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Theorem sadcaddlem 16396

Description: Lemma for sadcadd 16397. (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 1619	. . . . 5 ⊢ (∅ ∈ (𝐶‘𝑁) → (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) ↔ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)))
2	1	adantl 481	. . . 4 ⊢ ((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) → (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) ↔ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)))
3		2nn 12230	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
4	3	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℕ)
5		sadcp1.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6	4, 5	nnexpcld 14180	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝑁) ∈ ℕ)
7	6	nnred 12172	. . . . . . . 8 ⊢ (𝜑 → (2↑𝑁) ∈ ℝ)
8	7	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)) → (2↑𝑁) ∈ ℝ)
9		inss1 4191	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ 𝐴
10		sadval.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ℕ₀)
11	9, 10	sstrid 3947	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀)
12		fzofi 13909	. . . . . . . . . . . . . 14 ⊢ (0..^𝑁) ∈ Fin
13	12	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0..^𝑁) ∈ Fin)
14		inss2 4192	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
15		ssfi 9109	. . . . . . . . . . . . 13 ⊢ (((0..^𝑁) ∈ Fin ∧ (𝐴 ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (𝐴 ∩ (0..^𝑁)) ∈ Fin)
16	13, 14, 15	sylancl 587	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ∈ Fin)
17		elfpw 9266	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((𝐴 ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ (𝐴 ∩ (0..^𝑁)) ∈ Fin))
18	11, 16, 17	sylanbrc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
19		bitsf1o 16384	. . . . . . . . . . . . . . 15 ⊢ (bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin)
20		f1ocnv 6794	. . . . . . . . . . . . . . 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 6770	. . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . 13 ⊢ 𝐾:(𝒫 ℕ₀ ∩ Fin)–1-1-onto→ℕ₀
26		f1of 6782	. . . . . . . . . . . . 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 7037	. . . . . . . . . . 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 4191	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ (0..^𝑁)) ⊆ 𝐵
31		sadval.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ⊆ ℕ₀)
32	30, 31	sstrid 3947	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ₀)
33		inss2 4192	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
34		ssfi 9109	. . . . . . . . . . . . 13 ⊢ (((0..^𝑁) ∈ Fin ∧ (𝐵 ∩ (0..^𝑁)) ⊆ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ∈ Fin)
35	13, 33, 34	sylancl 587	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∩ (0..^𝑁)) ∈ Fin)
36		elfpw 9266	. . . . . . . . . . . 12 ⊢ ((𝐵 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin) ↔ ((𝐵 ∩ (0..^𝑁)) ⊆ ℕ₀ ∧ (𝐵 ∩ (0..^𝑁)) ∈ Fin))
37	32, 35, 36	sylanbrc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ∩ (0..^𝑁)) ∈ (𝒫 ℕ₀ ∩ Fin))
38	27	ffvelcdmi 7037	. . . . . . . . . . 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 12478	. . . . . . . . 9 ⊢ (𝜑 → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℕ₀)
41	40	nn0red 12475	. . . . . . . 8 ⊢ (𝜑 → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ)
42	41	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ)
43		2nn0 12430	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
44	43	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐴) → 2 ∈ ℕ₀)
45	5	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐴) → 𝑁 ∈ ℕ₀)
46	44, 45	nn0expcld 14181	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ∈ ℕ₀)
47		0nn0 12428	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
48	47	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ 𝐴) → 0 ∈ ℕ₀)
49	46, 48	ifclda 4517	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℕ₀)
50	43	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐵) → 2 ∈ ℕ₀)
51	5	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐵) → 𝑁 ∈ ℕ₀)
52	50, 51	nn0expcld 14181	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐵) → (2↑𝑁) ∈ ℕ₀)
53	47	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ 𝐵) → 0 ∈ ℕ₀)
54	52, 53	ifclda 4517	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℕ₀)
55	49, 54	nn0addcld 12478	. . . . . . . . 9 ⊢ (𝜑 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ∈ ℕ₀)
56	55	nn0red 12475	. . . . . . . 8 ⊢ (𝜑 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ∈ ℝ)
57	56	ad2antrr 727	. . . . . . 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 476	. . . . . . . 8 ⊢ ((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) → (2↑𝑁) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
60	59	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵)) → (2↑𝑁) ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
61	6	nnnn0d 12474	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2↑𝑁) ∈ ℕ₀)
62		ifcl 4527	. . . . . . . . . . . . 13 ⊢ (((2↑𝑁) ∈ ℕ₀ ∧ 0 ∈ ℕ₀) → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℕ₀)
63	61, 47, 62	sylancl 587	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℕ₀)
64	63	nn0ge0d 12477	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ if(𝑁 ∈ 𝐵, (2↑𝑁), 0))
65	7	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐵) → (2↑𝑁) ∈ ℝ)
66		0red 11147	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ 𝐵) → 0 ∈ ℝ)
67	65, 66	ifclda 4517	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℝ)
68	7, 67	addge01d 11737	. . . . . . . . . . 11 ⊢ (𝜑 → (0 ≤ if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ↔ (2↑𝑁) ≤ ((2↑𝑁) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0))))
69	64, 68	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝑁) ≤ ((2↑𝑁) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
70	69	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ≤ ((2↑𝑁) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
71		iftrue 4487	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝐴 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) = (2↑𝑁))
72	71	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐴) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) = (2↑𝑁))
73	72	oveq1d 7383	. . . . . . . . 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 5128	. . . . . . . 8 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ≤ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
75		ifcl 4527	. . . . . . . . . . . . 13 ⊢ (((2↑𝑁) ∈ ℕ₀ ∧ 0 ∈ ℕ₀) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℕ₀)
76	61, 47, 75	sylancl 587	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℕ₀)
77	76	nn0ge0d 12477	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ if(𝑁 ∈ 𝐴, (2↑𝑁), 0))
78	7	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝐴) → (2↑𝑁) ∈ ℝ)
79		0red 11147	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ 𝐴) → 0 ∈ ℝ)
80	78, 79	ifclda 4517	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℝ)
81	7, 80	addge02d 11738	. . . . . . . . . . 11 ⊢ (𝜑 → (0 ≤ if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ↔ (2↑𝑁) ≤ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + (2↑𝑁))))
82	77, 81	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝑁) ≤ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + (2↑𝑁)))
83	82	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐵) → (2↑𝑁) ≤ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + (2↑𝑁)))
84		iftrue 4487	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝐵 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) = (2↑𝑁))
85	84	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐵) → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) = (2↑𝑁))
86	85	oveq2d 7384	. . . . . . . . 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 5128	. . . . . . . 8 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ 𝑁 ∈ 𝐵) → (2↑𝑁) ≤ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
88	74, 87	jaodan 960	. . . . . . 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 11768	. . . . . 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 412	. . . . 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 986	. . . . . 6 ⊢ (¬ (𝑁 ∈ 𝐴 ∨ 𝑁 ∈ 𝐵) ↔ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵))
92		iffalse 4490	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑁 ∈ 𝐴 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) = 0)
93	92	ad2antrl 729	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) = 0)
94		iffalse 4490	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑁 ∈ 𝐵 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) = 0)
95	94	ad2antll 730	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) = 0)
96	93, 95	oveq12d 7386	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = (0 + 0))
97		00id 11320	. . . . . . . . . . . 12 ⊢ (0 + 0) = 0
98	96, 97	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = 0)
99	98	oveq2d 7384	. . . . . . . . . 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 12475	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℝ)
101	100	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℝ)
102	39	nn0red 12475	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℝ)
103	102	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℝ)
104	101, 103	readdcld 11173	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ)
105	104	recnd 11172	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℂ)
106	105	addridd 11345	. . . . . . . . . 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 2772	. . . . . . . . 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 6843	. . . . . . . . . . . . . . . 16 ⊢ (𝐾‘(𝐴 ∩ (0..^𝑁))) = (^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁)))
109	108	fveq2i 6845	. . . . . . . . . . . . . . 15 ⊢ ((bits ↾ ℕ₀)‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) = ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁))))
110	29	fvresd 6862	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) = (bits‘(𝐾‘(𝐴 ∩ (0..^𝑁)))))
111		f1ocnvfv2 7233	. . . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(𝐴 ∩ (0..^𝑁)))) = (𝐴 ∩ (0..^𝑁)))
113	109, 110, 112	3eqtr3a 2796	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (bits‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) = (𝐴 ∩ (0..^𝑁)))
114	113, 14	eqsstrdi 3980	. . . . . . . . . . . . 13 ⊢ (𝜑 → (bits‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) ⊆ (0..^𝑁))
115	29	nn0zd 12525	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℤ)
116		bitsfzo 16374	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) ↔ (bits‘(𝐾‘(𝐴 ∩ (0..^𝑁)))) ⊆ (0..^𝑁)))
118	114, 117	mpbird 257	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)))
119		elfzolt2 13596	. . . . . . . . . . . 12 ⊢ ((𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) → (𝐾‘(𝐴 ∩ (0..^𝑁))) < (2↑𝑁))
120	118, 119	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^𝑁))) < (2↑𝑁))
121	22	fveq1i 6843	. . . . . . . . . . . . . . . 16 ⊢ (𝐾‘(𝐵 ∩ (0..^𝑁))) = (^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁)))
122	121	fveq2i 6845	. . . . . . . . . . . . . . 15 ⊢ ((bits ↾ ℕ₀)‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) = ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁))))
123	39	fvresd 6862	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) = (bits‘(𝐾‘(𝐵 ∩ (0..^𝑁)))))
124		f1ocnvfv2 7233	. . . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((bits ↾ ℕ₀)‘(^◡(bits ↾ ℕ₀)‘(𝐵 ∩ (0..^𝑁)))) = (𝐵 ∩ (0..^𝑁)))
126	122, 123, 125	3eqtr3a 2796	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (bits‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) = (𝐵 ∩ (0..^𝑁)))
127	126, 33	eqsstrdi 3980	. . . . . . . . . . . . 13 ⊢ (𝜑 → (bits‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) ⊆ (0..^𝑁))
128	39	nn0zd 12525	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℤ)
129		bitsfzo 16374	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) ↔ (bits‘(𝐾‘(𝐵 ∩ (0..^𝑁)))) ⊆ (0..^𝑁)))
131	127, 130	mpbird 257	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)))
132		elfzolt2 13596	. . . . . . . . . . . 12 ⊢ ((𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ (0..^(2↑𝑁)) → (𝐾‘(𝐵 ∩ (0..^𝑁))) < (2↑𝑁))
133	131, 132	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^𝑁))) < (2↑𝑁))
134	100, 102, 7, 7, 120, 133	lt2addd 11772	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) < ((2↑𝑁) + (2↑𝑁)))
135	134	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) < ((2↑𝑁) + (2↑𝑁)))
136	107, 135	eqbrtrd 5122	. . . . . . . 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 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℝ)
138	67	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℝ)
139	137, 138	readdcld 11173	. . . . . . . . . 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 11173	. . . . . . . . 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 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → (2↑𝑁) ∈ ℝ)
142	141, 141	readdcld 11173	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∅ ∈ (𝐶‘𝑁)) ∧ (¬ 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ 𝐵)) → ((2↑𝑁) + (2↑𝑁)) ∈ ℝ)
143	140, 142	ltnled 11292	. . . . . . . 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 232	. . . . . . 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 412	. . . . . 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 242	. . . . 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 227	. . . 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 279	. . 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 1620	. . . . 5 ⊢ (¬ ∅ ∈ (𝐶‘𝑁) → (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) ↔ (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)))
150	149	adantl 481	. . . 4 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) ↔ (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)))
151	40	nn0ge0d 12477	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
152	7, 7	readdcld 11173	. . . . . . . . . 10 ⊢ (𝜑 → ((2↑𝑁) + (2↑𝑁)) ∈ ℝ)
153	152, 41	addge02d 11738	. . . . . . . . 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 232	. . . . . . . 8 ⊢ (𝜑 → ((2↑𝑁) + (2↑𝑁)) ≤ (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + ((2↑𝑁) + (2↑𝑁))))
155	154	ad2antrr 727	. . . . . . 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 7387	. . . . . . . . 9 ⊢ ((𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = ((2↑𝑁) + (2↑𝑁)))
157	156	adantl 481	. . . . . . . 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 7384	. . . . . . 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 5128	. . . . . 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 412	. . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℝ)
162	102	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℝ)
163	161, 162	readdcld 11173	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ)
164	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (2↑𝑁) ∈ ℝ)
165	7, 41	lenltd 11291	. . . . . . . . . . . 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 279	. . . . . . . . . . 11 ⊢ (𝜑 → (∅ ∈ (𝐶‘𝑁) ↔ ¬ ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) < (2↑𝑁)))
167	166	con2bid 354	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) < (2↑𝑁) ↔ ¬ ∅ ∈ (𝐶‘𝑁)))
168	167	biimpar 477	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) < (2↑𝑁))
169	163, 164, 164, 168	ltadd1dd 11760	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (2↑𝑁)) < ((2↑𝑁) + (2↑𝑁)))
170	163, 164	readdcld 11173	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) + (2↑𝑁)) ∈ ℝ)
171	152	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → ((2↑𝑁) + (2↑𝑁)) ∈ ℝ)
172	41, 56	readdcld 11173	. . . . . . . . . 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 480	. . . . . . . . 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 11237	. . . . . . . . 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 1374	. . . . . . . 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 696	. . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ∈ ℝ)
178	41	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ ∅ ∈ (𝐶‘𝑁)) → ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))) ∈ ℝ)
179	164, 177, 178	ltadd2d 11301	. . . . . . 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 259	. . . . . 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 11292	. . . . . . . 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 12476	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ∈ ℂ)
183	182	addlidd 11346	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) = if(𝑁 ∈ 𝐵, (2↑𝑁), 0))
184	7	leidd 11715	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2↑𝑁) ≤ (2↑𝑁))
185	61	nn0ge0d 12477	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (2↑𝑁))
186		breq1 5103	. . . . . . . . . . . . . 14 ⊢ ((2↑𝑁) = if(𝑁 ∈ 𝐵, (2↑𝑁), 0) → ((2↑𝑁) ≤ (2↑𝑁) ↔ if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ≤ (2↑𝑁)))
187		breq1 5103	. . . . . . . . . . . . . 14 ⊢ (0 = if(𝑁 ∈ 𝐵, (2↑𝑁), 0) → (0 ≤ (2↑𝑁) ↔ if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ≤ (2↑𝑁)))
188	186, 187	ifboth 4521	. . . . . . . . . . . . 13 ⊢ (((2↑𝑁) ≤ (2↑𝑁) ∧ 0 ≤ (2↑𝑁)) → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ≤ (2↑𝑁))
189	184, 185, 188	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐵, (2↑𝑁), 0) ≤ (2↑𝑁))
190	183, 189	eqbrtrd 5122	. . . . . . . . . . 11 ⊢ (𝜑 → (0 + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁))
191	92	oveq1d 7383	. . . . . . . . . . . 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 5110	. . . . . . . . . . 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 247	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 𝑁 ∈ 𝐴 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁)))
194	193	con1d 145	. . . . . . . . 9 ⊢ (𝜑 → (¬ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁) → 𝑁 ∈ 𝐴))
195	76	nn0cnd 12476	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ∈ ℂ)
196	195	addridd 11345	. . . . . . . . . . . 12 ⊢ (𝜑 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + 0) = if(𝑁 ∈ 𝐴, (2↑𝑁), 0))
197		breq1 5103	. . . . . . . . . . . . . 14 ⊢ ((2↑𝑁) = if(𝑁 ∈ 𝐴, (2↑𝑁), 0) → ((2↑𝑁) ≤ (2↑𝑁) ↔ if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ≤ (2↑𝑁)))
198		breq1 5103	. . . . . . . . . . . . . 14 ⊢ (0 = if(𝑁 ∈ 𝐴, (2↑𝑁), 0) → (0 ≤ (2↑𝑁) ↔ if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ≤ (2↑𝑁)))
199	197, 198	ifboth 4521	. . . . . . . . . . . . 13 ⊢ (((2↑𝑁) ≤ (2↑𝑁) ∧ 0 ≤ (2↑𝑁)) → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ≤ (2↑𝑁))
200	184, 185, 199	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑁 ∈ 𝐴, (2↑𝑁), 0) ≤ (2↑𝑁))
201	196, 200	eqbrtrd 5122	. . . . . . . . . . 11 ⊢ (𝜑 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + 0) ≤ (2↑𝑁))
202	94	oveq2d 7384	. . . . . . . . . . . 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 5110	. . . . . . . . . . 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 247	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 𝑁 ∈ 𝐵 → (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁)))
205	204	con1d 145	. . . . . . . . 9 ⊢ (𝜑 → (¬ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁) → 𝑁 ∈ 𝐵))
206	194, 205	jcad 512	. . . . . . . 8 ⊢ (𝜑 → (¬ (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) ≤ (2↑𝑁) → (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)))
207	181, 206	sylbid 240	. . . . . . 7 ⊢ (𝜑 → ((2↑𝑁) < (if(𝑁 ∈ 𝐴, (2↑𝑁), 0) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)) → (𝑁 ∈ 𝐴 ∧ 𝑁 ∈ 𝐵)))
208	207	adantr 480	. . . . . 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 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)))
210	160, 209	impbid 212	. . . 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 279	. . 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 813	. 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 16394	. 2 ⊢ (𝜑 → (∅ ∈ (𝐶‘(𝑁 + 1)) ↔ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁))))
215		2cnd 12235	. . . . 5 ⊢ (𝜑 → 2 ∈ ℂ)
216	215, 5	expp1d 14082	. . . 4 ⊢ (𝜑 → (2↑(𝑁 + 1)) = ((2↑𝑁) · 2))
217	6	nncnd 12173	. . . . 5 ⊢ (𝜑 → (2↑𝑁) ∈ ℂ)
218	217	times2d 12397	. . . 4 ⊢ (𝜑 → ((2↑𝑁) · 2) = ((2↑𝑁) + (2↑𝑁)))
219	216, 218	eqtrd 2772	. . 3 ⊢ (𝜑 → (2↑(𝑁 + 1)) = ((2↑𝑁) + (2↑𝑁)))
220	22	bitsinvp1 16388	. . . . . 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 585	. . . . 5 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^(𝑁 + 1)))) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐴, (2↑𝑁), 0)))
222	22	bitsinvp1 16388	. . . . . 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 585	. . . . 5 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^(𝑁 + 1)))) = ((𝐾‘(𝐵 ∩ (0..^𝑁))) + if(𝑁 ∈ 𝐵, (2↑𝑁), 0)))
224	221, 223	oveq12d 7386	. . . 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 12476	. . . . 5 ⊢ (𝜑 → (𝐾‘(𝐴 ∩ (0..^𝑁))) ∈ ℂ)
226	39	nn0cnd 12476	. . . . 5 ⊢ (𝜑 → (𝐾‘(𝐵 ∩ (0..^𝑁))) ∈ ℂ)
227	225, 195, 226, 182	add4d 11374	. . . 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 2772	. . 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 5113	. 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 311	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 206 ∧ wa 395 ∨ wo 848 = wceq 1542 caddwcad 1608 ∈ wcel 2114 ∩ cin 3902 ⊆ wss 3903 ∅c0 4287 ifcif 4481 𝒫 cpw 4556 class class class wbr 5100 ↦ cmpt 5181 ^◡ccnv 5631 ↾ cres 5634 ⟶wf 6496 –1-1-onto→wf1o 6499 ‘cfv 6500 (class class class)co 7368 ∈ cmpo 7370 1_oc1o 8400 2_oc2o 8401 Fincfn 8895 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11178 ≤ cle 11179 − cmin 11376 ℕcn 12157 2c2 12212 ℕ₀cn0 12413 ℤcz 12500 ..^cfzo 13582 seqcseq 13936 ↑cexp 13996 bitscbits 16358

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-xor 1514 df-tru 1545 df-fal 1555 df-cad 1609 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-disj 5068 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-map 8777 df-pm 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-oi 9427 df-dju 9825 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-xnn0 12487 df-z 12501 df-uz 12764 df-rp 12918 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-sum 15622 df-dvds 16192 df-bits 16361

This theorem is referenced by: sadcadd 16397

W3C validator