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Theorem sadadd2 15020

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 6557	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (0..^𝑥) = (0..^0))
3		fzo0 12361	. . . . . . . . . . 11 ⊢ (0..^0) = ∅
4	2, 3	syl6eq 2660	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (0..^𝑥) = ∅)
5	4	ineq2d 3776	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ ∅))
6		in0 3920	. . . . . . . . 9 ⊢ ((𝐴 sadd 𝐵) ∩ ∅) = ∅
7	5, 6	syl6eq 2660	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ∅)
8	7	fveq2d 6107	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘∅))
9		sadcadd.k	. . . . . . . . 9 ⊢ 𝐾 = ^◡(bits ↾ ℕ₀)
10		0nn0 11184	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ₀
11		fvres 6117	. . . . . . . . . . 11 ⊢ (0 ∈ ℕ₀ → ((bits ↾ ℕ₀)‘0) = (bits‘0))
12	10, 11	ax-mp 5	. . . . . . . . . 10 ⊢ ((bits ↾ ℕ₀)‘0) = (bits‘0)
13		0bits 14999	. . . . . . . . . 10 ⊢ (bits‘0) = ∅
14	12, 13	eqtr2i 2633	. . . . . . . . 9 ⊢ ∅ = ((bits ↾ ℕ₀)‘0)
15	9, 14	fveq12i 6108	. . . . . . . 8 ⊢ (𝐾‘∅) = (^◡(bits ↾ ℕ₀)‘((bits ↾ ℕ₀)‘0))
16		bitsf1o 15005	. . . . . . . . 9 ⊢ (bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin)
17		f1ocnvfv1 6432	. . . . . . . . 9 ⊢ (((bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin) ∧ 0 ∈ ℕ₀) → (^◡(bits ↾ ℕ₀)‘((bits ↾ ℕ₀)‘0)) = 0)
18	16, 10, 17	mp2an 704	. . . . . . . 8 ⊢ (^◡(bits ↾ ℕ₀)‘((bits ↾ ℕ₀)‘0)) = 0
19	15, 18	eqtri 2632	. . . . . . 7 ⊢ (𝐾‘∅) = 0
20	8, 19	syl6eq 2660	. . . . . 6 ⊢ (𝑥 = 0 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = 0)
21		fveq2 6103	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐶‘𝑥) = (𝐶‘0))
22	21	eleq2d 2673	. . . . . . 7 ⊢ (𝑥 = 0 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘0)))
23		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = 0 → (2↑𝑥) = (2↑0))
24	22, 23	ifbieq1d 4059	. . . . . 6 ⊢ (𝑥 = 0 → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘0), (2↑0), 0))
25	20, 24	oveq12d 6567	. . . . 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 3776	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ ∅))
27		in0 3920	. . . . . . . . . 10 ⊢ (𝐴 ∩ ∅) = ∅
28	26, 27	syl6eq 2660	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐴 ∩ (0..^𝑥)) = ∅)
29	28	fveq2d 6107	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘∅))
30	29, 19	syl6eq 2660	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = 0)
31	4	ineq2d 3776	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ ∅))
32		in0 3920	. . . . . . . . . 10 ⊢ (𝐵 ∩ ∅) = ∅
33	31, 32	syl6eq 2660	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐵 ∩ (0..^𝑥)) = ∅)
34	33	fveq2d 6107	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘∅))
35	34, 19	syl6eq 2660	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = 0)
36	30, 35	oveq12d 6567	. . . . . 6 ⊢ (𝑥 = 0 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = (0 + 0))
37		00id 10090	. . . . . 6 ⊢ (0 + 0) = 0
38	36, 37	syl6eq 2660	. . . . 5 ⊢ (𝑥 = 0 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = 0)
39	25, 38	eqeq12d 2625	. . . 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 329	. . 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 6557	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (0..^𝑥) = (0..^𝑘))
42	41	ineq2d 3776	. . . . . . 7 ⊢ (𝑥 = 𝑘 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^𝑘)))
43	42	fveq2d 6107	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑘))))
44		fveq2 6103	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝐶‘𝑥) = (𝐶‘𝑘))
45	44	eleq2d 2673	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘𝑘)))
46		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (2↑𝑥) = (2↑𝑘))
47	45, 46	ifbieq1d 4059	. . . . . 6 ⊢ (𝑥 = 𝑘 → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘𝑘), (2↑𝑘), 0))
48	43, 47	oveq12d 6567	. . . . 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 3776	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^𝑘)))
50	49	fveq2d 6107	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^𝑘))))
51	41	ineq2d 3776	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^𝑘)))
52	51	fveq2d 6107	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^𝑘))))
53	50, 52	oveq12d 6567	. . . . 5 ⊢ (𝑥 = 𝑘 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = ((𝐾‘(𝐴 ∩ (0..^𝑘))) + (𝐾‘(𝐵 ∩ (0..^𝑘)))))
54	48, 53	eqeq12d 2625	. . . 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 329	. . 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 6557	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (0..^𝑥) = (0..^(𝑘 + 1)))
57	56	ineq2d 3776	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1))))
58	57	fveq2d 6107	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^(𝑘 + 1)))))
59		fveq2 6103	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝐶‘𝑥) = (𝐶‘(𝑘 + 1)))
60	59	eleq2d 2673	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘(𝑘 + 1))))
61		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (2↑𝑥) = (2↑(𝑘 + 1)))
62	60, 61	ifbieq1d 4059	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘(𝑘 + 1)), (2↑(𝑘 + 1)), 0))
63	58, 62	oveq12d 6567	. . . . 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 3776	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^(𝑘 + 1))))
65	64	fveq2d 6107	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^(𝑘 + 1)))))
66	56	ineq2d 3776	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^(𝑘 + 1))))
67	66	fveq2d 6107	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^(𝑘 + 1)))))
68	65, 67	oveq12d 6567	. . . . 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 2625	. . . 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 329	. . 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 6557	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (0..^𝑥) = (0..^𝑁))
72	71	ineq2d 3776	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝐴 sadd 𝐵) ∩ (0..^𝑥)) = ((𝐴 sadd 𝐵) ∩ (0..^𝑁)))
73	72	fveq2d 6107	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑥))) = (𝐾‘((𝐴 sadd 𝐵) ∩ (0..^𝑁))))
74		fveq2 6103	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐶‘𝑥) = (𝐶‘𝑁))
75	74	eleq2d 2673	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (∅ ∈ (𝐶‘𝑥) ↔ ∅ ∈ (𝐶‘𝑁)))
76		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (2↑𝑥) = (2↑𝑁))
77	75, 76	ifbieq1d 4059	. . . . . 6 ⊢ (𝑥 = 𝑁 → if(∅ ∈ (𝐶‘𝑥), (2↑𝑥), 0) = if(∅ ∈ (𝐶‘𝑁), (2↑𝑁), 0))
78	73, 77	oveq12d 6567	. . . . 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 3776	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐴 ∩ (0..^𝑥)) = (𝐴 ∩ (0..^𝑁)))
80	79	fveq2d 6107	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐾‘(𝐴 ∩ (0..^𝑥))) = (𝐾‘(𝐴 ∩ (0..^𝑁))))
81	71	ineq2d 3776	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐵 ∩ (0..^𝑥)) = (𝐵 ∩ (0..^𝑁)))
82	81	fveq2d 6107	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐾‘(𝐵 ∩ (0..^𝑥))) = (𝐾‘(𝐵 ∩ (0..^𝑁))))
83	80, 82	oveq12d 6567	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝐾‘(𝐴 ∩ (0..^𝑥))) + (𝐾‘(𝐵 ∩ (0..^𝑥)))) = ((𝐾‘(𝐴 ∩ (0..^𝑁))) + (𝐾‘(𝐵 ∩ (0..^𝑁)))))
84	78, 83	eqeq12d 2625	. . . 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 329	. . 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 15014	. . . . . 6 ⊢ (𝜑 → ¬ ∅ ∈ (𝐶‘0))
90	89	iffalsed 4047	. . . . 5 ⊢ (𝜑 → if(∅ ∈ (𝐶‘0), (2↑0), 0) = 0)
91	90	oveq2d 6565	. . . 4 ⊢ (𝜑 → (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)) = (0 + 0))
92	91, 37	syl6eq 2660	. . 3 ⊢ (𝜑 → (0 + if(∅ ∈ (𝐶‘0), (2↑0), 0)) = 0)
93	86	ad2antrr 758	. . . . . . 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 758	. . . . . . 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 788	. . . . . . 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 476	. . . . . . 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 15019	. . . . . 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 449	. . . . 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 450	. . . 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 11348	. 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 37	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 383 = wceq 1475 caddwcad 1536 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ifcif 4036 𝒫 cpw 4108 ↦ cmpt 4643 ^◡ccnv 5037 ↾ cres 5040 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1_𝑜c1o 7440 2_𝑜c2o 7441 Fincfn 7841 0cc0 9815 1c1 9816 + caddc 9818 − cmin 10145 2c2 10947 ℕ₀cn0 11169 ..^cfzo 12334 seqcseq 12663 ↑cexp 12722 bitscbits 14979 sadd csad 14980

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-xor 1457 df-tru 1478 df-fal 1481 df-had 1524 df-cad 1537 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-dvds 14822 df-bits 14982 df-sad 15011

This theorem is referenced by: sadadd3 15021

W3C validator