relexpmulg - Mathbox for Richard Penner

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Theorem relexpmulg 44154

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 12430	. . . 4 ⊢ (𝐽 ∈ ℕ₀ ↔ (𝐽 ∈ ℕ ∨ 𝐽 = 0))
2		elnn0 12430	. . . . . 6 ⊢ (𝐾 ∈ ℕ₀ ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
3		relexpmulnn 44153	. . . . . . . . . 10 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾)) ∧ (𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))
4	3	3adantl3 1175	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) ∧ (𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))
5	4	expcom 414	. . . . . . . 8 ⊢ ((𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼)))
6	5	expcom 414	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → (𝐽 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))))
7		simprr 778	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐼 = (𝐽 · 𝐾))
8		simpll 772	. . . . . . . . . . . . . 14 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐾 = 0)
9	8	oveq2d 7372	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → (𝐽 · 𝐾) = (𝐽 · 0))
10		simplr 774	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐽 ∈ ℕ)
11	10	nncnd 12181	. . . . . . . . . . . . . 14 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐽 ∈ ℂ)
12	11	mul01d 11336	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → (𝐽 · 0) = 0)
13	7, 9, 12	3eqtrd 2778	. . . . . . . . . . . 12 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → 𝐼 = 0)
14		simpl 483	. . . . . . . . . . . . 13 ⊢ (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾))) → (𝐾 = 0 ∧ 𝐽 ∈ ℕ))
15		nnnle0 12201	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ ℕ → ¬ 𝐽 ≤ 0)
16	15	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → ¬ 𝐽 ≤ 0)
17		simpl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → 𝐾 = 0)
18	17	breq2d 5084	. . . . . . . . . . . . . 14 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → (𝐽 ≤ 𝐾 ↔ 𝐽 ≤ 0))
19	16, 18	mtbird 326	. . . . . . . . . . . . 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 421	. . . . . . . . 9 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → (𝑅 ∈ 𝑉 → (𝐼 = (𝐽 · 𝐾) → ((𝐼 = 0 → 𝐽 ≤ 𝐾) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼)))))
24	23	3impd 1355	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼)))
25	24	ex 413	. . . . . . 7 ⊢ (𝐾 = 0 → (𝐽 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))))
26	6, 25	jaoi 863	. . . . . 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 218	. . . . 5 ⊢ (𝐾 ∈ ℕ₀ → (𝐽 ∈ ℕ → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))))
28		simplr 774	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐽 = 0)
29	28	oveq2d 7372	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (𝑅↑_𝑟𝐽) = (𝑅↑_𝑟0))
30		simpr1 1201	. . . . . . . . . . 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 2774	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (𝑅↑_𝑟𝐽) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
34	33	oveq1d 7371	. . . . . . . 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 7841	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
36		rnexg 7842	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
37	35, 36	unexd 7697	. . . . . . . . . 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 772	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐾 ∈ ℕ₀)
40		relexpiidm 44148	. . . . . . . . 9 ⊢ (((dom 𝑅 ∪ ran 𝑅) ∈ V ∧ 𝐾 ∈ ℕ₀) → (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑_𝑟𝐾) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
41	38, 39, 40	syl2anc 590	. . . . . . . 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 1202	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐼 = (𝐽 · 𝐾))
43	28	oveq1d 7371	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (𝐽 · 𝐾) = (0 · 𝐾))
44	39	nn0cnd 12491	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐾 ∈ ℂ)
45	44	mul02d 11335	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (0 · 𝐾) = 0)
46	42, 43, 45	3eqtrd 2778	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → 𝐼 = 0)
47	46	oveq2d 7372	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → (𝑅↑_𝑟𝐼) = (𝑅↑_𝑟0))
48	47, 32	eqtr2d 2775	. . . . . . . 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 2778	. . . . . . 7 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾))) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))
50	49	ex 413	. . . . . 6 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐽 = 0) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼)))
51	50	ex 413	. . . . 5 ⊢ (𝐾 ∈ ℕ₀ → (𝐽 = 0 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))))
52	27, 51	jaod 865	. . . 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 243	. . 3 ⊢ (𝐾 ∈ ℕ₀ → (𝐽 ∈ ℕ₀ → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼))))
54	53	impcom 408	. 2 ⊢ ((𝐽 ∈ ℕ₀ ∧ 𝐾 ∈ ℕ₀) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) → ((𝑅↑_𝑟𝐽)↑_𝑟𝐾) = (𝑅↑_𝑟𝐼)))
55	54	impcom 408	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 396 ∨ wo 853 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 Vcvv 3431 ∪ cun 3881 class class class wbr 5072 I cid 5512 dom cdm 5618 ran crn 5619 ↾ cres 5620 (class class class)co 7356 0cc0 11029 · cmul 11034 ≤ cle 11171 ℕcn 12165 ℕ₀cn0 12428 ↑_𝑟crelexp 14972

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-n0 12429 df-z 12516 df-uz 12780 df-seq 13955 df-relexp 14973

This theorem is referenced by: (None)

W3C validator