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Theorem smueqlem 16401

Description: Any element of a sequence multiplication only depends on the values of the argument sequences up to and including that point. (Contributed by Mario Carneiro, 20-Sep-2016.)

**Hypotheses**
Ref	Expression
smueq.a	⊢ (𝜑 → 𝐴 ⊆ ℕ₀)
smueq.b	⊢ (𝜑 → 𝐵 ⊆ ℕ₀)
smueq.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
smueq.p	⊢ 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ₀, 𝑚 ∈ ℕ₀ ↦ (𝑝 sadd {𝑛 ∈ ℕ₀ ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
smueq.q	⊢ 𝑄 = seq0((𝑝 ∈ 𝒫 ℕ₀, 𝑚 ∈ ℕ₀ ↦ (𝑝 sadd {𝑛 ∈ ℕ₀ ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ (𝐵 ∩ (0..^𝑁)))})), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))

**Assertion**
Ref	Expression
smueqlem	⊢ (𝜑 → ((𝐴 smul 𝐵) ∩ (0..^𝑁)) = (((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∩ (0..^𝑁)))

Distinct variable groups: 𝑚,𝑛,𝑝,𝐴 𝐵,𝑚,𝑛,𝑝 𝑚,𝑁,𝑛,𝑝 𝜑,𝑛 Allowed substitution hints: 𝜑(𝑚,𝑝) 𝑃(𝑚,𝑛,𝑝) 𝑄(𝑚,𝑛,𝑝)

Proof of Theorem smueqlem Dummy variables 𝑘 𝑖 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smueq.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℕ₀)
2	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝐴 ⊆ ℕ₀)
3		smueq.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ ℕ₀)
4	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ₀)
5		smueq.p	. . . . . . 7 ⊢ 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ₀, 𝑚 ∈ ℕ₀ ↦ (𝑝 sadd {𝑛 ∈ ℕ₀ ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
6		elfzouz 13566	. . . . . . . . 9 ⊢ (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ (ℤ_≥‘0))
7	6	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ (ℤ_≥‘0))
8		nn0uz 12777	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
9	7, 8	eleqtrrdi 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℕ₀)
10	9	nn0zd 12497	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℤ)
11	10	peano2zd 12583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ∈ ℤ)
12		smueq.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
13	12	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ₀)
14	13	nn0zd 12497	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
15		elfzolt2 13571	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^𝑁) → 𝑘 < 𝑁)
16	15	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 < 𝑁)
17		nn0ltp1le 12534	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑘 < 𝑁 ↔ (𝑘 + 1) ≤ 𝑁))
18	9, 13, 17	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 < 𝑁 ↔ (𝑘 + 1) ≤ 𝑁))
19	16, 18	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
20		eluz2 12741	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑘 + 1)) ↔ ((𝑘 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑁))
21	11, 14, 19, 20	syl3anbrc 1344	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ (ℤ_≥‘(𝑘 + 1)))
22	2, 4, 5, 9, 21	smuval2 16393	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ (𝑃‘𝑁)))
23	12, 8	eleqtrdi 2838	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
24		eluzfz2b 13436	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘0) ↔ 𝑁 ∈ (0...𝑁))
25	23, 24	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
26		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → (𝑃‘𝑥) = (𝑃‘0))
27	26	ineq1d 4170	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘0) ∩ (0..^𝑁)))
28		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → (𝑄‘𝑥) = (𝑄‘0))
29	28	ineq1d 4170	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁)))
30	27, 29	eqeq12d 2745	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘0) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁))))
31	30	imbi2d 340	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → ((𝜑 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃‘0) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁)))))
32		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑖 → (𝑃‘𝑥) = (𝑃‘𝑖))
33	32	ineq1d 4170	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑖 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘𝑖) ∩ (0..^𝑁)))
34		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑖 → (𝑄‘𝑥) = (𝑄‘𝑖))
35	34	ineq1d 4170	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑖 → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)))
36	33, 35	eqeq12d 2745	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑖 → (((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁))))
37	36	imbi2d 340	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑖 → ((𝜑 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)))))
38		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑖 + 1) → (𝑃‘𝑥) = (𝑃‘(𝑖 + 1)))
39	38	ineq1d 4170	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑖 + 1) → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)))
40		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑖 + 1) → (𝑄‘𝑥) = (𝑄‘(𝑖 + 1)))
41	40	ineq1d 4170	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑖 + 1) → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))
42	39, 41	eqeq12d 2745	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑖 + 1) → (((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁))))
43	42	imbi2d 340	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑖 + 1) → ((𝜑 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))))
44		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑁 → (𝑃‘𝑥) = (𝑃‘𝑁))
45	44	ineq1d 4170	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑁 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘𝑁) ∩ (0..^𝑁)))
46		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑁 → (𝑄‘𝑥) = (𝑄‘𝑁))
47	46	ineq1d 4170	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑁 → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁)))
48	45, 47	eqeq12d 2745	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑁 → (((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁))))
49	48	imbi2d 340	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑁 → ((𝜑 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁)))))
50	1, 3, 5	smup0 16390	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃‘0) = ∅)
51		inss1 4188	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∩ (0..^𝑁)) ⊆ 𝐵
52	51, 3	sstrid 3947	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ₀)
53		smueq.q	. . . . . . . . . . . . . . 15 ⊢ 𝑄 = seq0((𝑝 ∈ 𝒫 ℕ₀, 𝑚 ∈ ℕ₀ ↦ (𝑝 sadd {𝑛 ∈ ℕ₀ ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ (𝐵 ∩ (0..^𝑁)))})), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
54	1, 52, 53	smup0 16390	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄‘0) = ∅)
55	50, 54	eqtr4d 2767	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃‘0) = (𝑄‘0))
56	55	ineq1d 4170	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑃‘0) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁)))
57	56	a1i 11	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘0) → (𝜑 → ((𝑃‘0) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁))))
58		oveq1 7356	. . . . . . . . . . . . . . 15 ⊢ (((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)) → (((𝑃‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) = (((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))))
59	58	ineq1d 4170	. . . . . . . . . . . . . 14 ⊢ (((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)) → ((((𝑃‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)) = ((((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
60	1	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝐴 ⊆ ℕ₀)
61	3	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ₀)
62		elfzonn0 13610	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ₀)
63	62	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ₀)
64	60, 61, 5, 63	smupp1 16391	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑃‘(𝑖 + 1)) = ((𝑃‘𝑖) sadd {𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)}))
65	64	ineq1d 4170	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = (((𝑃‘𝑖) sadd {𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)}) ∩ (0..^𝑁)))
66	1, 3, 5	smupf 16389	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃:ℕ₀⟶𝒫 ℕ₀)
67		ffvelcdm 7015	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:ℕ₀⟶𝒫 ℕ₀ ∧ 𝑖 ∈ ℕ₀) → (𝑃‘𝑖) ∈ 𝒫 ℕ₀)
68	66, 62, 67	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑃‘𝑖) ∈ 𝒫 ℕ₀)
69	68	elpwid 4560	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑃‘𝑖) ⊆ ℕ₀)
70		ssrab2 4031	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ⊆ ℕ₀
71	70	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → {𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ⊆ ℕ₀)
72	12	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ₀)
73	69, 71, 72	sadeq 16383	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (((𝑃‘𝑖) sadd {𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)}) ∩ (0..^𝑁)) = ((((𝑃‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
74	65, 73	eqtrd 2764	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((((𝑃‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
75	52	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ₀)
76	60, 75, 53, 63	smupp1 16391	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑄‘(𝑖 + 1)) = ((𝑄‘𝑖) sadd {𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}))
77	76	ineq1d 4170	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)) = (((𝑄‘𝑖) sadd {𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}) ∩ (0..^𝑁)))
78	1, 52, 53	smupf 16389	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑄:ℕ₀⟶𝒫 ℕ₀)
79		ffvelcdm 7015	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑄:ℕ₀⟶𝒫 ℕ₀ ∧ 𝑖 ∈ ℕ₀) → (𝑄‘𝑖) ∈ 𝒫 ℕ₀)
80	78, 62, 79	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑄‘𝑖) ∈ 𝒫 ℕ₀)
81	80	elpwid 4560	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑄‘𝑖) ⊆ ℕ₀)
82		ssrab2 4031	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ⊆ ℕ₀
83	82	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → {𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ⊆ ℕ₀)
84	81, 83, 72	sadeq 16383	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (((𝑄‘𝑖) sadd {𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}) ∩ (0..^𝑁)) = ((((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
85		elinel2 4153	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (ℕ₀ ∩ (0..^𝑁)) → 𝑛 ∈ (0..^𝑁))
86	61	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ₀)
87	86	sseld 3934	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ 𝐵 → (𝑛 − 𝑖) ∈ ℕ₀))
88		elfzo0 13603	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 ∈ (0..^𝑁) ↔ (𝑛 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 𝑛 < 𝑁))
89	88	simp2bi 1146	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ (0..^𝑁) → 𝑁 ∈ ℕ)
90	89	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ)
91		elfzonn0 13610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ ℕ₀)
92	91	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℕ₀)
93	92	nn0red 12446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℝ)
94	63	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ₀)
95	94	nn0red 12446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑖 ∈ ℝ)
96	93, 95	resubcld 11548	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛 − 𝑖) ∈ ℝ)
97	90	nnred 12143	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑁 ∈ ℝ)
98	94	nn0ge0d 12448	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 0 ≤ 𝑖)
99	93, 95	subge02d 11712	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (0 ≤ 𝑖 ↔ (𝑛 − 𝑖) ≤ 𝑛))
100	98, 99	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛 − 𝑖) ≤ 𝑛)
101		elfzolt2 13571	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 ∈ (0..^𝑁) → 𝑛 < 𝑁)
102	101	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 < 𝑁)
103	96, 93, 97, 100, 102	lelttrd 11274	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛 − 𝑖) < 𝑁)
104	90, 103	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁))
105		elfzo0 13603	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑛 − 𝑖) ∈ (0..^𝑁) ↔ ((𝑛 − 𝑖) ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁))
106		3anass 1094	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑛 − 𝑖) ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁) ↔ ((𝑛 − 𝑖) ∈ ℕ₀ ∧ (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁)))
107	105, 106	bitri 275	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑛 − 𝑖) ∈ (0..^𝑁) ↔ ((𝑛 − 𝑖) ∈ ℕ₀ ∧ (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁)))
108	107	baib 535	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑛 − 𝑖) ∈ ℕ₀ → ((𝑛 − 𝑖) ∈ (0..^𝑁) ↔ (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁)))
109	104, 108	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ ℕ₀ → (𝑛 − 𝑖) ∈ (0..^𝑁)))
110	87, 109	syld 47	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ 𝐵 → (𝑛 − 𝑖) ∈ (0..^𝑁)))
111	110	pm4.71rd 562	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ 𝐵 ↔ ((𝑛 − 𝑖) ∈ (0..^𝑁) ∧ (𝑛 − 𝑖) ∈ 𝐵)))
112		ancom 460	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 − 𝑖) ∈ (0..^𝑁) ∧ (𝑛 − 𝑖) ∈ 𝐵) ↔ ((𝑛 − 𝑖) ∈ 𝐵 ∧ (𝑛 − 𝑖) ∈ (0..^𝑁)))
113		elin 3919	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)) ↔ ((𝑛 − 𝑖) ∈ 𝐵 ∧ (𝑛 − 𝑖) ∈ (0..^𝑁)))
114	112, 113	bitr4i 278	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 − 𝑖) ∈ (0..^𝑁) ∧ (𝑛 − 𝑖) ∈ 𝐵) ↔ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))
115	111, 114	bitr2di 288	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)) ↔ (𝑛 − 𝑖) ∈ 𝐵))
116	115	anbi2d 630	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁))) ↔ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)))
117	85, 116	sylan2 593	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (ℕ₀ ∩ (0..^𝑁))) → ((𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁))) ↔ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)))
118	117	rabbidva 3401	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → {𝑛 ∈ (ℕ₀ ∩ (0..^𝑁)) ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} = {𝑛 ∈ (ℕ₀ ∩ (0..^𝑁)) ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)})
119		inrab2 4268	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ∩ (0..^𝑁)) = {𝑛 ∈ (ℕ₀ ∩ (0..^𝑁)) ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}
120		inrab2 4268	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁)) = {𝑛 ∈ (ℕ₀ ∩ (0..^𝑁)) ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)}
121	118, 119, 120	3eqtr4g 2789	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ∩ (0..^𝑁)) = ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁)))
122	121	oveq2d 7365	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ∩ (0..^𝑁))) = (((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))))
123	122	ineq1d 4170	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ∩ (0..^𝑁))) ∩ (0..^𝑁)) = ((((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
124	77, 84, 123	3eqtrd 2768	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
125	74, 124	eqeq12d 2745	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)) ↔ ((((𝑃‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)) = ((((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁))))
126	59, 125	imbitrrid 246	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁))))
127	126	expcom 413	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^𝑁) → (𝜑 → (((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))))
128	127	a2d 29	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^𝑁) → ((𝜑 → ((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁))) → (𝜑 → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))))
129	31, 37, 43, 49, 57, 128	fzind2 13688	. . . . . . . . . 10 ⊢ (𝑁 ∈ (0...𝑁) → (𝜑 → ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁))))
130	25, 129	mpcom 38	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁)))
131	130	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁)))
132	131	eleq2d 2814	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁))))
133		elin 3919	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝑃‘𝑁) ∧ 𝑘 ∈ (0..^𝑁)))
134	133	rbaib 538	. . . . . . . 8 ⊢ (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑃‘𝑁)))
135	134	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑃‘𝑁)))
136		elin 3919	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝑄‘𝑁) ∧ 𝑘 ∈ (0..^𝑁)))
137	136	rbaib 538	. . . . . . . 8 ⊢ (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑄‘𝑁)))
138	137	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑄‘𝑁)))
139	132, 135, 138	3bitr3d 309	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝑃‘𝑁) ↔ 𝑘 ∈ (𝑄‘𝑁)))
140	52	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ₀)
141	2, 140, 53, 13	smupval 16399	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑄‘𝑁) = ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))))
142	141	eleq2d 2814	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝑄‘𝑁) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁)))))
143	22, 139, 142	3bitrd 305	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁)))))
144	143	ex 412	. . . 4 ⊢ (𝜑 → (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))))))
145	144	pm5.32rd 578	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 smul 𝐵) ∧ 𝑘 ∈ (0..^𝑁)) ↔ (𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∧ 𝑘 ∈ (0..^𝑁))))
146		elin 3919	. . 3 ⊢ (𝑘 ∈ ((𝐴 smul 𝐵) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝐴 smul 𝐵) ∧ 𝑘 ∈ (0..^𝑁)))
147		elin 3919	. . 3 ⊢ (𝑘 ∈ (((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∩ (0..^𝑁)) ↔ (𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∧ 𝑘 ∈ (0..^𝑁)))
148	145, 146, 147	3bitr4g 314	. 2 ⊢ (𝜑 → (𝑘 ∈ ((𝐴 smul 𝐵) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∩ (0..^𝑁))))
149	148	eqrdv 2727	1 ⊢ (𝜑 → ((𝐴 smul 𝐵) ∩ (0..^𝑁)) = (((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∩ (0..^𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 {crab 3394 ∩ cin 3902 ⊆ wss 3903 ∅c0 4284 ifcif 4476 𝒫 cpw 4551 class class class wbr 5092 ↦ cmpt 5173 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ∈ cmpo 7351 0cc0 11009 1c1 11010 + caddc 11012 < clt 11149 ≤ cle 11150 − cmin 11347 ℕcn 12128 ℕ₀cn0 12384 ℤcz 12471 ℤ_≥cuz 12735 ...cfz 13410 ..^cfzo 13557 seqcseq 13908 sadd csad 16331 smul csmu 16332

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 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1512 df-tru 1543 df-fal 1553 df-had 1594 df-cad 1607 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-disj 5060 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-er 8625 df-map 8755 df-pm 8756 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9797 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-n0 12385 df-xnn0 12458 df-z 12472 df-uz 12736 df-rp 12894 df-fz 13411 df-fzo 13558 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-dvds 16164 df-bits 16333 df-sad 16362 df-smu 16387

This theorem is referenced by: smueq 16402

W3C validator