Theorem sadcadd 15180

Description: Non-recursive definition of the carry sequence. (Contributed by Mario Carneiro, 8-Sep-2016.)

Hypotheses

Ref	Expression
sadval.a	⊢ (𝜑 → 𝐴 ⊆ ℕ0)
sadval.b	⊢ (𝜑 → 𝐵 ⊆ ℕ0)
sadval.c	⊢ 𝐶 = seq0((𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
sadcp1.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
sadcadd.k	⊢ 𝐾 = ◡(bits ↾ ℕ0)

Assertion

Ref	Expression
sadcadd	⊢ (𝜑 → (∅ ∈ (𝐶‘𝑁) ↔ (2↑𝑁) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^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 sadcadd

Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sadcp1.n	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
2		fveq2 6191	. . . . . 6 ⊢ (𝑥 = 0 → (𝐶‘𝑥) = (𝐶‘0))
3	2	eleq2d 2687	. . . . 5 ⊢ (𝑥 = 0 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘0)))
4		oveq2 6658	. . . . . . 7 ⊢ (𝑥 = 0 → (2↑𝑥) = (2↑0))
5		2cn 11091	. . . . . . . 8 ⊢ 2 ∈ ℂ
6		exp0 12864	. . . . . . . 8 ⊢ (2 ∈ ℂ → (2↑0) = 1)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ (2↑0) = 1
8	4, 7	syl6eq 2672	. . . . . 6 ⊢ (𝑥 = 0 → (2↑𝑥) = 1)
9		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (0..^𝑥) = (0..^0))
10		fzo0 12492	. . . . . . . . . . . . 13 ⊢ (0..^0) = ∅
11	9, 10	syl6eq 2672	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (0..^𝑥) = ∅)
12	11	ineq2d 3814	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ ∅))
13		in0 3968	. . . . . . . . . . 11 ⊢ (𝐴 ∩ ∅) = ∅
14	12, 13	syl6eq 2672	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐴 ∩ (0..^𝑥)) = ∅)
15	14	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘∅))
16		sadcadd.k	. . . . . . . . . . 11 ⊢ 𝐾 = ◡(bits ↾ ℕ0)
17		0nn0 11307	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
18		fvres 6207	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℕ0 → ((bits ↾ ℕ0)‘0) = (bits‘0))
19	17, 18	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((bits ↾ ℕ0)‘0) = (bits‘0)
20		0bits 15161	. . . . . . . . . . . 12 ⊢ (bits‘0) = ∅
21	19, 20	eqtr2i 2645	. . . . . . . . . . 11 ⊢ ∅ = ((bits ↾ ℕ0)‘0)
22	16, 21	fveq12i 6196	. . . . . . . . . 10 ⊢ (𝐾‘∅) = (◡(bits ↾ ℕ0)‘((bits ↾ ℕ0)‘0))
23		bitsf1o 15167	. . . . . . . . . . 11 ⊢ (bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin)
24		f1ocnvfv1 6532	. . . . . . . . . . 11 ⊢ (((bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin) ∧ 0 ∈ ℕ0) → (◡(bits ↾ ℕ0)‘((bits ↾ ℕ0)‘0)) = 0)
25	23, 17, 24	mp2an 708	. . . . . . . . . 10 ⊢ (◡(bits ↾ ℕ0)‘((bits ↾ ℕ0)‘0)) = 0
26	22, 25	eqtri 2644	. . . . . . . . 9 ⊢ (𝐾‘∅) = 0
27	15, 26	syl6eq 2672	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = 0)
28	11	ineq2d 3814	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ ∅))
29		in0 3968	. . . . . . . . . . 11 ⊢ (𝐵 ∩ ∅) = ∅
30	28, 29	syl6eq 2672	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐵 ∩ (0..^𝑥)) = ∅)
31	30	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘∅))
32	31, 26	syl6eq 2672	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = 0)
33	27, 32	oveq12d 6668	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = (0 + 0))
34		00id 10211	. . . . . . 7 ⊢ (0 + 0) = 0
35	33, 34	syl6eq 2672	. . . . . 6 ⊢ (𝑥 = 0 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = 0)
36	8, 35	breq12d 4666	. . . . 5 ⊢ (𝑥 = 0 → ((2↑𝑥) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) ↔ 1 ≤ 0))
37	3, 36	bibi12d 335	. . . 4 ⊢ (𝑥 = 0 → ((∅ ∈ (𝐶‘𝑥) ↔ (2↑𝑥) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥))))) ↔ (∅ ∈ (𝐶‘0) ↔ 1 ≤ 0)))
38	37	imbi2d 330	. . 3 ⊢ (𝑥 = 0 → ((𝜑 → (∅ ∈ (𝐶‘𝑥) ↔ (2↑𝑥) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))))) ↔ (𝜑 → (∅ ∈ (𝐶‘0) ↔ 1 ≤ 0))))
39		fveq2 6191	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐶‘𝑥) = (𝐶‘𝑘))
40	39	eleq2d 2687	. . . . 5 ⊢ (𝑥 = 𝑘 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘𝑘)))
41		oveq2 6658	. . . . . 6 ⊢ (𝑥 = 𝑘 → (2↑𝑥) = (2↑𝑘))
42		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (0..^𝑥) = (0..^𝑘))
43	42	ineq2d 3814	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^𝑘)))
44	43	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^𝑘))))
45	42	ineq2d 3814	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^𝑘)))
46	45	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^𝑘))))
47	44, 46	oveq12d 6668	. . . . . 6 ⊢ (𝑥 = 𝑘 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))
48	41, 47	breq12d 4666	. . . . 5 ⊢ (𝑥 = 𝑘 → ((2↑𝑥) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) ↔ (2↑𝑘) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))))
49	40, 48	bibi12d 335	. . . 4 ⊢ (𝑥 = 𝑘 → ((∅ ∈ (𝐶‘𝑥) ↔ (2↑𝑥) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥))))) ↔ (∅ ∈ (𝐶‘𝑘) ↔ (2↑𝑘) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))))
50	49	imbi2d 330	. . 3 ⊢ (𝑥 = 𝑘 → ((𝜑 → (∅ ∈ (𝐶‘𝑥) ↔ (2↑𝑥) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))))) ↔ (𝜑 → (∅ ∈ (𝐶‘𝑘) ↔ (2↑𝑘) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))))))
51		fveq2 6191	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐶‘𝑥) = (𝐶‘(𝑘 + 1)))
52	51	eleq2d 2687	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘(𝑘 + 1))))
53		oveq2 6658	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (2↑𝑥) = (2↑(𝑘 + 1)))
54		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = (𝑘 + 1) → (0..^𝑥) = (0..^(𝑘 + 1)))
55	54	ineq2d 3814	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^(𝑘 + 1))))
56	55	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))))
57	54	ineq2d 3814	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^(𝑘 + 1))))
58	57	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1)))))
59	56, 58	oveq12d 6668	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1))))))
60	53, 59	breq12d 4666	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → ((2↑𝑥) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) ↔ (2↑(𝑘 + 1)) ≤ ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1)))))))
61	52, 60	bibi12d 335	. . . 4 ⊢ (𝑥 = (𝑘 + 1) → ((∅ ∈ (𝐶‘𝑥) ↔ (2↑𝑥) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥))))) ↔ (∅ ∈ (𝐶‘(𝑘 + 1)) ↔ (2↑(𝑘 + 1)) ≤ ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1))))))))
62	61	imbi2d 330	. . 3 ⊢ (𝑥 = (𝑘 + 1) → ((𝜑 → (∅ ∈ (𝐶‘𝑥) ↔ (2↑𝑥) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))))) ↔ (𝜑 → (∅ ∈ (𝐶‘(𝑘 + 1)) ↔ (2↑(𝑘 + 1)) ≤ ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1)))))))))
63		fveq2 6191	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐶‘𝑥) = (𝐶‘𝑁))
64	63	eleq2d 2687	. . . . 5 ⊢ (𝑥 = 𝑁 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘𝑁)))
65		oveq2 6658	. . . . . 6 ⊢ (𝑥 = 𝑁 → (2↑𝑥) = (2↑𝑁))
66		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (0..^𝑥) = (0..^𝑁))
67	66	ineq2d 3814	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^𝑁)))
68	67	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^𝑁))))
69	66	ineq2d 3814	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^𝑁)))
70	69	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^𝑁))))
71	68, 70	oveq12d 6668	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
72	65, 71	breq12d 4666	. . . . 5 ⊢ (𝑥 = 𝑁 → ((2↑𝑥) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) ↔ (2↑𝑁) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁))))))
73	64, 72	bibi12d 335	. . . 4 ⊢ (𝑥 = 𝑁 → ((∅ ∈ (𝐶‘𝑥) ↔ (2↑𝑥) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥))))) ↔ (∅ ∈ (𝐶‘𝑁) ↔ (2↑𝑁) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))))
74	73	imbi2d 330	. . 3 ⊢ (𝑥 = 𝑁 → ((𝜑 → (∅ ∈ (𝐶‘𝑥) ↔ (2↑𝑥) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))))) ↔ (𝜑 → (∅ ∈ (𝐶‘𝑁) ↔ (2↑𝑁) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁))))))))
75		sadval.a	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℕ0)
76		sadval.b	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ℕ0)
77		sadval.c	. . . . 5 ⊢ 𝐶 = seq0((𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
78	75, 76, 77	sadc0 15176	. . . 4 ⊢ (𝜑 → ¬ ∅ ∈ (𝐶‘0))
79		0lt1 10550	. . . . . 6 ⊢ 0 < 1
80		0re 10040	. . . . . . 7 ⊢ 0 ∈ ℝ
81		1re 10039	. . . . . . 7 ⊢ 1 ∈ ℝ
82	80, 81	ltnlei 10158	. . . . . 6 ⊢ (0 < 1 ↔ ¬ 1 ≤ 0)
83	79, 82	mpbi 220	. . . . 5 ⊢ ¬ 1 ≤ 0
84	83	a1i 11	. . . 4 ⊢ (𝜑 → ¬ 1 ≤ 0)
85	78, 84	2falsed 366	. . 3 ⊢ (𝜑 → (∅ ∈ (𝐶‘0) ↔ 1 ≤ 0))
86	75	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (∅ ∈ (𝐶‘𝑘) ↔ (2↑𝑘) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))) → 𝐴 ⊆ ℕ0)
87	76	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (∅ ∈ (𝐶‘𝑘) ↔ (2↑𝑘) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))) → 𝐵 ⊆ ℕ0)
88		simplr 792	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (∅ ∈ (𝐶‘𝑘) ↔ (2↑𝑘) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))) → 𝑘 ∈ ℕ0)
89		simpr 477	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (∅ ∈ (𝐶‘𝑘) ↔ (2↑𝑘) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))) → (∅ ∈ (𝐶‘𝑘) ↔ (2↑𝑘) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))))
90	86, 87, 77, 88, 16, 89	sadcaddlem 15179	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (∅ ∈ (𝐶‘𝑘) ↔ (2↑𝑘) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))) → (∅ ∈ (𝐶‘(𝑘 + 1)) ↔ (2↑(𝑘 + 1)) ≤ ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1)))))))
91	90	ex 450	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((∅ ∈ (𝐶‘𝑘) ↔ (2↑𝑘) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))) → (∅ ∈ (𝐶‘(𝑘 + 1)) ↔ (2↑(𝑘 + 1)) ≤ ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1))))))))
92	91	expcom 451	. . . 4 ⊢ (𝑘 ∈ ℕ0 → (𝜑 → ((∅ ∈ (𝐶‘𝑘) ↔ (2↑𝑘) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))) → (∅ ∈ (𝐶‘(𝑘 + 1)) ↔ (2↑(𝑘 + 1)) ≤ ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1)))))))))
93	92	a2d 29	. . 3 ⊢ (𝑘 ∈ ℕ0 → ((𝜑 → (∅ ∈ (𝐶‘𝑘) ↔ (2↑𝑘) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))) → (𝜑 → (∅ ∈ (𝐶‘(𝑘 + 1)) ↔ (2↑(𝑘 + 1)) ≤ ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1)))))))))
94	38, 50, 62, 74, 85, 93	nn0ind 11472	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝜑 → (∅ ∈ (𝐶‘𝑁) ↔ (2↑𝑁) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))))
95	1, 94	mpcom 38	1 ⊢ (𝜑 → (∅ ∈ (𝐶‘𝑁) ↔ (2↑𝑁) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁))))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  caddwcad 1545   ∈ wcel 1990   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ifcif 4086  𝒫 cpw 4158   class class class wbr 4653   ↦ cmpt 4729  ^◡ccnv 5113   ↾ cres 5116  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1_𝑜c1o 7553  2_𝑜c2o 7554  Fincfn 7955  ℂcc 9934  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075   − cmin 10266  2c2 11070  ℕ₀cn0 11292  ..^cfzo 12465  seqcseq 12801  ↑cexp 12860  bitscbits 15141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-xor 1465  df-tru 1486  df-fal 1489  df-cad 1546  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-bits 15144

This theorem is referenced by: (None)