bitsfzo 16617	. . . . . . . 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 596	. . . . . . 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 260	. . . . . 6 ⊢ (𝜑 → (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)))
121		elfzolt2 13813	. . . . . 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 18	. . . . 5 ⊢ (𝜑 → (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))) < (2↑𝑁))
123		modid 14046	. . . . 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 852	. . . 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 2804	. . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) mod (2↑𝑁)) = (𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁))))
126	125	fveq2d 6897	. 2 ⊢ (𝜑 → (bits‘((𝐴 + 𝐵) mod (2↑𝑁))) = (bits‘(𝐾‘(((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)))))
127	126, 115	eqtr2d 2799	1 ⊢ (𝜑 → (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^𝑁)) = (bits‘((𝐴 + 𝐵) mod (2↑𝑁))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 = wceq 1569 caddwcad 1638 ∈ wcel 2146 ≠ wne 2958 ⊆ wss 3899 ∩ cin 4091 ∅c0 4280 ifcif 4484 𝒫 cpw 4559 class class class wbr 5105 ↦ cmpt 5188 ^◡ccnv 5653 ↾ cres 5656 ⟶wf 6542 –1-1-onto→wf1o 6545 ‘cfv 6546 (class class class)co 7427 ∈ cmpo 7429 1_oc1o 8480 2_oc2o 8481 Fincfn 8984 ℝcr 11209 0cc0 11210 1c1 11211 + caddc 11213 < clt 11353 ≤ cle 11354 − cmin 11551 / cdiv 11983 ℕcn 12345 2c2 12407 ℕ₀cn0 12616 ℤcz 12703 ℝ⁺crp 13130 ..^cfzo 13798 mod cmo 14019 seqcseq 14154 ↑cexp 14214 ∥ cdvds 16434 bitscbits 16601 sadd csad 16602

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-inf2 9652 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-pre-sup 11288

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-xor 1541 df-tru 1572 df-fal 1582 df-had 1623 df-cad 1639 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-int 4910 df-iun 4955 df-disj 5073 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-se 5608 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-om 7885 df-1st 8008 df-2nd 8009 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-1o 8487 df-2o 8488 df-oadd 8491 df-er 8728 df-map 8860 df-pm 8861 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9444 df-inf 9445 df-oi 9514 df-dju 9992 df-card 10030 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-div 11984 df-nn 12346 df-2 12415 df-3 12416 df-n0 12617 df-xnn0 12690 df-z 12704 df-uz 12976 df-rp 13131 df-fz 13650 df-fzo 13799 df-fl 13942 df-mod 14020 df-seq 14155 df-exp 14215 df-hash 14485 df-cj 15278 df-re 15279 df-im 15280 df-sqrt 15414 df-abs 15415 df-clim 15667 df-sum 15866 df-dvds 16435 df-bits 16604 df-sad 16633

This theorem is used by: sadadd 16649

W3C validator