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Theorem sadadd2 16674

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 7459	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (0..^𝑥) = (0..^0))
3		fzo0 13858	. . . . . . . . . . 11 ⊢ (0..^0) = ∅
4	2, 3	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (0..^𝑥) = ∅)
5	4	ineq2d 4120	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ ∅))
6		in0 4348	. . . . . . . . 9 ⊢ ((𝐴 sadd 𝐵) ∩ ∅) = ∅
7	5, 6	eqtrdi 2814	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ∅)
8	7	fveq2d 6920	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘∅))
9		sadcadd.k	. . . . . . . . 9 ⊢ 𝐾 = ^◡(bits ↾ ℕ₀)
10		0nn0 12661	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ₀
11		fvres 6935	. . . . . . . . . . 11 ⊢ (0 ∈ ℕ₀ → ((bits ↾ ℕ₀)‘0) = (bits‘0))
12	10, 11	ax-mp 5	. . . . . . . . . 10 ⊢ ((bits ↾ ℕ₀)‘0) = (bits‘0)
13		0bits 16653	. . . . . . . . . 10 ⊢ (bits‘0) = ∅
14	12, 13	eqtr2i 2787	. . . . . . . . 9 ⊢ ∅ = ((bits ↾ ℕ₀)‘0)
15	9, 14	fveq12i 6922	. . . . . . . 8 ⊢ (𝐾‘∅) = (^◡(bits ↾ ℕ₀)‘((bits ↾ ℕ₀)‘0))
16		bitsf1o 16659	. . . . . . . . 9 ⊢ (bits ↾ ℕ₀):ℕ₀⟶1-1onto(𝒫 ℕ₀ ∩ Fin)
17		f1ocnvfv1 7315	. . . . . . . . 9 ⊢ (((bits ↾ ℕ₀):ℕ₀⟶1-1onto(𝒫 ℕ₀ ∩ Fin) ∧ 0 ∈ ℕ₀) → (^◡(bits ↾ ℕ₀)‘((bits ↾ ℕ₀)‘0)) = 0)
18	16, 10, 17	mp2an 705	. . . . . . . 8 ⊢ (^◡(bits ↾ ℕ₀)‘((bits ↾ ℕ₀)‘0)) = 0
19	15, 18	eqtri 2786	. . . . . . 7 ⊢ (𝐾‘∅) = 0
20	8, 19	eqtrdi 2814	. . . . . 6 ⊢ (𝑥 = 0 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = 0)
21		fveq2 6916	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐶‘𝑥) = (𝐶‘0))
22	21	eleq2d 2850	. . . . . . 7 ⊢ (𝑥 = 0 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘0)))
23		oveq2 7459	. . . . . . 7 ⊢ (𝑥 = 0 → (2↑𝑥) = (2↑0))
24	22, 23	ifbieq1d 4510	. . . . . 6 ⊢ (𝑥 = 0 → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘0), (2↑0), 0))
25	20, 24	oveq12d 7469	. . . . 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 4120	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ ∅))
27		in0 4348	. . . . . . . . . 10 ⊢ (𝐴 ∩ ∅) = ∅
28	26, 27	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐴 ∩ (0..^𝑥)) = ∅)
29	28	fveq2d 6920	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘∅))
30	29, 19	eqtrdi 2814	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = 0)
31	4	ineq2d 4120	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ ∅))
32		in0 4348	. . . . . . . . . 10 ⊢ (𝐵 ∩ ∅) = ∅
33	31, 32	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐵 ∩ (0..^𝑥)) = ∅)
34	33	fveq2d 6920	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘∅))
35	34, 19	eqtrdi 2814	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = 0)
36	30, 35	oveq12d 7469	. . . . . 6 ⊢ (𝑥 = 0 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = (0 + 0))
37		00id 11524	. . . . . 6 ⊢ (0 + 0) = 0
38	36, 37	eqtrdi 2814	. . . . 5 ⊢ (𝑥 = 0 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = 0)
39	25, 38	eqeq12d 2779	. . . 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 343	. . 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 7459	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (0..^𝑥) = (0..^𝑘))
42	41	ineq2d 4120	. . . . . . 7 ⊢ (𝑥 = 𝑘 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^𝑘)))
43	42	fveq2d 6920	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))))
44		fveq2 6916	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝐶‘𝑥) = (𝐶‘𝑘))
45	44	eleq2d 2850	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘𝑘)))
46		oveq2 7459	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (2↑𝑥) = (2↑𝑘))
47	45, 46	ifbieq1d 4510	. . . . . 6 ⊢ (𝑥 = 𝑘 → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘𝑘), (2↑𝑘), 0))
48	43, 47	oveq12d 7469	. . . . 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 4120	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^𝑘)))
50	49	fveq2d 6920	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^𝑘))))
51	41	ineq2d 4120	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^𝑘)))
52	51	fveq2d 6920	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^𝑘))))
53	50, 52	oveq12d 7469	. . . . 5 ⊢ (𝑥 = 𝑘 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))
54	48, 53	eqeq12d 2779	. . . 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 343	. . 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 7459	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (0..^𝑥) = (0..^(𝑘 + 1)))
57	56	ineq2d 4120	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1))))
58	57	fveq2d 6920	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))))
59		fveq2 6916	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝐶‘𝑥) = (𝐶‘(𝑘 + 1)))
60	59	eleq2d 2850	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘(𝑘 + 1))))
61		oveq2 7459	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (2↑𝑥) = (2↑(𝑘 + 1)))
62	60, 61	ifbieq1d 4510	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0))
63	58, 62	oveq12d 7469	. . . . 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 4120	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^(𝑘 + 1))))
65	64	fveq2d 6920	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))))
66	56	ineq2d 4120	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^(𝑘 + 1))))
67	66	fveq2d 6920	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1)))))
68	65, 67	oveq12d 7469	. . . . 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 2779	. . . 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 343	. . 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 7459	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (0..^𝑥) = (0..^𝑁))
72	71	ineq2d 4120	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^𝑁)))
73	72	fveq2d 6920	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))))
74		fveq2 6916	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐶‘𝑥) = (𝐶‘𝑁))
75	74	eleq2d 2850	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘𝑁)))
76		oveq2 7459	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (2↑𝑥) = (2↑𝑁))
77	75, 76	ifbieq1d 4510	. . . . . 6 ⊢ (𝑥 = 𝑁 → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘𝑁), (2↑𝑁), 0))
78	73, 77	oveq12d 7469	. . . . 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 4120	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^𝑁)))
80	79	fveq2d 6920	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^𝑁))))
81	71	ineq2d 4120	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^𝑁)))
82	81	fveq2d 6920	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^𝑁))))
83	80, 82	oveq12d 7469	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
84	78, 83	eqeq12d 2779	. . . 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 343	. . 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 16668	. . . . . 6 ⊢ (𝜑 → ¬ ∅ ∈ (𝐶‘0))
90	89	iffalsed 4496	. . . . 5 ⊢ (𝜑 → if(∅ ∈ (𝐶‘0), (2↑0), 0) = 0)
91	90	oveq2d 7467	. . . 4 ⊢ (𝜑 → (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)) = (0 + 0))
92	91, 37	eqtrdi 2814	. . 3 ⊢ (𝜑 → (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)) = 0)
93	86	ad2antrr 739	. . . . . . 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 739	. . . . . . 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 781	. . . . . . 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 490	. . . . . . 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 16673	. . . . . 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 418	. . . . 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 419	. . . 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 30	. . 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 12834	. 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 39	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 caddwcad 1639 ∈ wcel 2147 ⊆ wss 3900 ∩ cin 4092 ∅c0 4281 ifcif 4485 𝒫 cpw 4560 ↦ cmpt 5192 ^◡ccnv 5657 ↾ cres 5660 ⟶1-1ontowf1o 6565 ‘cfv 6566 (class class class)co 7451 ∈ cmpo 7453 1_oc1o 8507 2_oc2o 8508 Fincfn 9011 0cc0 11239 1c1 11240 + caddc 11242 − cmin 11580 2c2 12437 ℕ₀cn0 12646 ..^cfzo 13828 seqcseq 14184 ↑cexp 14244 bitscbits 16633 sadd csad 16634

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

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-had 1624 df-cad 1640 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-disj 5076 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-oadd 8518 df-er 8755 df-map 8887 df-pm 8888 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-sup 9471 df-inf 9472 df-oi 9541 df-dju 10019 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-n0 12647 df-xnn0 12720 df-z 12734 df-uz 13006 df-rp 13161 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-hash 14516 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-clim 15699 df-sum 15898 df-dvds 16467 df-bits 16636 df-sad 16665

This theorem is used by: sadadd3 16675

W3C validator