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Theorem sadcaddlem 16547

Description: Lemma for sadcadd 16548. (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 1650	. . . . 5 ⊢ (∅ ∈ (𝐶‘𝑁) → (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) ↔ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)))
2	1	adantl 487	. . . 4 ⊢ ((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) → (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) ↔ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)))
3		2nn 12338	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
4	3	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℕ)
5		sadcp1.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6	4, 5	nnexpcld 14309	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝑁) ∈ ℕ)
7	6	nnred 12272	. . . . . . . 8 ⊢ (𝜑 → (2↑𝑁) ∈ ℝ)
8	7	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)) → (2↑𝑁) ∈ ℝ)
9		inss1 4182	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ 𝐴
10		sadval.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ℕ₀)
11	9, 10	sstrid 3942	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀)
12		fzofi 14038	. . . . . . . . . . . . . 14 ⊢ (0..^𝑁) ∈ Fin
13	12	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0..^𝑁) ∈ Fin)
14		inss2 4183	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
15		ssfi 9167	. . . . . . . . . . . . 13 ⊢ (((0..^𝑁) ∈ Fin ∧ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (𝐴 ∩ (0..^𝑁)) ∈ Fin)
16	13, 14, 15	sylancl 598	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ∈ Fin)
17		elfpw 9321	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ (𝐴 ∩ (0..^𝑁)) ∈ Fin))
18	11, 16, 17	sylanbrc 595	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
19		bitsf1o 16535	. . . . . . . . . . . . . . 15 ⊢ (bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin)
20		f1ocnv 6830	. . . . . . . . . . . . . . 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 6805	. . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 13 ⊢ 𝐾:(𝒫 ℕ₀ ∩ Fin)–1-1-onto→ℕ₀
26		f1of 6817	. . . . . . . . . . . . 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	ffvelcdmi 7076	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℕ₀)
29	18, 28	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℕ₀)
30		inss1 4182	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ (0..^𝑁)) ⊆ 𝐵
31		sadval.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ⊆ ℕ₀)
32	30, 31	sstrid 3942	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ₀)
33		inss2 4183	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
34		ssfi 9167	. . . . . . . . . . . . 13 ⊢ (((0..^𝑁) ∈ Fin ∧ (𝐵 ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ∈ Fin)
35	13, 33, 34	sylancl 598	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∩ (0..^𝑁)) ∈ Fin)
36		elfpw 9321	. . . . . . . . . . . 12 ⊢ ((𝐵 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((𝐵 ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ (𝐵 ∩ (0..^𝑁)) ∈ Fin))
37	32, 35, 36	sylanbrc 595	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
38	27	ffvelcdmi 7076	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℕ₀)
39	37, 38	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℕ₀)
40	29, 39	nn0addcld 12593	. . . . . . . . 9 ⊢ (𝜑 → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℕ₀)
41	40	nn0red 12590	. . . . . . . 8 ⊢ (𝜑 → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ)
42	41	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ)
43		2nn0 12545	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
44	43	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐴) → 2 ∈ ℕ₀)
45	5	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐴) → 𝑁 ∈ ℕ₀)
46	44, 45	nn0expcld 14310	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ∈ ℕ₀)
47		0nn0 12543	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
48	47	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ 𝐴) → 0 ∈ ℕ₀)
49	46, 48	ifclda 4518	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℕ₀)
50	43	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐵) → 2 ∈ ℕ₀)
51	5	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐵) → 𝑁 ∈ ℕ₀)
52	50, 51	nn0expcld 14310	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐵) → (2↑𝑁) ∈ ℕ₀)
53	47	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ 𝐵) → 0 ∈ ℕ₀)
54	52, 53	ifclda 4518	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℕ₀)
55	49, 54	nn0addcld 12593	. . . . . . . . 9 ⊢ (𝜑 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ∈ ℕ₀)
56	55	nn0red 12590	. . . . . . . 8 ⊢ (𝜑 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ∈ ℝ)
57	56	ad2antrr 739	. . . . . . 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 482	. . . . . . . 8 ⊢ ((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) → (2↑𝑁) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
60	59	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)) → (2↑𝑁) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
61	6	nnnn0d 12589	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2↑𝑁) ∈ ℕ₀)
62		ifcl 4528	. . . . . . . . . . . . 13 ⊢ (((2↑𝑁) ∈ ℕ₀ ∧ 0 ∈ ℕ₀) → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℕ₀)
63	61, 47, 62	sylancl 598	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℕ₀)
64	63	nn0ge0d 12592	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ if(𝑁 ∈ 𝐵, (2↑𝑁), 0))
65	7	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐵) → (2↑𝑁) ∈ ℝ)
66		0red 11235	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ 𝐵) → 0 ∈ ℝ)
67	65, 66	ifclda 4518	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℝ)
68	7, 67	addge01d 11826	. . . . . . . . . . 11 ⊢ (𝜑 → (0 ≤ if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ↔ (2↑𝑁) ≤ ((2↑𝑁) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))))
69	64, 68	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝑁) ≤ ((2↑𝑁) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
70	69	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ≤ ((2↑𝑁) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
71		iftrue 4488	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝐴 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) = (2↑𝑁))
72	71	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐴) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) = (2↑𝑁))
73	72	oveq1d 7428	. . . . . . . . 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 5133	. . . . . . . 8 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ≤ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
75		ifcl 4528	. . . . . . . . . . . . 13 ⊢ (((2↑𝑁) ∈ ℕ₀ ∧ 0 ∈ ℕ₀) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℕ₀)
76	61, 47, 75	sylancl 598	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℕ₀)
77	76	nn0ge0d 12592	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ if(𝑁 ∈ 𝐴, (2↑𝑁), 0))
78	7	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ∈ ℝ)
79		0red 11235	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ 𝐴) → 0 ∈ ℝ)
80	78, 79	ifclda 4518	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℝ)
81	7, 80	addge02d 11827	. . . . . . . . . . 11 ⊢ (𝜑 → (0 ≤ if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ↔ (2↑𝑁) ≤ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + (2↑𝑁))))
82	77, 81	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝑁) ≤ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + (2↑𝑁)))
83	82	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐵) → (2↑𝑁) ≤ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + (2↑𝑁)))
84		iftrue 4488	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝐵 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) = (2↑𝑁))
85	84	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐵) → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) = (2↑𝑁))
86	85	oveq2d 7429	. . . . . . . . 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 5133	. . . . . . . 8 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐵) → (2↑𝑁) ≤ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
88	74, 87	jaodan 972	. . . . . . 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 11857	. . . . . 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 418	. . . . 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 999	. . . . . 6 ⊢ (¬ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵) ↔ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵))
92		iffalse 4491	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑁 ∈ 𝐴 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) = 0)
93	92	ad2antrl 741	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) = 0)
94		iffalse 4491	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑁 ∈ 𝐵 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) = 0)
95	94	ad2antll 742	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) = 0)
96	93, 95	oveq12d 7431	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = (0 + 0))
97		00id 11409	. . . . . . . . . . . 12 ⊢ (0 + 0) = 0
98	96, 97	eqtrdi 2811	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = 0)
99	98	oveq2d 7429	. . . . . . . . . 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 12590	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℝ)
101	100	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℝ)
102	39	nn0red 12590	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℝ)
103	102	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℝ)
104	101, 103	readdcld 11262	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ)
105	104	recnd 11261	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℂ)
106	105	addridd 11434	. . . . . . . . . 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 2795	. . . . . . . . 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 6879	. . . . . . . . . . . . . . . 16 ⊢ (𝐾‘(𝐴 ∩ (0..^𝑁))) = (^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁)))
109	108	fveq2i 6881	. . . . . . . . . . . . . . 15 ⊢ ((bits ↾ ℕ₀)‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) = ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁))))
110	29	fvresd 6898	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) = (bits‘(𝐾‘(𝐴 ∩ (0..^𝑁)))))
111		f1ocnvfv2 7278	. . . . . . . . . . . . . . . 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)))
112	19, 18, 111	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁)))) = (𝐴 ∩ (0..^𝑁)))
113	109, 110, 112	3eqtr3a 2819	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (bits‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) = (𝐴 ∩ (0..^𝑁)))
114	113, 14	eqsstrdi 3975	. . . . . . . . . . . . 13 ⊢ (𝜑 → (bits‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) ⊆ (0..^𝑁))
115	29	nn0zd 12640	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℤ)
116		bitsfzo 16525	. . . . . . . . . . . . . 14 ⊢ (((𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) ↔ (bits‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) ⊆ (0..^𝑁)))
117	115, 5, 116	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) ↔ (bits‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) ⊆ (0..^𝑁)))
118	114, 117	mpbird 260	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)))
119		elfzolt2 13724	. . . . . . . . . . . 12 ⊢ ((𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) → (𝐾‘(𝐴 ∩ (0..^𝑁))) < (2↑𝑁))
120	118, 119	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^𝑁))) < (2↑𝑁))
121	22	fveq1i 6879	. . . . . . . . . . . . . . . 16 ⊢ (𝐾‘(𝐵 ∩ (0..^𝑁))) = (^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁)))
122	121	fveq2i 6881	. . . . . . . . . . . . . . 15 ⊢ ((bits ↾ ℕ₀)‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) = ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁))))
123	39	fvresd 6898	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) = (bits‘(𝐾‘(𝐵 ∩ (0..^𝑁)))))
124		f1ocnvfv2 7278	. . . . . . . . . . . . . . . 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)))
125	19, 37, 124	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁)))) = (𝐵 ∩ (0..^𝑁)))
126	122, 123, 125	3eqtr3a 2819	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (bits‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) = (𝐵 ∩ (0..^𝑁)))
127	126, 33	eqsstrdi 3975	. . . . . . . . . . . . 13 ⊢ (𝜑 → (bits‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) ⊆ (0..^𝑁))
128	39	nn0zd 12640	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℤ)
129		bitsfzo 16525	. . . . . . . . . . . . . 14 ⊢ (((𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) ↔ (bits‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) ⊆ (0..^𝑁)))
130	128, 5, 129	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) ↔ (bits‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) ⊆ (0..^𝑁)))
131	127, 130	mpbird 260	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)))
132		elfzolt2 13724	. . . . . . . . . . . 12 ⊢ ((𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) → (𝐾‘(𝐵 ∩ (0..^𝑁))) < (2↑𝑁))
133	131, 132	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^𝑁))) < (2↑𝑁))
134	100, 102, 7, 7, 120, 133	lt2addd 11861	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) < ((2↑𝑁) + (2↑𝑁)))
135	134	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) < ((2↑𝑁) + (2↑𝑁)))
136	107, 135	eqbrtrd 5127	. . . . . . . 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)))
137	80	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℝ)
138	67	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℝ)
139	137, 138	readdcld 11262	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ∈ ℝ)
140	104, 139	readdcld 11262	. . . . . . . . 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)
141	7	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (2↑𝑁) ∈ ℝ)
142	141, 141	readdcld 11262	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → ((2↑𝑁) + (2↑𝑁)) ∈ ℝ)
143	140, 142	ltnled 11381	. . . . . . . 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)))))
144	136, 143	mpbid 235	. . . . . . 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))))
145	144	ex 418	. . . . . 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)))))
146	91, 145	biimtrid 245	. . . . 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)))))
147	90, 146	impcon4bid 230	. . . 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)))))
148	2, 147	bitrd 282	. . 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)))))
149		cad0 1651	. . . . 5 ⊢ (¬ ∅ ∈ (𝐶‘𝑁) → (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) ↔ (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)))
150	149	adantl 487	. . . 4 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) ↔ (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)))
151	40	nn0ge0d 12592	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
152	7, 7	readdcld 11262	. . . . . . . . . 10 ⊢ (𝜑 → ((2↑𝑁) + (2↑𝑁)) ∈ ℝ)
153	152, 41	addge02d 11827	. . . . . . . . 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)))))
154	151, 153	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + ((2↑𝑁) + (2↑𝑁))))
155	154	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)) → ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + ((2↑𝑁) + (2↑𝑁))))
156	71, 84	oveqan12d 7432	. . . . . . . . 9 ⊢ ((𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = ((2↑𝑁) + (2↑𝑁)))
157	156	adantl 487	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = ((2↑𝑁) + (2↑𝑁)))
158	157	oveq2d 7429	. . . . . . 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))))
159	155, 158	breqtrrd 5133	. . . . . 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))))
160	159	ex 418	. . . . 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)))))
161	100	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℝ)
162	102	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℝ)
163	161, 162	readdcld 11262	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ)
164	7	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (2↑𝑁) ∈ ℝ)
165	7, 41	lenltd 11380	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2↑𝑁) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ↔ ¬ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) < (2↑𝑁)))
166	58, 165	bitrd 282	. . . . . . . . . . 11 ⊢ (𝜑 → (∅ ∈ (𝐶‘𝑁) ↔ ¬ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) < (2↑𝑁)))
167	166	con2bid 357	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) < (2↑𝑁) ↔ ¬ ∅ ∈ (𝐶‘𝑁)))
168	167	biimpar 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) < (2↑𝑁))
169	163, 164, 164, 168	ltadd1dd 11849	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (2↑𝑁)) < ((2↑𝑁) + (2↑𝑁)))
170	163, 164	readdcld 11262	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (2↑𝑁)) ∈ ℝ)
171	152	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → ((2↑𝑁) + (2↑𝑁)) ∈ ℝ)
172	41, 56	readdcld 11262	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))) ∈ ℝ)
173	172	adantr 486	. . . . . . . . 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)
174		ltletr 11326	. . . . . . . . 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)))))
175	170, 171, 173, 174	syl3anc 1398	. . . . . . . 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)))))
176	169, 175	mpand 708	. . . . . . 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)))))
177	56	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ∈ ℝ)
178	41	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ)
179	164, 177, 178	ltadd2d 11390	. . . . . . 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)))))
180	176, 179	sylibrd 262	. . . . . 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))))
181	7, 56	ltnled 11381	. . . . . . . 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)))
182	63	nn0cnd 12591	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℂ)
183	182	addlidd 11435	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = if(𝑁 ∈ 𝐵, (2↑𝑁), 0))
184	7	leidd 11804	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2↑𝑁) ≤ (2↑𝑁))
185	61	nn0ge0d 12592	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (2↑𝑁))
186		breq1 5106	. . . . . . . . . . . . . 14 ⊢ ((2↑𝑁) = if(𝑁 ∈ 𝐵, (2↑𝑁), 0) → ((2↑𝑁) ≤ (2↑𝑁) ↔ if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ≤ (2↑𝑁)))
187		breq1 5106	. . . . . . . . . . . . . 14 ⊢ (0 = if(𝑁 ∈ 𝐵, (2↑𝑁), 0) → (0 ≤ (2↑𝑁) ↔ if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ≤ (2↑𝑁)))
188	186, 187	ifboth 4522	. . . . . . . . . . . . 13 ⊢ (((2↑𝑁) ≤ (2↑𝑁) ∧ 0 ≤ (2↑𝑁)) → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ≤ (2↑𝑁))
189	184, 185, 188	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ≤ (2↑𝑁))
190	183, 189	eqbrtrd 5127	. . . . . . . . . . 11 ⊢ (𝜑 → (0 + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁))
191	92	oveq1d 7428	. . . . . . . . . . . 12 ⊢ (¬ 𝑁 ∈ 𝐴 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = (0 + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
192	191	breq1d 5113	. . . . . . . . . . 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)))
193	190, 192	syl5ibrcom 250	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 𝑁 ∈ 𝐴 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁)))
194	193	con1d 146	. . . . . . . . 9 ⊢ (𝜑 → (¬ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁) → 𝑁 ∈ 𝐴))
195	76	nn0cnd 12591	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℂ)
196	195	addridd 11434	. . . . . . . . . . . 12 ⊢ (𝜑 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + 0) = if(𝑁 ∈ 𝐴, (2↑𝑁), 0))
197		breq1 5106	. . . . . . . . . . . . . 14 ⊢ ((2↑𝑁) = if(𝑁 ∈ 𝐴, (2↑𝑁), 0) → ((2↑𝑁) ≤ (2↑𝑁) ↔ if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ≤ (2↑𝑁)))
198		breq1 5106	. . . . . . . . . . . . . 14 ⊢ (0 = if(𝑁 ∈ 𝐴, (2↑𝑁), 0) → (0 ≤ (2↑𝑁) ↔ if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ≤ (2↑𝑁)))
199	197, 198	ifboth 4522	. . . . . . . . . . . . 13 ⊢ (((2↑𝑁) ≤ (2↑𝑁) ∧ 0 ≤ (2↑𝑁)) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ≤ (2↑𝑁))
200	184, 185, 199	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ≤ (2↑𝑁))
201	196, 200	eqbrtrd 5127	. . . . . . . . . . 11 ⊢ (𝜑 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + 0) ≤ (2↑𝑁))
202	94	oveq2d 7429	. . . . . . . . . . . 12 ⊢ (¬ 𝑁 ∈ 𝐵 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + 0))
203	202	breq1d 5113	. . . . . . . . . . 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)))
204	201, 203	syl5ibrcom 250	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 𝑁 ∈ 𝐵 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁)))
205	204	con1d 146	. . . . . . . . 9 ⊢ (𝜑 → (¬ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁) → 𝑁 ∈ 𝐵))
206	194, 205	jcad 522	. . . . . . . 8 ⊢ (𝜑 → (¬ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁) → (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)))
207	181, 206	sylbid 243	. . . . . . 7 ⊢ (𝜑 → ((2↑𝑁) < (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) → (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)))
208	207	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → ((2↑𝑁) < (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) → (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)))
209	180, 208	syld 48	. . . . 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)))
210	160, 209	impbid 215	. . . 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)))))
211	150, 210	bitrd 282	. . 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)))))
212	148, 211	pm2.61dan 825	. 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)))))
213		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))))
214	10, 31, 213, 5	sadcp1 16545	. 2 ⊢ (𝜑 → (∅ ∈ (𝐶‘(𝑁 + 1)) ↔ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁))))
215		2cnd 12343	. . . . 5 ⊢ (𝜑 → 2 ∈ ℂ)
216	215, 5	expp1d 14211	. . . 4 ⊢ (𝜑 → (2↑(𝑁 + 1)) = ((2↑𝑁) · 2))
217	6	nncnd 12273	. . . . 5 ⊢ (𝜑 → (2↑𝑁) ∈ ℂ)
218	217	times2d 12512	. . . 4 ⊢ (𝜑 → ((2↑𝑁) · 2) = ((2↑𝑁) + (2↑𝑁)))
219	216, 218	eqtrd 2795	. . 3 ⊢ (𝜑 → (2↑(𝑁 + 1)) = ((2↑𝑁) + (2↑𝑁)))
220	22	bitsinvp1 16539	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐴, (2↑𝑁), 0)))
221	10, 5, 220	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐴, (2↑𝑁), 0)))
222	22	bitsinvp1 16539	. . . . . 6 ⊢ ((𝐵 ⊆ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐾‘(𝐵 ∩ (0..^(𝑁 + 1)))) = ((𝐾‘(𝐵 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
223	31, 5, 222	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^(𝑁 + 1)))) = ((𝐾‘(𝐵 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
224	221, 223	oveq12d 7431	. . . 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))))
225	29	nn0cnd 12591	. . . . 5 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℂ)
226	39	nn0cnd 12591	. . . . 5 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℂ)
227	225, 195, 226, 182	add4d 11463	. . . 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))))
228	224, 227	eqtrd 2795	. . 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))))
229	219, 228	breq12d 5116	. 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)))))
230	212, 214, 229	3bitr4d 314	1 ⊢ (𝜑 → (∅ ∈ (𝐶‘(𝑁 + 1)) ↔ (2↑(𝑁 + 1)) ≤ ((𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑁 + 1)))))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 caddwcad 1639 ∈ wcel 2145 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 ifcif 4482 𝒫 cpw 4557 class class class wbr 5103 ↦ cmpt 5186 ^◡ccnv 5654 ↾ cres 5657 ⟶wf 6529 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7413 ∈ cmpo 7415 1_oc1o 8448 2_oc2o 8449 Fincfn 8952 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 · cmul 11129 < clt 11267 ≤ cle 11268 − cmin 11465 ℕcn 12257 2c2 12319 ℕ₀cn0 12528 ℤcz 12615 ..^cfzo 13709 seqcseq 14065 ↑cexp 14125 bitscbits 16509

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-xor 1542 df-tru 1573 df-fal 1583 df-cad 1640 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-er 8696 df-map 8828 df-pm 8829 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9906 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12888 df-rp 13043 df-fz 13562 df-fzo 13710 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-clim 15575 df-sum 15774 df-dvds 16343 df-bits 16512

This theorem is used by: sadcadd 16548

W3C validator