sadadd2 - Metamath Proof Explorer

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Theorem sadadd2 15812

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 7167	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (0..^𝑥) = (0..^0))
3		fzo0 13064	. . . . . . . . . . 11 ⊢ (0..^0) = ∅
4	2, 3	syl6eq 2875	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (0..^𝑥) = ∅)
5	4	ineq2d 4192	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ ∅))
6		in0 4348	. . . . . . . . 9 ⊢ ((𝐴 sadd 𝐵) ∩ ∅) = ∅
7	5, 6	syl6eq 2875	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ∅)
8	7	fveq2d 6677	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘∅))
9		sadcadd.k	. . . . . . . . 9 ⊢ 𝐾 = ^◡(bits ↾ ℕ₀)
10		0nn0 11915	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ₀
11		fvres 6692	. . . . . . . . . . 11 ⊢ (0 ∈ ℕ₀ → ((bits ↾ ℕ₀)‘0) = (bits‘0))
12	10, 11	ax-mp 5	. . . . . . . . . 10 ⊢ ((bits ↾ ℕ₀)‘0) = (bits‘0)
13		0bits 15791	. . . . . . . . . 10 ⊢ (bits‘0) = ∅
14	12, 13	eqtr2i 2848	. . . . . . . . 9 ⊢ ∅ = ((bits ↾ ℕ₀)‘0)
15	9, 14	fveq12i 6679	. . . . . . . 8 ⊢ (𝐾‘∅) = (^◡(bits ↾ ℕ₀)‘((bits ↾ ℕ₀)‘0))
16		bitsf1o 15797	. . . . . . . . 9 ⊢ (bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin)
17		f1ocnvfv1 7036	. . . . . . . . 9 ⊢ (((bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin) ∧ 0 ∈ ℕ₀) → (^◡(bits ↾ ℕ₀)‘((bits ↾ ℕ₀)‘0)) = 0)
18	16, 10, 17	mp2an 690	. . . . . . . 8 ⊢ (^◡(bits ↾ ℕ₀)‘((bits ↾ ℕ₀)‘0)) = 0
19	15, 18	eqtri 2847	. . . . . . 7 ⊢ (𝐾‘∅) = 0
20	8, 19	syl6eq 2875	. . . . . 6 ⊢ (𝑥 = 0 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = 0)
21		fveq2 6673	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐶‘𝑥) = (𝐶‘0))
22	21	eleq2d 2901	. . . . . . 7 ⊢ (𝑥 = 0 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘0)))
23		oveq2 7167	. . . . . . 7 ⊢ (𝑥 = 0 → (2↑𝑥) = (2↑0))
24	22, 23	ifbieq1d 4493	. . . . . 6 ⊢ (𝑥 = 0 → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘0), (2↑0), 0))
25	20, 24	oveq12d 7177	. . . . 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 4192	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ ∅))
27		in0 4348	. . . . . . . . . 10 ⊢ (𝐴 ∩ ∅) = ∅
28	26, 27	syl6eq 2875	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐴 ∩ (0..^𝑥)) = ∅)
29	28	fveq2d 6677	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘∅))
30	29, 19	syl6eq 2875	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = 0)
31	4	ineq2d 4192	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ ∅))
32		in0 4348	. . . . . . . . . 10 ⊢ (𝐵 ∩ ∅) = ∅
33	31, 32	syl6eq 2875	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐵 ∩ (0..^𝑥)) = ∅)
34	33	fveq2d 6677	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘∅))
35	34, 19	syl6eq 2875	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = 0)
36	30, 35	oveq12d 7177	. . . . . 6 ⊢ (𝑥 = 0 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = (0 + 0))
37		00id 10818	. . . . . 6 ⊢ (0 + 0) = 0
38	36, 37	syl6eq 2875	. . . . 5 ⊢ (𝑥 = 0 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = 0)
39	25, 38	eqeq12d 2840	. . . 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 7167	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (0..^𝑥) = (0..^𝑘))
42	41	ineq2d 4192	. . . . . . 7 ⊢ (𝑥 = 𝑘 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^𝑘)))
43	42	fveq2d 6677	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))))
44		fveq2 6673	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝐶‘𝑥) = (𝐶‘𝑘))
45	44	eleq2d 2901	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘𝑘)))
46		oveq2 7167	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (2↑𝑥) = (2↑𝑘))
47	45, 46	ifbieq1d 4493	. . . . . 6 ⊢ (𝑥 = 𝑘 → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘𝑘), (2↑𝑘), 0))
48	43, 47	oveq12d 7177	. . . . 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 4192	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^𝑘)))
50	49	fveq2d 6677	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^𝑘))))
51	41	ineq2d 4192	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^𝑘)))
52	51	fveq2d 6677	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^𝑘))))
53	50, 52	oveq12d 7177	. . . . 5 ⊢ (𝑥 = 𝑘 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))
54	48, 53	eqeq12d 2840	. . . 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 7167	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (0..^𝑥) = (0..^(𝑘 + 1)))
57	56	ineq2d 4192	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1))))
58	57	fveq2d 6677	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))))
59		fveq2 6673	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝐶‘𝑥) = (𝐶‘(𝑘 + 1)))
60	59	eleq2d 2901	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘(𝑘 + 1))))
61		oveq2 7167	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (2↑𝑥) = (2↑(𝑘 + 1)))
62	60, 61	ifbieq1d 4493	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0))
63	58, 62	oveq12d 7177	. . . . 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 4192	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^(𝑘 + 1))))
65	64	fveq2d 6677	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))))
66	56	ineq2d 4192	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^(𝑘 + 1))))
67	66	fveq2d 6677	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1)))))
68	65, 67	oveq12d 7177	. . . . 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 2840	. . . 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 7167	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (0..^𝑥) = (0..^𝑁))
72	71	ineq2d 4192	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^𝑁)))
73	72	fveq2d 6677	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))))
74		fveq2 6673	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐶‘𝑥) = (𝐶‘𝑁))
75	74	eleq2d 2901	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘𝑁)))
76		oveq2 7167	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (2↑𝑥) = (2↑𝑁))
77	75, 76	ifbieq1d 4493	. . . . . 6 ⊢ (𝑥 = 𝑁 → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘𝑁), (2↑𝑁), 0))
78	73, 77	oveq12d 7177	. . . . 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 4192	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^𝑁)))
80	79	fveq2d 6677	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^𝑁))))
81	71	ineq2d 4192	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^𝑁)))
82	81	fveq2d 6677	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^𝑁))))
83	80, 82	oveq12d 7177	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
84	78, 83	eqeq12d 2840	. . . 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 15806	. . . . . 6 ⊢ (𝜑 → ¬ ∅ ∈ (𝐶‘0))
90	89	iffalsed 4481	. . . . 5 ⊢ (𝜑 → if(∅ ∈ (𝐶‘0), (2↑0), 0) = 0)
91	90	oveq2d 7175	. . . 4 ⊢ (𝜑 → (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)) = (0 + 0))
92	91, 37	syl6eq 2875	. . 3 ⊢ (𝜑 → (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)) = 0)
93	86	ad2antrr 724	. . . . . . 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 724	. . . . . . 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 767	. . . . . . 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 487	. . . . . . 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 15811	. . . . . 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 415	. . . . 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 416	. . . 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 12080	. 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 398 = wceq 1536 caddwcad 1606 ∈ wcel 2113 ∩ cin 3938 ⊆ wss 3939 ∅c0 4294 ifcif 4470 𝒫 cpw 4542 ↦ cmpt 5149 ^◡ccnv 5557 ↾ cres 5560 –1-1-onto→wf1o 6357 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 1_oc1o 8098 2_oc2o 8099 Fincfn 8512 0cc0 10540 1c1 10541 + caddc 10543 − cmin 10873 2c2 11695 ℕ₀cn0 11900 ..^cfzo 13036 seqcseq 13372 ↑cexp 13432 bitscbits 15771 sadd csad 15772

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1502 df-tru 1539 df-fal 1549 df-had 1593 df-cad 1607 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-disj 5035 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-inf 8910 df-oi 8977 df-dju 9333 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-sum 15046 df-dvds 15611 df-bits 15774 df-sad 15803

This theorem is referenced by: sadadd3 15813

W3C validator