smueqlem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > smueqlem		Structured version Visualization version GIF version

Theorem smueqlem 16572

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 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝐴 ⊆ ℕ₀)
3		smueq.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ ℕ₀)
4	3	adantr 486	. . . . . . 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 13711	. . . . . . . . 9 ⊢ (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ (ℤ_≥‘0))
7	6	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ (ℤ_≥‘0))
8		nn0uz 12918	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
9	7, 8	eleqtrrdi 2876	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℕ₀)
10	9	nn0zd 12633	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℤ)
11	10	peano2zd 12721	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ∈ ℤ)
12		smueq.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
13	12	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ₀)
14	13	nn0zd 12633	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
15		elfzolt2 13716	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^𝑁) → 𝑘 < 𝑁)
16	15	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 < 𝑁)
17		nn0ltp1le 12672	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑘 < 𝑁 ↔ (𝑘 + 1) ≤ 𝑁))
18	9, 13, 17	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 < 𝑁 ↔ (𝑘 + 1) ≤ 𝑁))
19	16, 18	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
20		eluz2 12886	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑘 + 1)) ↔ ((𝑘 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑁))
21	11, 14, 19, 20	syl3anbrc 1362	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ (ℤ_≥‘(𝑘 + 1)))
22	2, 4, 5, 9, 21	smuval2 16564	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ (𝑃‘𝑁)))
23	12, 8	eleqtrdi 2875	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
24		eluzfz2b 13579	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘0) ↔ 𝑁 ∈ (0...𝑁))
25	23, 24	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
26		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → (𝑃‘𝑥) = (𝑃‘0))
27	26	ineq1d 4172	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘0) ∩ (0..^𝑁)))
28		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → (𝑄‘𝑥) = (𝑄‘0))
29	28	ineq1d 4172	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁)))
30	27, 29	eqeq12d 2781	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘0) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁))))
31	30	imbi2d 343	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → ((𝜑 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃‘0) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁)))))
32		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑖 → (𝑃‘𝑥) = (𝑃‘𝑖))
33	32	ineq1d 4172	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑖 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘𝑖) ∩ (0..^𝑁)))
34		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑖 → (𝑄‘𝑥) = (𝑄‘𝑖))
35	34	ineq1d 4172	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑖 → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)))
36	33, 35	eqeq12d 2781	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑖 → (((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁))))
37	36	imbi2d 343	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑖 → ((𝜑 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)))))
38		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑖 + 1) → (𝑃‘𝑥) = (𝑃‘(𝑖 + 1)))
39	38	ineq1d 4172	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑖 + 1) → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)))
40		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑖 + 1) → (𝑄‘𝑥) = (𝑄‘(𝑖 + 1)))
41	40	ineq1d 4172	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑖 + 1) → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))
42	39, 41	eqeq12d 2781	. . . . . . . . . . . 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 343	. . . . . . . . . . 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 6885	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑁 → (𝑃‘𝑥) = (𝑃‘𝑁))
45	44	ineq1d 4172	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑁 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘𝑁) ∩ (0..^𝑁)))
46		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑁 → (𝑄‘𝑥) = (𝑄‘𝑁))
47	46	ineq1d 4172	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑁 → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁)))
48	45, 47	eqeq12d 2781	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑁 → (((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁))))
49	48	imbi2d 343	. . . . . . . . . . 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 16561	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃‘0) = ∅)
51		inss1 4189	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∩ (0..^𝑁)) ⊆ 𝐵
52	51, 3	sstrid 3949	. . . . . . . . . . . . . . 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 16561	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄‘0) = ∅)
55	50, 54	eqtr4d 2803	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃‘0) = (𝑄‘0))
56	55	ineq1d 4172	. . . . . . . . . . . 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 7426	. . . . . . . . . . . . . . 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 4172	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝐴 ⊆ ℕ₀)
61	3	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ₀)
62		elfzonn0 13755	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ₀)
63	62	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ₀)
64	60, 61, 5, 63	smupp1 16562	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑃‘(𝑖 + 1)) = ((𝑃‘𝑖) sadd {𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)}))
65	64	ineq1d 4172	. . . . . . . . . . . . . . . 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 16560	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃:ℕ₀⟶𝒫 ℕ₀)
67		ffvelcdm 7080	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:ℕ₀⟶𝒫 ℕ₀ ∧ 𝑖 ∈ ℕ₀) → (𝑃‘𝑖) ∈ 𝒫 ℕ₀)
68	66, 62, 67	syl2an 608	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑃‘𝑖) ∈ 𝒫 ℕ₀)
69	68	elpwid 4573	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑃‘𝑖) ⊆ ℕ₀)
70		ssrab2 4035	. . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ₀)
73	69, 71, 72	sadeq 16554	. . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ₀)
76	60, 75, 53, 63	smupp1 16562	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑄‘(𝑖 + 1)) = ((𝑄‘𝑖) sadd {𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}))
77	76	ineq1d 4172	. . . . . . . . . . . . . . . 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 16560	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑄:ℕ₀⟶𝒫 ℕ₀)
79		ffvelcdm 7080	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑄:ℕ₀⟶𝒫 ℕ₀ ∧ 𝑖 ∈ ℕ₀) → (𝑄‘𝑖) ∈ 𝒫 ℕ₀)
80	78, 62, 79	syl2an 608	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑄‘𝑖) ∈ 𝒫 ℕ₀)
81	80	elpwid 4573	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑄‘𝑖) ⊆ ℕ₀)
82		ssrab2 4035	. . . . . . . . . . . . . . . . . 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 16554	. . . . . . . . . . . . . . . 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 4155	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (ℕ₀ ∩ (0..^𝑁)) → 𝑛 ∈ (0..^𝑁))
86	61	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ₀)
87	86	sseld 3937	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ 𝐵 → (𝑛 − 𝑖) ∈ ℕ₀))
88		elfzo0 13748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 ∈ (0..^𝑁) ↔ (𝑛 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 𝑛 < 𝑁))
89	88	simp2bi 1164	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ (0..^𝑁) → 𝑁 ∈ ℕ)
90	89	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ)
91		elfzonn0 13755	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ ℕ₀)
92	91	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℕ₀)
93	92	nn0red 12583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℝ)
94	63	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ₀)
95	94	nn0red 12583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑖 ∈ ℝ)
96	93, 95	resubcld 11659	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛 − 𝑖) ∈ ℝ)
97	90	nnred 12265	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑁 ∈ ℝ)
98	94	nn0ge0d 12585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 0 ≤ 𝑖)
99	93, 95	subge02d 11823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (0 ≤ 𝑖 ↔ (𝑛 − 𝑖) ≤ 𝑛))
100	98, 99	mpbid 235	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛 − 𝑖) ≤ 𝑛)
101		elfzolt2 13716	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 ∈ (0..^𝑁) → 𝑛 < 𝑁)
102	101	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 < 𝑁)
103	96, 93, 97, 100, 102	lelttrd 11385	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛 − 𝑖) < 𝑁)
104	90, 103	jca 521	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁))
105		elfzo0 13748	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑛 − 𝑖) ∈ (0..^𝑁) ↔ ((𝑛 − 𝑖) ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁))
106		3anass 1111	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑛 − 𝑖) ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁) ↔ ((𝑛 − 𝑖) ∈ ℕ₀ ∧ (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁)))
107	105, 106	bitri 278	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑛 − 𝑖) ∈ (0..^𝑁) ↔ ((𝑛 − 𝑖) ∈ ℕ₀ ∧ (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁)))
108	107	baib 545	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑛 − 𝑖) ∈ ℕ₀ → ((𝑛 − 𝑖) ∈ (0..^𝑁) ↔ (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁)))
109	104, 108	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ ℕ₀ → (𝑛 − 𝑖) ∈ (0..^𝑁)))
110	87, 109	syld 48	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ 𝐵 → (𝑛 − 𝑖) ∈ (0..^𝑁)))
111	110	pm4.71rd 572	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ 𝐵 ↔ ((𝑛 − 𝑖) ∈ (0..^𝑁) ∧ (𝑛 − 𝑖) ∈ 𝐵)))
112		ancom 466	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 − 𝑖) ∈ (0..^𝑁) ∧ (𝑛 − 𝑖) ∈ 𝐵) ↔ ((𝑛 − 𝑖) ∈ 𝐵 ∧ (𝑛 − 𝑖) ∈ (0..^𝑁)))
113		elin 3922	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)) ↔ ((𝑛 − 𝑖) ∈ 𝐵 ∧ (𝑛 − 𝑖) ∈ (0..^𝑁)))
114	112, 113	bitr4i 281	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 − 𝑖) ∈ (0..^𝑁) ∧ (𝑛 − 𝑖) ∈ 𝐵) ↔ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))
115	111, 114	bitr2di 291	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)) ↔ (𝑛 − 𝑖) ∈ 𝐵))
116	115	anbi2d 642	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁))) ↔ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)))
117	85, 116	sylan2 605	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (ℕ₀ ∩ (0..^𝑁))) → ((𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁))) ↔ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)))
118	117	rabbidva 3424	. . . . . . . . . . . . . . . . . . 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 4270	. . . . . . . . . . . . . . . . . . 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 4270	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑛 ∈ ℕ₀ ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁)) = {𝑛 ∈ (ℕ₀ ∩ (0..^𝑁)) ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)}
121	118, 119, 120	3eqtr4g 2825	. . . . . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . . . . 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 4172	. . . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . 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 249	. . . . . . . . . . . . 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 419	. . . . . . . . . . . 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 30	. . . . . . . . . . 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 13836	. . . . . . . . . 10 ⊢ (𝑁 ∈ (0...𝑁) → (𝜑 → ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁))))
130	25, 129	mpcom 39	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁)))
131	130	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁)))
132	131	eleq2d 2851	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁))))
133		elin 3922	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝑃‘𝑁) ∧ 𝑘 ∈ (0..^𝑁)))
134	133	rbaib 548	. . . . . . . 8 ⊢ (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑃‘𝑁)))
135	134	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑃‘𝑁)))
136		elin 3922	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝑄‘𝑁) ∧ 𝑘 ∈ (0..^𝑁)))
137	136	rbaib 548	. . . . . . . 8 ⊢ (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑄‘𝑁)))
138	137	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑄‘𝑁)))
139	132, 135, 138	3bitr3d 312	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝑃‘𝑁) ↔ 𝑘 ∈ (𝑄‘𝑁)))
140	52	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ₀)
141	2, 140, 53, 13	smupval 16570	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑄‘𝑁) = ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))))
142	141	eleq2d 2851	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝑄‘𝑁) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁)))))
143	22, 139, 142	3bitrd 308	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁)))))
144	143	ex 418	. . . 4 ⊢ (𝜑 → (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))))))
145	144	pm5.32rd 589	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 smul 𝐵) ∧ 𝑘 ∈ (0..^𝑁)) ↔ (𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∧ 𝑘 ∈ (0..^𝑁))))
146		elin 3922	. . 3 ⊢ (𝑘 ∈ ((𝐴 smul 𝐵) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝐴 smul 𝐵) ∧ 𝑘 ∈ (0..^𝑁)))
147		elin 3922	. . 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 317	. 2 ⊢ (𝜑 → (𝑘 ∈ ((𝐴 smul 𝐵) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∩ (0..^𝑁))))
149	148	eqrdv 2763	1 ⊢ (𝜑 → ((𝐴 smul 𝐵) ∩ (0..^𝑁)) = (((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∩ (0..^𝑁)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 {crab 3418 ∩ cin 3905 ⊆ wss 3906 ∅c0 4286 ifcif 4489 𝒫 cpw 4564 class class class wbr 5111 ↦ cmpt 5194 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ∈ cmpo 7421 0cc0 11117 1c1 11118 + caddc 11120 < clt 11260 ≤ cle 11261 − cmin 11458 ℕcn 12250 ℕ₀cn0 12521 ℤcz 12608 ℤ_≥cuz 12880 ...cfz 13553 ..^cfzo 13701 seqcseq 14057 sadd csad 16502 smul csmu 16503

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-pre-sup 11195

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 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-disj 5079 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-er 8700 df-map 8832 df-pm 8833 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-sup 9409 df-inf 9410 df-oi 9479 df-dju 9903 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-div 11889 df-nn 12251 df-2 12320 df-3 12321 df-n0 12522 df-xnn0 12595 df-z 12609 df-uz 12881 df-rp 13035 df-fz 13554 df-fzo 13702 df-fl 13845 df-mod 13923 df-seq 14058 df-exp 14118 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-sum 15764 df-dvds 16335 df-bits 16504 df-sad 16533 df-smu 16558

This theorem is used by: smueq 16573

W3C validator