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Theorem sadadd2 16167

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 7283	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (0..^𝑥) = (0..^0))
3		fzo0 13411	. . . . . . . . . . 11 ⊢ (0..^0) = ∅
4	2, 3	eqtrdi 2794	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (0..^𝑥) = ∅)
5	4	ineq2d 4146	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ ∅))
6		in0 4325	. . . . . . . . 9 ⊢ ((𝐴 sadd 𝐵) ∩ ∅) = ∅
7	5, 6	eqtrdi 2794	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ∅)
8	7	fveq2d 6778	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘∅))
9		sadcadd.k	. . . . . . . . 9 ⊢ 𝐾 = ^◡(bits ↾ ℕ₀)
10		0nn0 12248	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ₀
11		fvres 6793	. . . . . . . . . . 11 ⊢ (0 ∈ ℕ₀ → ((bits ↾ ℕ₀)‘0) = (bits‘0))
12	10, 11	ax-mp 5	. . . . . . . . . 10 ⊢ ((bits ↾ ℕ₀)‘0) = (bits‘0)
13		0bits 16146	. . . . . . . . . 10 ⊢ (bits‘0) = ∅
14	12, 13	eqtr2i 2767	. . . . . . . . 9 ⊢ ∅ = ((bits ↾ ℕ₀)‘0)
15	9, 14	fveq12i 6780	. . . . . . . 8 ⊢ (𝐾‘∅) = (^◡(bits ↾ ℕ₀)‘((bits ↾ ℕ₀)‘0))
16		bitsf1o 16152	. . . . . . . . 9 ⊢ (bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin)
17		f1ocnvfv1 7148	. . . . . . . . 9 ⊢ (((bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin) ∧ 0 ∈ ℕ₀) → (^◡(bits ↾ ℕ₀)‘((bits ↾ ℕ₀)‘0)) = 0)
18	16, 10, 17	mp2an 689	. . . . . . . 8 ⊢ (^◡(bits ↾ ℕ₀)‘((bits ↾ ℕ₀)‘0)) = 0
19	15, 18	eqtri 2766	. . . . . . 7 ⊢ (𝐾‘∅) = 0
20	8, 19	eqtrdi 2794	. . . . . 6 ⊢ (𝑥 = 0 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = 0)
21		fveq2 6774	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐶‘𝑥) = (𝐶‘0))
22	21	eleq2d 2824	. . . . . . 7 ⊢ (𝑥 = 0 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘0)))
23		oveq2 7283	. . . . . . 7 ⊢ (𝑥 = 0 → (2↑𝑥) = (2↑0))
24	22, 23	ifbieq1d 4483	. . . . . 6 ⊢ (𝑥 = 0 → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘0), (2↑0), 0))
25	20, 24	oveq12d 7293	. . . . 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 4146	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ ∅))
27		in0 4325	. . . . . . . . . 10 ⊢ (𝐴 ∩ ∅) = ∅
28	26, 27	eqtrdi 2794	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐴 ∩ (0..^𝑥)) = ∅)
29	28	fveq2d 6778	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘∅))
30	29, 19	eqtrdi 2794	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = 0)
31	4	ineq2d 4146	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ ∅))
32		in0 4325	. . . . . . . . . 10 ⊢ (𝐵 ∩ ∅) = ∅
33	31, 32	eqtrdi 2794	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐵 ∩ (0..^𝑥)) = ∅)
34	33	fveq2d 6778	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘∅))
35	34, 19	eqtrdi 2794	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = 0)
36	30, 35	oveq12d 7293	. . . . . 6 ⊢ (𝑥 = 0 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = (0 + 0))
37		00id 11150	. . . . . 6 ⊢ (0 + 0) = 0
38	36, 37	eqtrdi 2794	. . . . 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 341	. . 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 7283	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (0..^𝑥) = (0..^𝑘))
42	41	ineq2d 4146	. . . . . . 7 ⊢ (𝑥 = 𝑘 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^𝑘)))
43	42	fveq2d 6778	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))))
44		fveq2 6774	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝐶‘𝑥) = (𝐶‘𝑘))
45	44	eleq2d 2824	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘𝑘)))
46		oveq2 7283	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (2↑𝑥) = (2↑𝑘))
47	45, 46	ifbieq1d 4483	. . . . . 6 ⊢ (𝑥 = 𝑘 → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘𝑘), (2↑𝑘), 0))
48	43, 47	oveq12d 7293	. . . . 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 4146	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^𝑘)))
50	49	fveq2d 6778	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^𝑘))))
51	41	ineq2d 4146	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^𝑘)))
52	51	fveq2d 6778	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^𝑘))))
53	50, 52	oveq12d 7293	. . . . 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 341	. . 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 7283	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (0..^𝑥) = (0..^(𝑘 + 1)))
57	56	ineq2d 4146	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1))))
58	57	fveq2d 6778	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))))
59		fveq2 6774	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝐶‘𝑥) = (𝐶‘(𝑘 + 1)))
60	59	eleq2d 2824	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘(𝑘 + 1))))
61		oveq2 7283	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (2↑𝑥) = (2↑(𝑘 + 1)))
62	60, 61	ifbieq1d 4483	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0))
63	58, 62	oveq12d 7293	. . . . 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 4146	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^(𝑘 + 1))))
65	64	fveq2d 6778	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))))
66	56	ineq2d 4146	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^(𝑘 + 1))))
67	66	fveq2d 6778	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1)))))
68	65, 67	oveq12d 7293	. . . . 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 341	. . 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 7283	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (0..^𝑥) = (0..^𝑁))
72	71	ineq2d 4146	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^𝑁)))
73	72	fveq2d 6778	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))))
74		fveq2 6774	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐶‘𝑥) = (𝐶‘𝑁))
75	74	eleq2d 2824	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘𝑁)))
76		oveq2 7283	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (2↑𝑥) = (2↑𝑁))
77	75, 76	ifbieq1d 4483	. . . . . 6 ⊢ (𝑥 = 𝑁 → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘𝑁), (2↑𝑁), 0))
78	73, 77	oveq12d 7293	. . . . 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 4146	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^𝑁)))
80	79	fveq2d 6778	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^𝑁))))
81	71	ineq2d 4146	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^𝑁)))
82	81	fveq2d 6778	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^𝑁))))
83	80, 82	oveq12d 7293	. . . . 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 341	. . 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 16161	. . . . . 6 ⊢ (𝜑 → ¬ ∅ ∈ (𝐶‘0))
90	89	iffalsed 4470	. . . . 5 ⊢ (𝜑 → if(∅ ∈ (𝐶‘0), (2↑0), 0) = 0)
91	90	oveq2d 7291	. . . 4 ⊢ (𝜑 → (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)) = (0 + 0))
92	91, 37	eqtrdi 2794	. . 3 ⊢ (𝜑 → (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)) = 0)
93	86	ad2antrr 723	. . . . . . 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 723	. . . . . . 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 766	. . . . . . 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 485	. . . . . . 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 16166	. . . . . 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 413	. . . . 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 414	. . . 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 12415	. 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 396 = wceq 1539 caddwcad 1608 ∈ wcel 2106 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 ifcif 4459 𝒫 cpw 4533 ↦ cmpt 5157 ^◡ccnv 5588 ↾ cres 5591 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 1_oc1o 8290 2_oc2o 8291 Fincfn 8733 0cc0 10871 1c1 10872 + caddc 10874 − cmin 11205 2c2 12028 ℕ₀cn0 12233 ..^cfzo 13382 seqcseq 13721 ↑cexp 13782 bitscbits 16126 sadd csad 16127

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-xor 1507 df-tru 1542 df-fal 1552 df-had 1595 df-cad 1609 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 df-dvds 15964 df-bits 16129 df-sad 16158

This theorem is referenced by: sadadd3 16168

W3C validator