relexpmulg - Mathbox for Richard Penner

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Theorem relexpmulg 44419

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 12507	. . . 4 ⊢ (𝐽 ∈ ℕ₀ ↔ (𝐽 ∈ ℕ ∨ 𝐽 = 0))
2		elnn0 12507	. . . . . 6 ⊢ (𝐾 ∈ ℕ₀ ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
3		relexpmulnn 44418	. . . . . . . . . 10 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾)) ∧ (𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))
4	3	3adantl3 1187	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) ∧ (𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))
5	4	expcom 418	. . . . . . . 8 ⊢ ((𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼)))
6	5	expcom 418	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → (𝐽 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))))
7		simprr 784	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐼 = (𝐽 · 𝐾))
8		simpll 778	. . . . . . . . . . . . . 14 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐾 = 0)
9	8	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → (𝐽 · 𝐾) = (𝐽 · 0))
10		simplr 780	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐽 ∈ ℕ)
11	10	nncnd 12250	. . . . . . . . . . . . . 14 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐽 ∈ ℂ)
12	11	mul01d 11410	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → (𝐽 · 0) = 0)
13	7, 9, 12	3eqtrd 2802	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐼 = 0)
14		simpl 487	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → (𝐾 = 0 ∧ 𝐽 ∈ ℕ))
15		nnnle0 12270	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ ℕ → ¬ 𝐽 ≤ 0)
16	15	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → ¬ 𝐽 ≤ 0)
17		simpl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → 𝐾 = 0)
18	17	breq2d 5122	. . . . . . . . . . . . . 14 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → (𝐽 ≤ 𝐾 ↔ 𝐽 ≤ 0))
19	16, 18	mtbird 328	. . . . . . . . . . . . 13 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → ¬ 𝐽 ≤ 𝐾)
20	14, 19	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → ¬ 𝐽 ≤ 𝐾)
21	13, 20	jcnd 164	. . . . . . . . . . 11 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → ¬ (𝐼 = 0 → 𝐽 ≤ 𝐾))
22	21	pm2.21d 122	. . . . . . . . . 10 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → ((𝐼 = 0 → 𝐽 ≤ 𝐾) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼)))
23	22	exp32 425	. . . . . . . . 9 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → (𝑅 ∈ 𝑉 → (𝐼 = (𝐽 · 𝐾) → ((𝐼 = 0 → 𝐽 ≤ 𝐾) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼)))))
24	23	3impd 1367	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼)))
25	24	ex 417	. . . . . . 7 ⊢ (𝐾 = 0 → (𝐽 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))))
26	6, 25	jaoi 870	. . . . . 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 220	. . . . 5 ⊢ (𝐾 ∈ ℕ₀ → (𝐽 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))))
28		simplr 780	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐽 = 0)
29	28	oveq2d 7428	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (𝑅↑_𝑟𝐽) = (𝑅↑_𝑟0))
30		simpr1 1213	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝑅 ∈ 𝑉)
31		relexp0g 15061	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑_𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
32	30, 31	syl 18	. . . . . . . . . 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 2798	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (𝑅↑_𝑟𝐽) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
34	33	oveq1d 7427	. . . . . . . 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 7899	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
36		rnexg 7900	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
37	35, 36	unexd 7754	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
38	30, 37	syl 18	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
39		simpll 778	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐾 ∈ ℕ₀)
40		relexpiidm 44413	. . . . . . . . 9 ⊢ (((dom 𝑅 ∪ ran 𝑅) ∈ V ∧ 𝐾 ∈ ℕ₀) → (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑_𝑟𝐾) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
41	38, 39, 40	syl2anc 595	. . . . . . . 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 1214	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐼 = (𝐽 · 𝐾))
43	28	oveq1d 7427	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (𝐽 · 𝐾) = (0 · 𝐾))
44	39	nn0cnd 12568	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐾 ∈ ℂ)
45	44	mul02d 11409	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (0 · 𝐾) = 0)
46	42, 43, 45	3eqtrd 2802	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐼 = 0)
47	46	oveq2d 7428	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (𝑅↑_𝑟𝐼) = (𝑅↑_𝑟0))
48	47, 32	eqtr2d 2799	. . . . . . . 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 2802	. . . . . . 7 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))
50	49	ex 417	. . . . . 6 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼)))
51	50	ex 417	. . . . 5 ⊢ (𝐾 ∈ ℕ₀ → (𝐽 = 0 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))))
52	27, 51	jaod 872	. . . 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 245	. . 3 ⊢ (𝐾 ∈ ℕ₀ → (𝐽 ∈ ℕ₀ → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))))
54	53	impcom 412	. 2 ⊢ ((𝐽 ∈ ℕ₀ ∧ 𝐾 ∈ ℕ₀) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼)))
55	54	impcom 412	1 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) ∧ (𝐽 ∈ ℕ₀ ∧ 𝐾 ∈ ℕ₀)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∨ wo 860 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 Vcvv 3455 ∪ cun 3904 class class class wbr 5110 I cid 5557 dom cdm 5663 ran crn 5664 ↾ cres 5665 (class class class)co 7412 0cc0 11101 · cmul 11106 ≤ cle 11245 ℕcn 12234 ℕ₀cn0 12505 ↑_𝑟crelexp 15058

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 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 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-n0 12506 df-z 12593 df-uz 12864 df-seq 14040 df-relexp 15059

This theorem is referenced by: (None)

W3C validator