Theorem sadcadd 16483

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 6862	. . . . . 6 ⊢ (𝑥 = 0 → (𝐶‘𝑥) = (𝐶‘0))
3	2	eleq2d 2847	. . . . 5 ⊢ (𝑥 = 0 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘0)))
4		oveq2 7399	. . . . . . 7 ⊢ (𝑥 = 0 → (2↑𝑥) = (2↑0))
5		2cn 12287	. . . . . . . 8 ⊢ 2 ∈ ℂ
6		exp0 14072	. . . . . . . 8 ⊢ (2 ∈ ℂ → (2↑0) = 1)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ (2↑0) = 1
8	4, 7	eqtrdi 2812	. . . . . 6 ⊢ (𝑥 = 0 → (2↑𝑥) = 1)
9		oveq2 7399	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (0..^𝑥) = (0..^0))
10		fzo0 13683	. . . . . . . . . . . . 13 ⊢ (0..^0) = ∅
11	9, 10	eqtrdi 2812	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (0..^𝑥) = ∅)
12	11	ineq2d 4170	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ ∅))
13		in0 4346	. . . . . . . . . . 11 ⊢ (𝐴 ∩ ∅) = ∅
14	12, 13	eqtrdi 2812	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐴 ∩ (0..^𝑥)) = ∅)
15	14	fveq2d 6866	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘∅))
16		sadcadd.k	. . . . . . . . . . 11 ⊢ 𝐾 = ◡(bits ↾ ℕ0)
17		0nn0 12490	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
18		fvres 6881	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℕ0 → ((bits ↾ ℕ0)‘0) = (bits‘0))
19	17, 18	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((bits ↾ ℕ0)‘0) = (bits‘0)
20		0bits 16464	. . . . . . . . . . . 12 ⊢ (bits‘0) = ∅
21	19, 20	eqtr2i 2785	. . . . . . . . . . 11 ⊢ ∅ = ((bits ↾ ℕ0)‘0)
22	16, 21	fveq12i 6868	. . . . . . . . . 10 ⊢ (𝐾‘∅) = (◡(bits ↾ ℕ0)‘((bits ↾ ℕ0)‘0))
23		bitsf1o 16470	. . . . . . . . . . 11 ⊢ (bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin)
24		f1ocnvfv1 7255	. . . . . . . . . . 11 ⊢ (((bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin) ∧ 0 ∈ ℕ0) → (◡(bits ↾ ℕ0)‘((bits ↾ ℕ0)‘0)) = 0)
25	23, 17, 24	mp2an 702	. . . . . . . . . 10 ⊢ (◡(bits ↾ ℕ0)‘((bits ↾ ℕ0)‘0)) = 0
26	22, 25	eqtri 2784	. . . . . . . . 9 ⊢ (𝐾‘∅) = 0
27	15, 26	eqtrdi 2812	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = 0)
28	11	ineq2d 4170	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ ∅))
29		in0 4346	. . . . . . . . . . 11 ⊢ (𝐵 ∩ ∅) = ∅
30	28, 29	eqtrdi 2812	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐵 ∩ (0..^𝑥)) = ∅)
31	30	fveq2d 6866	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘∅))
32	31, 26	eqtrdi 2812	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = 0)
33	27, 32	oveq12d 7409	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = (0 + 0))
34		00id 11352	. . . . . . 7 ⊢ (0 + 0) = 0
35	33, 34	eqtrdi 2812	. . . . . 6 ⊢ (𝑥 = 0 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = 0)
36	8, 35	breq12d 5110	. . . . 5 ⊢ (𝑥 = 0 → ((2↑𝑥) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) ↔ 1 ≤ 0))
37	3, 36	bibi12d 347	. . . 4 ⊢ (𝑥 = 0 → ((∅ ∈ (𝐶‘𝑥) ↔ (2↑𝑥) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥))))) ↔ (∅ ∈ (𝐶‘0) ↔ 1 ≤ 0)))
38	37	imbi2d 342	. . 3 ⊢ (𝑥 = 0 → ((𝜑 → (∅ ∈ (𝐶‘𝑥) ↔ (2↑𝑥) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))))) ↔ (𝜑 → (∅ ∈ (𝐶‘0) ↔ 1 ≤ 0))))
39		fveq2 6862	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐶‘𝑥) = (𝐶‘𝑘))
40	39	eleq2d 2847	. . . . 5 ⊢ (𝑥 = 𝑘 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘𝑘)))
41		oveq2 7399	. . . . . 6 ⊢ (𝑥 = 𝑘 → (2↑𝑥) = (2↑𝑘))
42		oveq2 7399	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (0..^𝑥) = (0..^𝑘))
43	42	ineq2d 4170	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^𝑘)))
44	43	fveq2d 6866	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^𝑘))))
45	42	ineq2d 4170	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^𝑘)))
46	45	fveq2d 6866	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^𝑘))))
47	44, 46	oveq12d 7409	. . . . . 6 ⊢ (𝑥 = 𝑘 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))
48	41, 47	breq12d 5110	. . . . 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 347	. . . 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 342	. . 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 6862	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐶‘𝑥) = (𝐶‘(𝑘 + 1)))
52	51	eleq2d 2847	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘(𝑘 + 1))))
53		oveq2 7399	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (2↑𝑥) = (2↑(𝑘 + 1)))
54		oveq2 7399	. . . . . . . . 9 ⊢ (𝑥 = (𝑘 + 1) → (0..^𝑥) = (0..^(𝑘 + 1)))
55	54	ineq2d 4170	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^(𝑘 + 1))))
56	55	fveq2d 6866	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))))
57	54	ineq2d 4170	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^(𝑘 + 1))))
58	57	fveq2d 6866	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1)))))
59	56, 58	oveq12d 7409	. . . . . 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 5110	. . . . 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 347	. . . 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 342	. . 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 6862	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐶‘𝑥) = (𝐶‘𝑁))
64	63	eleq2d 2847	. . . . 5 ⊢ (𝑥 = 𝑁 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘𝑁)))
65		oveq2 7399	. . . . . 6 ⊢ (𝑥 = 𝑁 → (2↑𝑥) = (2↑𝑁))
66		oveq2 7399	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (0..^𝑥) = (0..^𝑁))
67	66	ineq2d 4170	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^𝑁)))
68	67	fveq2d 6866	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^𝑁))))
69	66	ineq2d 4170	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^𝑁)))
70	69	fveq2d 6866	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^𝑁))))
71	68, 70	oveq12d 7409	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
72	65, 71	breq12d 5110	. . . . 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 347	. . . 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 342	. . 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 16479	. . . 4 ⊢ (𝜑 → ¬ ∅ ∈ (𝐶‘0))
79		0lt1 11703	. . . . . 6 ⊢ 0 < 1
80		0re 11177	. . . . . . 7 ⊢ 0 ∈ ℝ
81		1re 11175	. . . . . . 7 ⊢ 1 ∈ ℝ
82	80, 81	ltnlei 11298	. . . . . 6 ⊢ (0 < 1 ↔ ¬ 1 ≤ 0)
83	79, 82	mpbi 232	. . . . 5 ⊢ ¬ 1 ≤ 0
84	83	a1i 11	. . . 4 ⊢ (𝜑 → ¬ 1 ≤ 0)
85	78, 84	2falsed 378	. . 3 ⊢ (𝜑 → (∅ ∈ (𝐶‘0) ↔ 1 ≤ 0))
86	75	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (∅ ∈ (𝐶‘𝑘) ↔ (2↑𝑘) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))) → 𝐴 ⊆ ℕ0)
87	76	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (∅ ∈ (𝐶‘𝑘) ↔ (2↑𝑘) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))) → 𝐵 ⊆ ℕ0)
88		simplr 778	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (∅ ∈ (𝐶‘𝑘) ↔ (2↑𝑘) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))) → 𝑘 ∈ ℕ0)
89		simpr 488	. . . . . . 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 16482	. . . . . 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 416	. . . . 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 417	. . . 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 12662	. 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 208   ∧ wa 399   = wceq 1559  caddwcad 1625   ∈ wcel 2141   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283  ifcif 4477  𝒫 cpw 4552   class class class wbr 5097   ↦ cmpt 5178  ^◡ccnv 5642   ↾ cres 5645  –1-1-onto→wf1o 6515  ‘cfv 6516  (class class class)co 7391   ∈ cmpo 7393  1_oc1o 8424  2_oc2o 8425  Fincfn 8921  ℂcc 11065  0cc0 11067  1c1 11068   + caddc 11070   < clt 11210   ≤ cle 11211   − cmin 11408  2c2 12266  ℕ₀cn0 12475  ..^cfzo 13653  seqcseq 14008  ↑cexp 14068  bitscbits 16444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-inf2 9590  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144  ax-pre-sup 11145

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-xor 1531  df-tru 1562  df-fal 1572  df-cad 1626  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-disj 5065  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-isom 6525  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-2o 8432  df-oadd 8435  df-er 8672  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9382  df-inf 9383  df-oi 9452  df-dju 9853  df-card 9891  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-div 11839  df-nn 12205  df-2 12274  df-3 12275  df-n0 12476  df-xnn0 12549  df-z 12563  df-uz 12834  df-rp 12988  df-fz 13507  df-fzo 13654  df-fl 13796  df-mod 13874  df-seq 14009  df-exp 14069  df-hash 14338  df-cj 15117  df-re 15118  df-im 15119  df-sqrt 15253  df-abs 15254  df-clim 15506  df-sum 15705  df-dvds 16278  df-bits 16447

This theorem is referenced by: (None)