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Theorem sadcaddlem 15179

Description: Lemma for sadcadd 15180. (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 ∈ 2_𝑜, 𝑚 ∈ ℕ₀ ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1_𝑜, ∅)), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
sadcp1.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
sadcadd.k	⊢ 𝐾 = ^◡(bits ↾ ℕ₀)
sadcaddlem.1	⊢ (𝜑 → (∅ ∈ (𝐶‘𝑁) ↔ (2↑𝑁) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁))))))

**Assertion**
Ref	Expression
sadcaddlem	⊢ (𝜑 → (∅ ∈ (𝐶‘(𝑁 + 1)) ↔ (2↑(𝑁 + 1)) ≤ ((𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑁 + 1)))))))

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 sadcaddlem**
Step	Hyp	Ref	Expression
1		cad1 1555	. . . . 5 ⊢ (∅ ∈ (𝐶‘𝑁) → (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) ↔ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)))
2	1	adantl 482	. . . 4 ⊢ ((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) → (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) ↔ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)))
3		2nn 11185	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
4	3	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℕ)
5		sadcp1.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6	4, 5	nnexpcld 13030	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝑁) ∈ ℕ)
7	6	nnred 11035	. . . . . . . 8 ⊢ (𝜑 → (2↑𝑁) ∈ ℝ)
8	7	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)) → (2↑𝑁) ∈ ℝ)
9		inss1 3833	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ 𝐴
10		sadval.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ℕ₀)
11	9, 10	syl5ss 3614	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀)
12		fzofi 12773	. . . . . . . . . . . . . 14 ⊢ (0..^𝑁) ∈ Fin
13	12	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0..^𝑁) ∈ Fin)
14		inss2 3834	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
15		ssfi 8180	. . . . . . . . . . . . 13 ⊢ (((0..^𝑁) ∈ Fin ∧ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (𝐴 ∩ (0..^𝑁)) ∈ Fin)
16	13, 14, 15	sylancl 694	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ∈ Fin)
17		elfpw 8268	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ (𝐴 ∩ (0..^𝑁)) ∈ Fin))
18	11, 16, 17	sylanbrc 698	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
19		bitsf1o 15167	. . . . . . . . . . . . . . 15 ⊢ (bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin)
20		f1ocnv 6149	. . . . . . . . . . . . . . 15 ⊢ ((bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin) → ^◡(bits ↾ ℕ₀):(𝒫 ℕ₀ ∩ Fin)–1-1-onto→ℕ₀)
21	19, 20	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ^◡(bits ↾ ℕ₀):(𝒫 ℕ₀ ∩ Fin)–1-1-onto→ℕ₀
22		sadcadd.k	. . . . . . . . . . . . . . 15 ⊢ 𝐾 = ^◡(bits ↾ ℕ₀)
23		f1oeq1 6127	. . . . . . . . . . . . . . 15 ⊢ (𝐾 = ^◡(bits ↾ ℕ₀) → (𝐾:(𝒫 ℕ₀ ∩ Fin)–1-1-onto→ℕ₀ ↔ ^◡(bits ↾ ℕ₀):(𝒫 ℕ₀ ∩ Fin)–1-1-onto→ℕ₀))
24	22, 23	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝐾:(𝒫 ℕ₀ ∩ Fin)–1-1-onto→ℕ₀ ↔ ^◡(bits ↾ ℕ₀):(𝒫 ℕ₀ ∩ Fin)–1-1-onto→ℕ₀)
25	21, 24	mpbir 221	. . . . . . . . . . . . 13 ⊢ 𝐾:(𝒫 ℕ₀ ∩ Fin)–1-1-onto→ℕ₀
26		f1of 6137	. . . . . . . . . . . . 13 ⊢ (𝐾:(𝒫 ℕ₀ ∩ Fin)–1-1-onto→ℕ₀ → 𝐾:(𝒫 ℕ₀ ∩ Fin)⟶ℕ₀)
27	25, 26	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝐾:(𝒫 ℕ₀ ∩ Fin)⟶ℕ₀
28	27	ffvelrni 6358	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℕ₀)
29	18, 28	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℕ₀)
30		inss1 3833	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ (0..^𝑁)) ⊆ 𝐵
31		sadval.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ⊆ ℕ₀)
32	30, 31	syl5ss 3614	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ₀)
33		inss2 3834	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
34		ssfi 8180	. . . . . . . . . . . . 13 ⊢ (((0..^𝑁) ∈ Fin ∧ (𝐵 ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ∈ Fin)
35	13, 33, 34	sylancl 694	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∩ (0..^𝑁)) ∈ Fin)
36		elfpw 8268	. . . . . . . . . . . 12 ⊢ ((𝐵 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((𝐵 ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ (𝐵 ∩ (0..^𝑁)) ∈ Fin))
37	32, 35, 36	sylanbrc 698	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
38	27	ffvelrni 6358	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℕ₀)
39	37, 38	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℕ₀)
40	29, 39	nn0addcld 11355	. . . . . . . . 9 ⊢ (𝜑 → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℕ₀)
41	40	nn0red 11352	. . . . . . . 8 ⊢ (𝜑 → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ)
42	41	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ)
43		2nn0 11309	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
44	43	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐴) → 2 ∈ ℕ₀)
45	5	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐴) → 𝑁 ∈ ℕ₀)
46	44, 45	nn0expcld 13031	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ∈ ℕ₀)
47		0nn0 11307	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
48	47	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ 𝐴) → 0 ∈ ℕ₀)
49	46, 48	ifclda 4120	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℕ₀)
50	43	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐵) → 2 ∈ ℕ₀)
51	5	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐵) → 𝑁 ∈ ℕ₀)
52	50, 51	nn0expcld 13031	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐵) → (2↑𝑁) ∈ ℕ₀)
53	47	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ 𝐵) → 0 ∈ ℕ₀)
54	52, 53	ifclda 4120	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℕ₀)
55	49, 54	nn0addcld 11355	. . . . . . . . 9 ⊢ (𝜑 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ∈ ℕ₀)
56	55	nn0red 11352	. . . . . . . 8 ⊢ (𝜑 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ∈ ℝ)
57	56	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ∈ ℝ)
58		sadcaddlem.1	. . . . . . . . 9 ⊢ (𝜑 → (∅ ∈ (𝐶‘𝑁) ↔ (2↑𝑁) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁))))))
59	58	biimpa 501	. . . . . . . 8 ⊢ ((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) → (2↑𝑁) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
60	59	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)) → (2↑𝑁) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
61	6	nnnn0d 11351	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2↑𝑁) ∈ ℕ₀)
62		ifcl 4130	. . . . . . . . . . . . 13 ⊢ (((2↑𝑁) ∈ ℕ₀ ∧ 0 ∈ ℕ₀) → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℕ₀)
63	61, 47, 62	sylancl 694	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℕ₀)
64	63	nn0ge0d 11354	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ if(𝑁 ∈ 𝐵, (2↑𝑁), 0))
65	7	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐵) → (2↑𝑁) ∈ ℝ)
66		0red 10041	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ 𝐵) → 0 ∈ ℝ)
67	65, 66	ifclda 4120	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℝ)
68	7, 67	addge01d 10615	. . . . . . . . . . 11 ⊢ (𝜑 → (0 ≤ if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ↔ (2↑𝑁) ≤ ((2↑𝑁) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))))
69	64, 68	mpbid 222	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝑁) ≤ ((2↑𝑁) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
70	69	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ≤ ((2↑𝑁) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
71		iftrue 4092	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝐴 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) = (2↑𝑁))
72	71	adantl 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐴) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) = (2↑𝑁))
73	72	oveq1d 6665	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐴) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = ((2↑𝑁) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
74	70, 73	breqtrrd 4681	. . . . . . . 8 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ≤ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
75		ifcl 4130	. . . . . . . . . . . . 13 ⊢ (((2↑𝑁) ∈ ℕ₀ ∧ 0 ∈ ℕ₀) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℕ₀)
76	61, 47, 75	sylancl 694	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℕ₀)
77	76	nn0ge0d 11354	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ if(𝑁 ∈ 𝐴, (2↑𝑁), 0))
78	7	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ∈ ℝ)
79		0red 10041	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ 𝐴) → 0 ∈ ℝ)
80	78, 79	ifclda 4120	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℝ)
81	7, 80	addge02d 10616	. . . . . . . . . . 11 ⊢ (𝜑 → (0 ≤ if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ↔ (2↑𝑁) ≤ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + (2↑𝑁))))
82	77, 81	mpbid 222	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝑁) ≤ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + (2↑𝑁)))
83	82	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐵) → (2↑𝑁) ≤ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + (2↑𝑁)))
84		iftrue 4092	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝐵 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) = (2↑𝑁))
85	84	adantl 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐵) → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) = (2↑𝑁))
86	85	oveq2d 6666	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐵) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + (2↑𝑁)))
87	83, 86	breqtrrd 4681	. . . . . . . 8 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐵) → (2↑𝑁) ≤ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
88	74, 87	jaodan 826	. . . . . . 7 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)) → (2↑𝑁) ≤ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
89	8, 8, 42, 57, 60, 88	le2addd 10646	. . . . . 6 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)) → ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))))
90	89	ex 450	. . . . 5 ⊢ ((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) → ((𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵) → ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))))
91		ioran 511	. . . . . 6 ⊢ (¬ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵) ↔ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵))
92		iffalse 4095	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑁 ∈ 𝐴 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) = 0)
93	92	ad2antrl 764	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) = 0)
94		iffalse 4095	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑁 ∈ 𝐵 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) = 0)
95	94	ad2antll 765	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) = 0)
96	93, 95	oveq12d 6668	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = (0 + 0))
97		00id 10211	. . . . . . . . . . . 12 ⊢ (0 + 0) = 0
98	96, 97	syl6eq 2672	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = 0)
99	98	oveq2d 6666	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))) = (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + 0))
100	29	nn0red 11352	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℝ)
101	100	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℝ)
102	39	nn0red 11352	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℝ)
103	102	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℝ)
104	101, 103	readdcld 10069	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ)
105	104	recnd 10068	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℂ)
106	105	addid1d 10236	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + 0) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
107	99, 106	eqtrd 2656	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
108	22	fveq1i 6192	. . . . . . . . . . . . . . . 16 ⊢ (𝐾‘(𝐴 ∩ (0..^𝑁))) = (^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁)))
109	108	fveq2i 6194	. . . . . . . . . . . . . . 15 ⊢ ((bits ↾ ℕ₀)‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) = ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁))))
110		fvres 6207	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℕ₀ → ((bits ↾ ℕ₀)‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) = (bits‘(𝐾‘(𝐴 ∩ (0..^𝑁)))))
111	29, 110	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) = (bits‘(𝐾‘(𝐴 ∩ (0..^𝑁)))))
112		f1ocnvfv2 6533	. . . . . . . . . . . . . . . 16 ⊢ (((bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin) ∧ (𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin)) → ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁)))) = (𝐴 ∩ (0..^𝑁)))
113	19, 18, 112	sylancr 695	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁)))) = (𝐴 ∩ (0..^𝑁)))
114	109, 111, 113	3eqtr3a 2680	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (bits‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) = (𝐴 ∩ (0..^𝑁)))
115	114, 14	syl6eqss 3655	. . . . . . . . . . . . 13 ⊢ (𝜑 → (bits‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) ⊆ (0..^𝑁))
116	29	nn0zd 11480	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℤ)
117		bitsfzo 15157	. . . . . . . . . . . . . 14 ⊢ (((𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) ↔ (bits‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) ⊆ (0..^𝑁)))
118	116, 5, 117	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) ↔ (bits‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) ⊆ (0..^𝑁)))
119	115, 118	mpbird 247	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)))
120		elfzolt2 12479	. . . . . . . . . . . 12 ⊢ ((𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) → (𝐾‘(𝐴 ∩ (0..^𝑁))) < (2↑𝑁))
121	119, 120	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^𝑁))) < (2↑𝑁))
122	22	fveq1i 6192	. . . . . . . . . . . . . . . 16 ⊢ (𝐾‘(𝐵 ∩ (0..^𝑁))) = (^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁)))
123	122	fveq2i 6194	. . . . . . . . . . . . . . 15 ⊢ ((bits ↾ ℕ₀)‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) = ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁))))
124		fvres 6207	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℕ₀ → ((bits ↾ ℕ₀)‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) = (bits‘(𝐾‘(𝐵 ∩ (0..^𝑁)))))
125	39, 124	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) = (bits‘(𝐾‘(𝐵 ∩ (0..^𝑁)))))
126		f1ocnvfv2 6533	. . . . . . . . . . . . . . . 16 ⊢ (((bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin) ∧ (𝐵 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin)) → ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁)))) = (𝐵 ∩ (0..^𝑁)))
127	19, 37, 126	sylancr 695	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁)))) = (𝐵 ∩ (0..^𝑁)))
128	123, 125, 127	3eqtr3a 2680	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (bits‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) = (𝐵 ∩ (0..^𝑁)))
129	128, 33	syl6eqss 3655	. . . . . . . . . . . . 13 ⊢ (𝜑 → (bits‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) ⊆ (0..^𝑁))
130	39	nn0zd 11480	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℤ)
131		bitsfzo 15157	. . . . . . . . . . . . . 14 ⊢ (((𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) ↔ (bits‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) ⊆ (0..^𝑁)))
132	130, 5, 131	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) ↔ (bits‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) ⊆ (0..^𝑁)))
133	129, 132	mpbird 247	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)))
134		elfzolt2 12479	. . . . . . . . . . . 12 ⊢ ((𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) → (𝐾‘(𝐵 ∩ (0..^𝑁))) < (2↑𝑁))
135	133, 134	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^𝑁))) < (2↑𝑁))
136	100, 102, 7, 7, 121, 135	lt2addd 10650	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) < ((2↑𝑁) + (2↑𝑁)))
137	136	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) < ((2↑𝑁) + (2↑𝑁)))
138	107, 137	eqbrtrd 4675	. . . . . . . 8 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))) < ((2↑𝑁) + (2↑𝑁)))
139	80	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℝ)
140	67	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℝ)
141	139, 140	readdcld 10069	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ∈ ℝ)
142	104, 141	readdcld 10069	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))) ∈ ℝ)
143	7	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (2↑𝑁) ∈ ℝ)
144	143, 143	readdcld 10069	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → ((2↑𝑁) + (2↑𝑁)) ∈ ℝ)
145	142, 144	ltnled 10184	. . . . . . . 8 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → ((((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))) < ((2↑𝑁) + (2↑𝑁)) ↔ ¬ ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))))
146	138, 145	mpbid 222	. . . . . . 7 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → ¬ ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))))
147	146	ex 450	. . . . . 6 ⊢ ((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) → ((¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵) → ¬ ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))))
148	91, 147	syl5bi 232	. . . . 5 ⊢ ((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) → (¬ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵) → ¬ ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))))
149	90, 148	impcon4bid 217	. . . 4 ⊢ ((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) → ((𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵) ↔ ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))))
150	2, 149	bitrd 268	. . 3 ⊢ ((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) → (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) ↔ ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))))
151		cad0 1556	. . . . 5 ⊢ (¬ ∅ ∈ (𝐶‘𝑁) → (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) ↔ (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)))
152	151	adantl 482	. . . 4 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) ↔ (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)))
153	40	nn0ge0d 11354	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
154	7, 7	readdcld 10069	. . . . . . . . . 10 ⊢ (𝜑 → ((2↑𝑁) + (2↑𝑁)) ∈ ℝ)
155	154, 41	addge02d 10616	. . . . . . . . 9 ⊢ (𝜑 → (0 ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ↔ ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + ((2↑𝑁) + (2↑𝑁)))))
156	153, 155	mpbid 222	. . . . . . . 8 ⊢ (𝜑 → ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + ((2↑𝑁) + (2↑𝑁))))
157	156	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)) → ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + ((2↑𝑁) + (2↑𝑁))))
158	71, 84	oveqan12d 6669	. . . . . . . . 9 ⊢ ((𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = ((2↑𝑁) + (2↑𝑁)))
159	158	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = ((2↑𝑁) + (2↑𝑁)))
160	159	oveq2d 6666	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)) → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))) = (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + ((2↑𝑁) + (2↑𝑁))))
161	157, 160	breqtrrd 4681	. . . . . 6 ⊢ (((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)) → ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))))
162	161	ex 450	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → ((𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵) → ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))))
163	100	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℝ)
164	102	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℝ)
165	163, 164	readdcld 10069	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ)
166	7	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (2↑𝑁) ∈ ℝ)
167	7, 41	lenltd 10183	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2↑𝑁) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ↔ ¬ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) < (2↑𝑁)))
168	58, 167	bitrd 268	. . . . . . . . . . 11 ⊢ (𝜑 → (∅ ∈ (𝐶‘𝑁) ↔ ¬ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) < (2↑𝑁)))
169	168	con2bid 344	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) < (2↑𝑁) ↔ ¬ ∅ ∈ (𝐶‘𝑁)))
170	169	biimpar 502	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) < (2↑𝑁))
171	165, 166, 166, 170	ltadd1dd 10638	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (2↑𝑁)) < ((2↑𝑁) + (2↑𝑁)))
172	165, 166	readdcld 10069	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (2↑𝑁)) ∈ ℝ)
173	154	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → ((2↑𝑁) + (2↑𝑁)) ∈ ℝ)
174	41, 56	readdcld 10069	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))) ∈ ℝ)
175	174	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))) ∈ ℝ)
176		ltletr 10129	. . . . . . . . 9 ⊢ (((((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (2↑𝑁)) ∈ ℝ ∧ ((2↑𝑁) + (2↑𝑁)) ∈ ℝ ∧ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))) ∈ ℝ) → (((((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (2↑𝑁)) < ((2↑𝑁) + (2↑𝑁)) ∧ ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))) → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (2↑𝑁)) < (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))))
177	172, 173, 175, 176	syl3anc 1326	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (((((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (2↑𝑁)) < ((2↑𝑁) + (2↑𝑁)) ∧ ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))) → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (2↑𝑁)) < (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))))
178	171, 177	mpand 711	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))) → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (2↑𝑁)) < (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))))
179	56	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ∈ ℝ)
180	41	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ)
181	166, 179, 180	ltadd2d 10193	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → ((2↑𝑁) < (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ↔ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (2↑𝑁)) < (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))))
182	178, 181	sylibrd 249	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))) → (2↑𝑁) < (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))))
183	7, 56	ltnled 10184	. . . . . . . 8 ⊢ (𝜑 → ((2↑𝑁) < (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ↔ ¬ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁)))
184	63	nn0cnd 11353	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℂ)
185	184	addid2d 10237	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = if(𝑁 ∈ 𝐵, (2↑𝑁), 0))
186	7	leidd 10594	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2↑𝑁) ≤ (2↑𝑁))
187	61	nn0ge0d 11354	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (2↑𝑁))
188		breq1 4656	. . . . . . . . . . . . . 14 ⊢ ((2↑𝑁) = if(𝑁 ∈ 𝐵, (2↑𝑁), 0) → ((2↑𝑁) ≤ (2↑𝑁) ↔ if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ≤ (2↑𝑁)))
189		breq1 4656	. . . . . . . . . . . . . 14 ⊢ (0 = if(𝑁 ∈ 𝐵, (2↑𝑁), 0) → (0 ≤ (2↑𝑁) ↔ if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ≤ (2↑𝑁)))
190	188, 189	ifboth 4124	. . . . . . . . . . . . 13 ⊢ (((2↑𝑁) ≤ (2↑𝑁) ∧ 0 ≤ (2↑𝑁)) → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ≤ (2↑𝑁))
191	186, 187, 190	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ≤ (2↑𝑁))
192	185, 191	eqbrtrd 4675	. . . . . . . . . . 11 ⊢ (𝜑 → (0 + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁))
193	92	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (¬ 𝑁 ∈ 𝐴 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = (0 + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
194	193	breq1d 4663	. . . . . . . . . . 11 ⊢ (¬ 𝑁 ∈ 𝐴 → ((if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁) ↔ (0 + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁)))
195	192, 194	syl5ibrcom 237	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 𝑁 ∈ 𝐴 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁)))
196	195	con1d 139	. . . . . . . . 9 ⊢ (𝜑 → (¬ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁) → 𝑁 ∈ 𝐴))
197	76	nn0cnd 11353	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℂ)
198	197	addid1d 10236	. . . . . . . . . . . 12 ⊢ (𝜑 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + 0) = if(𝑁 ∈ 𝐴, (2↑𝑁), 0))
199		breq1 4656	. . . . . . . . . . . . . 14 ⊢ ((2↑𝑁) = if(𝑁 ∈ 𝐴, (2↑𝑁), 0) → ((2↑𝑁) ≤ (2↑𝑁) ↔ if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ≤ (2↑𝑁)))
200		breq1 4656	. . . . . . . . . . . . . 14 ⊢ (0 = if(𝑁 ∈ 𝐴, (2↑𝑁), 0) → (0 ≤ (2↑𝑁) ↔ if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ≤ (2↑𝑁)))
201	199, 200	ifboth 4124	. . . . . . . . . . . . 13 ⊢ (((2↑𝑁) ≤ (2↑𝑁) ∧ 0 ≤ (2↑𝑁)) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ≤ (2↑𝑁))
202	186, 187, 201	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ≤ (2↑𝑁))
203	198, 202	eqbrtrd 4675	. . . . . . . . . . 11 ⊢ (𝜑 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + 0) ≤ (2↑𝑁))
204	94	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (¬ 𝑁 ∈ 𝐵 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + 0))
205	204	breq1d 4663	. . . . . . . . . . 11 ⊢ (¬ 𝑁 ∈ 𝐵 → ((if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁) ↔ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + 0) ≤ (2↑𝑁)))
206	203, 205	syl5ibrcom 237	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 𝑁 ∈ 𝐵 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁)))
207	206	con1d 139	. . . . . . . . 9 ⊢ (𝜑 → (¬ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁) → 𝑁 ∈ 𝐵))
208	196, 207	jcad 555	. . . . . . . 8 ⊢ (𝜑 → (¬ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁) → (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)))
209	183, 208	sylbid 230	. . . . . . 7 ⊢ (𝜑 → ((2↑𝑁) < (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) → (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)))
210	209	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → ((2↑𝑁) < (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) → (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)))
211	182, 210	syld 47	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))) → (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)))
212	162, 211	impbid 202	. . . 4 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → ((𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵) ↔ ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))))
213	152, 212	bitrd 268	. . 3 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) ↔ ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))))
214	150, 213	pm2.61dan 832	. 2 ⊢ (𝜑 → (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) ↔ ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))))
215		sadval.c	. . 3 ⊢ 𝐶 = seq0((𝑐 ∈ 2_𝑜, 𝑚 ∈ ℕ₀ ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1_𝑜, ∅)), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
216	10, 31, 215, 5	sadcp1 15177	. 2 ⊢ (𝜑 → (∅ ∈ (𝐶‘(𝑁 + 1)) ↔ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁))))
217		2cnd 11093	. . . . 5 ⊢ (𝜑 → 2 ∈ ℂ)
218	217, 5	expp1d 13009	. . . 4 ⊢ (𝜑 → (2↑(𝑁 + 1)) = ((2↑𝑁) · 2))
219	6	nncnd 11036	. . . . 5 ⊢ (𝜑 → (2↑𝑁) ∈ ℂ)
220	219	times2d 11276	. . . 4 ⊢ (𝜑 → ((2↑𝑁) · 2) = ((2↑𝑁) + (2↑𝑁)))
221	218, 220	eqtrd 2656	. . 3 ⊢ (𝜑 → (2↑(𝑁 + 1)) = ((2↑𝑁) + (2↑𝑁)))
222	22	bitsinvp1 15171	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐴, (2↑𝑁), 0)))
223	10, 5, 222	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐴, (2↑𝑁), 0)))
224	22	bitsinvp1 15171	. . . . . 6 ⊢ ((𝐵 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐾‘(𝐵 ∩ (0..^(𝑁 + 1)))) = ((𝐾‘(𝐵 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
225	31, 5, 224	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^(𝑁 + 1)))) = ((𝐾‘(𝐵 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
226	223, 225	oveq12d 6668	. . . 4 ⊢ (𝜑 → ((𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑁 + 1))))) = (((𝐾‘(𝐴 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐴, (2↑𝑁), 0)) + ((𝐾‘(𝐵 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))))
227	29	nn0cnd 11353	. . . . 5 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℂ)
228	39	nn0cnd 11353	. . . . 5 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℂ)
229	227, 197, 228, 184	add4d 10264	. . . 4 ⊢ (𝜑 → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐴, (2↑𝑁), 0)) + ((𝐾‘(𝐵 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))) = (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))))
230	226, 229	eqtrd 2656	. . 3 ⊢ (𝜑 → ((𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑁 + 1))))) = (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))))
231	221, 230	breq12d 4666	. 2 ⊢ (𝜑 → ((2↑(𝑁 + 1)) ≤ ((𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑁 + 1))))) ↔ ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))))
232	214, 216, 231	3bitr4d 300	1 ⊢ (𝜑 → (∅ ∈ (𝐶‘(𝑁 + 1)) ↔ (2↑(𝑁 + 1)) ≤ ((𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑁 + 1)))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ 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 ⟶wf 5884 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1_𝑜c1o 7553 2_𝑜c2o 7554 Fincfn 7955 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ..^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: sadcadd 15180

W3C validator