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Theorem sadadd2 16095

Description: Sum of initial segments of the sadd 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 ∈ 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)

**Assertion**
Ref	Expression
sadadd2	⊢ (𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) + if(∅ ∈ (𝐶‘𝑁), (2↑𝑁), 0)) = ((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 sadadd2 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		oveq2 7263	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (0..^𝑥) = (0..^0))
3		fzo0 13339	. . . . . . . . . . 11 ⊢ (0..^0) = ∅
4	2, 3	eqtrdi 2795	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (0..^𝑥) = ∅)
5	4	ineq2d 4143	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ ∅))
6		in0 4322	. . . . . . . . 9 ⊢ ((𝐴 sadd 𝐵) ∩ ∅) = ∅
7	5, 6	eqtrdi 2795	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ∅)
8	7	fveq2d 6760	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘∅))
9		sadcadd.k	. . . . . . . . 9 ⊢ 𝐾 = ^◡(bits ↾ ℕ₀)
10		0nn0 12178	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ₀
11		fvres 6775	. . . . . . . . . . 11 ⊢ (0 ∈ ℕ₀ → ((bits ↾ ℕ₀)‘0) = (bits‘0))
12	10, 11	ax-mp 5	. . . . . . . . . 10 ⊢ ((bits ↾ ℕ₀)‘0) = (bits‘0)
13		0bits 16074	. . . . . . . . . 10 ⊢ (bits‘0) = ∅
14	12, 13	eqtr2i 2767	. . . . . . . . 9 ⊢ ∅ = ((bits ↾ ℕ₀)‘0)
15	9, 14	fveq12i 6762	. . . . . . . 8 ⊢ (𝐾‘∅) = (^◡(bits ↾ ℕ₀)‘((bits ↾ ℕ₀)‘0))
16		bitsf1o 16080	. . . . . . . . 9 ⊢ (bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin)
17		f1ocnvfv1 7129	. . . . . . . . 9 ⊢ (((bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin) ∧ 0 ∈ ℕ₀) → (^◡(bits ↾ ℕ₀)‘((bits ↾ ℕ₀)‘0)) = 0)
18	16, 10, 17	mp2an 688	. . . . . . . 8 ⊢ (^◡(bits ↾ ℕ₀)‘((bits ↾ ℕ₀)‘0)) = 0
19	15, 18	eqtri 2766	. . . . . . 7 ⊢ (𝐾‘∅) = 0
20	8, 19	eqtrdi 2795	. . . . . 6 ⊢ (𝑥 = 0 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = 0)
21		fveq2 6756	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐶‘𝑥) = (𝐶‘0))
22	21	eleq2d 2824	. . . . . . 7 ⊢ (𝑥 = 0 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘0)))
23		oveq2 7263	. . . . . . 7 ⊢ (𝑥 = 0 → (2↑𝑥) = (2↑0))
24	22, 23	ifbieq1d 4480	. . . . . 6 ⊢ (𝑥 = 0 → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘0), (2↑0), 0))
25	20, 24	oveq12d 7273	. . . . 5 ⊢ (𝑥 = 0 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0)) = (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)))
26	4	ineq2d 4143	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ ∅))
27		in0 4322	. . . . . . . . . 10 ⊢ (𝐴 ∩ ∅) = ∅
28	26, 27	eqtrdi 2795	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐴 ∩ (0..^𝑥)) = ∅)
29	28	fveq2d 6760	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘∅))
30	29, 19	eqtrdi 2795	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = 0)
31	4	ineq2d 4143	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ ∅))
32		in0 4322	. . . . . . . . . 10 ⊢ (𝐵 ∩ ∅) = ∅
33	31, 32	eqtrdi 2795	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐵 ∩ (0..^𝑥)) = ∅)
34	33	fveq2d 6760	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘∅))
35	34, 19	eqtrdi 2795	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = 0)
36	30, 35	oveq12d 7273	. . . . . 6 ⊢ (𝑥 = 0 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = (0 + 0))
37		00id 11080	. . . . . 6 ⊢ (0 + 0) = 0
38	36, 37	eqtrdi 2795	. . . . 5 ⊢ (𝑥 = 0 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = 0)
39	25, 38	eqeq12d 2754	. . . 4 ⊢ (𝑥 = 0 → (((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) ↔ (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)) = 0))
40	39	imbi2d 340	. . 3 ⊢ (𝑥 = 0 → ((𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥))))) ↔ (𝜑 → (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)) = 0)))
41		oveq2 7263	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (0..^𝑥) = (0..^𝑘))
42	41	ineq2d 4143	. . . . . . 7 ⊢ (𝑥 = 𝑘 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^𝑘)))
43	42	fveq2d 6760	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))))
44		fveq2 6756	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝐶‘𝑥) = (𝐶‘𝑘))
45	44	eleq2d 2824	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘𝑘)))
46		oveq2 7263	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (2↑𝑥) = (2↑𝑘))
47	45, 46	ifbieq1d 4480	. . . . . 6 ⊢ (𝑥 = 𝑘 → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘𝑘), (2↑𝑘), 0))
48	43, 47	oveq12d 7273	. . . . 5 ⊢ (𝑥 = 𝑘 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0)) = ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶‘𝑘), (2↑𝑘), 0)))
49	41	ineq2d 4143	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^𝑘)))
50	49	fveq2d 6760	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^𝑘))))
51	41	ineq2d 4143	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^𝑘)))
52	51	fveq2d 6760	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^𝑘))))
53	50, 52	oveq12d 7273	. . . . 5 ⊢ (𝑥 = 𝑘 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))
54	48, 53	eqeq12d 2754	. . . 4 ⊢ (𝑥 = 𝑘 → (((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) ↔ ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶‘𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))))
55	54	imbi2d 340	. . 3 ⊢ (𝑥 = 𝑘 → ((𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥))))) ↔ (𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶‘𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))))
56		oveq2 7263	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (0..^𝑥) = (0..^(𝑘 + 1)))
57	56	ineq2d 4143	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1))))
58	57	fveq2d 6760	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))))
59		fveq2 6756	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝐶‘𝑥) = (𝐶‘(𝑘 + 1)))
60	59	eleq2d 2824	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘(𝑘 + 1))))
61		oveq2 7263	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (2↑𝑥) = (2↑(𝑘 + 1)))
62	60, 61	ifbieq1d 4480	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0))
63	58, 62	oveq12d 7273	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0)) = ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))) + if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0)))
64	56	ineq2d 4143	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^(𝑘 + 1))))
65	64	fveq2d 6760	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))))
66	56	ineq2d 4143	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^(𝑘 + 1))))
67	66	fveq2d 6760	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1)))))
68	65, 67	oveq12d 7273	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1))))))
69	63, 68	eqeq12d 2754	. . . 4 ⊢ (𝑥 = (𝑘 + 1) → (((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) ↔ ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))) + if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0)) = ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1)))))))
70	69	imbi2d 340	. . 3 ⊢ (𝑥 = (𝑘 + 1) → ((𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥))))) ↔ (𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))) + if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0)) = ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1))))))))
71		oveq2 7263	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (0..^𝑥) = (0..^𝑁))
72	71	ineq2d 4143	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^𝑁)))
73	72	fveq2d 6760	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))))
74		fveq2 6756	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐶‘𝑥) = (𝐶‘𝑁))
75	74	eleq2d 2824	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘𝑁)))
76		oveq2 7263	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (2↑𝑥) = (2↑𝑁))
77	75, 76	ifbieq1d 4480	. . . . . 6 ⊢ (𝑥 = 𝑁 → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘𝑁), (2↑𝑁), 0))
78	73, 77	oveq12d 7273	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0)) = ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) + if(∅ ∈ (𝐶‘𝑁), (2↑𝑁), 0)))
79	71	ineq2d 4143	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^𝑁)))
80	79	fveq2d 6760	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^𝑁))))
81	71	ineq2d 4143	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^𝑁)))
82	81	fveq2d 6760	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^𝑁))))
83	80, 82	oveq12d 7273	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
84	78, 83	eqeq12d 2754	. . . 4 ⊢ (𝑥 = 𝑁 → (((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) ↔ ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) + if(∅ ∈ (𝐶‘𝑁), (2↑𝑁), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁))))))
85	84	imbi2d 340	. . 3 ⊢ (𝑥 = 𝑁 → ((𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) + if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥))))) ↔ (𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) + if(∅ ∈ (𝐶‘𝑁), (2↑𝑁), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))))
86		sadval.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℕ₀)
87		sadval.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ ℕ₀)
88		sadval.c	. . . . . . 7 ⊢ 𝐶 = seq0((𝑐 ∈ 2_o, 𝑚 ∈ ℕ₀ ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1_o, ∅)), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
89	86, 87, 88	sadc0 16089	. . . . . 6 ⊢ (𝜑 → ¬ ∅ ∈ (𝐶‘0))
90	89	iffalsed 4467	. . . . 5 ⊢ (𝜑 → if(∅ ∈ (𝐶‘0), (2↑0), 0) = 0)
91	90	oveq2d 7271	. . . 4 ⊢ (𝜑 → (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)) = (0 + 0))
92	91, 37	eqtrdi 2795	. . 3 ⊢ (𝜑 → (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)) = 0)
93	86	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶‘𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))) → 𝐴 ⊆ ℕ₀)
94	87	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶‘𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))) → 𝐵 ⊆ ℕ₀)
95		simplr 765	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶‘𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))) → 𝑘 ∈ ℕ₀)
96		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶‘𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))) → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶‘𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))
97	93, 94, 88, 95, 9, 96	sadadd2lem 16094	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶‘𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))) → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))) + if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0)) = ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1))))))
98	97	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶‘𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))) → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))) + if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0)) = ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1)))))))
99	98	expcom 413	. . . 4 ⊢ (𝑘 ∈ ℕ₀ → (𝜑 → (((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶‘𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))) → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))) + if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0)) = ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1))))))))
100	99	a2d 29	. . 3 ⊢ (𝑘 ∈ ℕ₀ → ((𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))) + if(∅ ∈ (𝐶‘𝑘), (2↑𝑘), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘))))) → (𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))) + if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0)) = ((𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))) + (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1))))))))
101	40, 55, 70, 85, 92, 100	nn0ind 12345	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) + if(∅ ∈ (𝐶‘𝑁), (2↑𝑁), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁))))))
102	1, 101	mpcom 38	1 ⊢ (𝜑 → ((𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))) + if(∅ ∈ (𝐶‘𝑁), (2↑𝑁), 0)) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 caddwcad 1609 ∈ wcel 2108 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 ifcif 4456 𝒫 cpw 4530 ↦ cmpt 5153 ^◡ccnv 5579 ↾ cres 5582 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 1_oc1o 8260 2_oc2o 8261 Fincfn 8691 0cc0 10802 1c1 10803 + caddc 10805 − cmin 11135 2c2 11958 ℕ₀cn0 12163 ..^cfzo 13311 seqcseq 13649 ↑cexp 13710 bitscbits 16054 sadd csad 16055

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-xor 1504 df-tru 1542 df-fal 1552 df-had 1596 df-cad 1610 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-dvds 15892 df-bits 16057 df-sad 16086

This theorem is referenced by: sadadd3 16096

W3C validator