relexpmulg - Mathbox for Richard Penner

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Theorem relexpmulg 43700

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 12511	. . . 4 ⊢ (𝐽 ∈ ℕ₀ ↔ (𝐽 ∈ ℕ ∨ 𝐽 = 0))
2		elnn0 12511	. . . . . 6 ⊢ (𝐾 ∈ ℕ₀ ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
3		relexpmulnn 43699	. . . . . . . . . 10 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾)) ∧ (𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))
4	3	3adantl3 1168	. . . . . . . . 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 772	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐼 = (𝐽 · 𝐾))
8		simpll 766	. . . . . . . . . . . . . 14 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐾 = 0)
9	8	oveq2d 7429	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → (𝐽 · 𝐾) = (𝐽 · 0))
10		simplr 768	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐽 ∈ ℕ)
11	10	nncnd 12264	. . . . . . . . . . . . . 14 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐽 ∈ ℂ)
12	11	mul01d 11442	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → (𝐽 · 0) = 0)
13	7, 9, 12	3eqtrd 2773	. . . . . . . . . . . 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 12281	. . . . . . . . . . . . . . 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 5135	. . . . . . . . . . . . . 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 1348	. . . . . . . 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 857	. . . . . 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 217	. . . . 5 ⊢ (𝐾 ∈ ℕ₀ → (𝐽 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))))
28		simplr 768	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐽 = 0)
29	28	oveq2d 7429	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (𝑅↑_𝑟𝐽) = (𝑅↑_𝑟0))
30		simpr1 1194	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝑅 ∈ 𝑉)
31		relexp0g 15044	. . . . . . . . . . 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 2769	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (𝑅↑_𝑟𝐽) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
34	33	oveq1d 7428	. . . . . . . 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 7905	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
36		rnexg 7906	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
37	35, 36	unexd 7756	. . . . . . . . . 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 766	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐾 ∈ ℕ₀)
40		relexpiidm 43694	. . . . . . . . 9 ⊢ (((dom 𝑅 ∪ ran 𝑅) ∈ V ∧ 𝐾 ∈ ℕ₀) → (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑_𝑟𝐾) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
41	38, 39, 40	syl2anc 584	. . . . . . . 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 1195	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐼 = (𝐽 · 𝐾))
43	28	oveq1d 7428	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (𝐽 · 𝐾) = (0 · 𝐾))
44	39	nn0cnd 12572	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐾 ∈ ℂ)
45	44	mul02d 11441	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (0 · 𝐾) = 0)
46	42, 43, 45	3eqtrd 2773	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐼 = 0)
47	46	oveq2d 7429	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (𝑅↑_𝑟𝐼) = (𝑅↑_𝑟0))
48	47, 32	eqtr2d 2770	. . . . . . . 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 2773	. . . . . . 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 859	. . . 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 242	. . 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))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 Vcvv 3463 ∪ cun 3929 class class class wbr 5123 I cid 5557 dom cdm 5665 ran crn 5666 ↾ cres 5667 (class class class)co 7413 0cc0 11137 · cmul 11142 ≤ cle 11278 ℕcn 12248 ℕ₀cn0 12509 ↑_𝑟crelexp 15041

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7870 df-2nd 7997 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8727 df-en 8968 df-dom 8969 df-sdom 8970 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-nn 12249 df-n0 12510 df-z 12597 df-uz 12861 df-seq 14025 df-relexp 15042

This theorem is referenced by: (None)

W3C validator