Theorem relexpmulg 43042

Description: With ordered exponents, the composition of powers of a relation is the relation raised to the product of exponents. (Contributed by RP, 13-Jun-2020.)

Assertion

Ref	Expression
relexpmulg	⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) ∧ (𝐽 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))

Proof of Theorem relexpmulg

Step	Hyp	Ref	Expression
1		elnn0 12478	. . . 4 ⊢ (𝐽 ∈ ℕ0 ↔ (𝐽 ∈ ℕ ∨ 𝐽 = 0))
2		elnn0 12478	. . . . . 6 ⊢ (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
3		relexpmulnn 43041	. . . . . . . . . 10 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾)) ∧ (𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
4	3	3adantl3 1165	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) ∧ (𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
5	4	expcom 413	. . . . . . . 8 ⊢ ((𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼)))
6	5	expcom 413	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → (𝐽 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
7		simprr 770	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐼 = (𝐽 · 𝐾))
8		simpll 764	. . . . . . . . . . . . . 14 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐾 = 0)
9	8	oveq2d 7421	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → (𝐽 · 𝐾) = (𝐽 · 0))
10		simplr 766	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐽 ∈ ℕ)
11	10	nncnd 12232	. . . . . . . . . . . . . 14 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐽 ∈ ℂ)
12	11	mul01d 11417	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → (𝐽 · 0) = 0)
13	7, 9, 12	3eqtrd 2770	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐼 = 0)
14		simpl 482	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → (𝐾 = 0 ∧ 𝐽 ∈ ℕ))
15		nnnle0 12249	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ ℕ → ¬ 𝐽 ≤ 0)
16	15	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → ¬ 𝐽 ≤ 0)
17		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → 𝐾 = 0)
18	17	breq2d 5153	. . . . . . . . . . . . . 14 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → (𝐽 ≤ 𝐾 ↔ 𝐽 ≤ 0))
19	16, 18	mtbird 325	. . . . . . . . . . . . 13 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → ¬ 𝐽 ≤ 𝐾)
20	14, 19	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → ¬ 𝐽 ≤ 𝐾)
21	13, 20	jcnd 163	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → ¬ (𝐼 = 0 → 𝐽 ≤ 𝐾))
22	21	pm2.21d 121	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → ((𝐼 = 0 → 𝐽 ≤ 𝐾) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼)))
23	22	exp32 420	. . . . . . . . 9 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → (𝑅 ∈ 𝑉 → (𝐼 = (𝐽 · 𝐾) → ((𝐼 = 0 → 𝐽 ≤ 𝐾) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼)))))
24	23	3impd 1345	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼)))
25	24	ex 412	. . . . . . 7 ⊢ (𝐾 = 0 → (𝐽 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
26	6, 25	jaoi 854	. . . . . 6 ⊢ ((𝐾 ∈ ℕ ∨ 𝐾 = 0) → (𝐽 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
27	2, 26	sylbi 216	. . . . 5 ⊢ (𝐾 ∈ ℕ0 → (𝐽 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
28		simplr 766	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐽 = 0)
29	28	oveq2d 7421	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (𝑅↑𝑟𝐽) = (𝑅↑𝑟0))
30		simpr1 1191	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝑅 ∈ 𝑉)
31		relexp0g 14975	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
32	30, 31	syl 17	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
33	29, 32	eqtrd 2766	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (𝑅↑𝑟𝐽) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
34	33	oveq1d 7420	. . . . . . . 8 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟𝐾))
35		dmexg 7891	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
36		rnexg 7892	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
37	35, 36	unexd 7738	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
38	30, 37	syl 17	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
39		simpll 764	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐾 ∈ ℕ0)
40		relexpiidm 43036	. . . . . . . . 9 ⊢ (((dom 𝑅 ∪ ran 𝑅) ∈ V ∧ 𝐾 ∈ ℕ0) → (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟𝐾) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
41	38, 39, 40	syl2anc 583	. . . . . . . 8 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟𝐾) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
42		simpr2 1192	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐼 = (𝐽 · 𝐾))
43	28	oveq1d 7420	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (𝐽 · 𝐾) = (0 · 𝐾))
44	39	nn0cnd 12538	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐾 ∈ ℂ)
45	44	mul02d 11416	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (0 · 𝐾) = 0)
46	42, 43, 45	3eqtrd 2770	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐼 = 0)
47	46	oveq2d 7421	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (𝑅↑𝑟𝐼) = (𝑅↑𝑟0))
48	47, 32	eqtr2d 2767	. . . . . . . 8 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟𝐼))
49	34, 41, 48	3eqtrd 2770	. . . . . . 7 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
50	49	ex 412	. . . . . 6 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐽 = 0) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼)))
51	50	ex 412	. . . . 5 ⊢ (𝐾 ∈ ℕ0 → (𝐽 = 0 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
52	27, 51	jaod 856	. . . 4 ⊢ (𝐾 ∈ ℕ0 → ((𝐽 ∈ ℕ ∨ 𝐽 = 0) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
53	1, 52	biimtrid 241	. . 3 ⊢ (𝐾 ∈ ℕ0 → (𝐽 ∈ ℕ0 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
54	53	impcom 407	. 2 ⊢ ((𝐽 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼)))
55	54	impcom 407	1 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) ∧ (𝐽 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3468   ∪ cun 3941   class class class wbr 5141   I cid 5566  dom cdm 5669  ran crn 5670   ↾ cres 5671  (class class class)co 7405  0cc0 11112   · cmul 11117   ≤ cle 11253  ℕcn 12216  ℕ₀cn0 12476  ↑_𝑟crelexp 14972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-seq 13973  df-relexp 14973

This theorem is referenced by: (None)