Theorem relexp01min 42449

Description: With exponents limited to 0 and 1, the composition of powers of a relation is the relation raised to the minimum of exponents. (Contributed by RP, 12-Jun-2020.)

Assertion

Ref	Expression
relexp01min	⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) ∧ (𝐽 ∈ {0, 1} ∧ 𝐾 ∈ {0, 1})) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))

Proof of Theorem relexp01min

Step	Hyp	Ref	Expression
1		elpri 4649	. . 3 ⊢ (𝐽 ∈ {0, 1} → (𝐽 = 0 ∨ 𝐽 = 1))
2		elpri 4649	. . 3 ⊢ (𝐾 ∈ {0, 1} → (𝐾 = 0 ∨ 𝐾 = 1))
3		dmresi 6049	. . . . . . . . . . 11 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
4		rnresi 6071	. . . . . . . . . . 11 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
5	3, 4	uneq12i 4160	. . . . . . . . . 10 ⊢ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅))
6		unidm 4151	. . . . . . . . . 10 ⊢ ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
7	5, 6	eqtri 2760	. . . . . . . . 9 ⊢ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (dom 𝑅 ∪ ran 𝑅)
8	7	reseq2i 5976	. . . . . . . 8 ⊢ ( I ↾ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
9		simp1 1136	. . . . . . . . . . . 12 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 0)
10	9	oveq2d 7421	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = (𝑅↑𝑟0))
11		simp3l 1201	. . . . . . . . . . . 12 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅 ∈ 𝑉)
12		relexp0g 14965	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
13	11, 12	syl 17	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
14	10, 13	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
15		simp2 1137	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 0)
16	14, 15	oveq12d 7423	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟0))
17		dmexg 7890	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
18		rnexg 7891	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
19	17, 18	unexd 7737	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
20	19	resiexd 7214	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
21		relexp0g 14965	. . . . . . . . . 10 ⊢ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V → (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟0) = ( I ↾ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))))
22	11, 20, 21	3syl 18	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟0) = ( I ↾ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))))
23	16, 22	eqtrd 2772	. . . . . . . 8 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = ( I ↾ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))))
24		simp3r 1202	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
25		0re 11212	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
26	25	ltnri 11319	. . . . . . . . . . . . 13 ⊢ ¬ 0 < 0
27	9, 15	breq12d 5160	. . . . . . . . . . . . 13 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 0 < 0))
28	26, 27	mtbiri 326	. . . . . . . . . . . 12 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
29	28	iffalsed 4538	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
30	24, 29, 15	3eqtrd 2776	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
31	30	oveq2d 7421	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = (𝑅↑𝑟0))
32	31, 13	eqtrd 2772	. . . . . . . 8 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
33	8, 23, 32	3eqtr4a 2798	. . . . . . 7 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
34	33	3exp 1119	. . . . . 6 ⊢ (𝐽 = 0 → (𝐾 = 0 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
35		simp1 1136	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 1)
36	35	oveq2d 7421	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = (𝑅↑𝑟1))
37		simp3l 1201	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅 ∈ 𝑉)
38	37	relexp1d 14972	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟1) = 𝑅)
39	36, 38	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = 𝑅)
40		simp2 1137	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 0)
41	39, 40	oveq12d 7423	. . . . . . . 8 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟0))
42		simp3r 1202	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
43		0lt1 11732	. . . . . . . . . . . . 13 ⊢ 0 < 1
44		1re 11210	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
45	25, 44	ltnsymi 11329	. . . . . . . . . . . . 13 ⊢ (0 < 1 → ¬ 1 < 0)
46	43, 45	mp1i 13	. . . . . . . . . . . 12 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 1 < 0)
47	35, 40	breq12d 5160	. . . . . . . . . . . 12 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 1 < 0))
48	46, 47	mtbird 324	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
49	48	iffalsed 4538	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
50	42, 49, 40	3eqtrd 2776	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
51	50	oveq2d 7421	. . . . . . . 8 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = (𝑅↑𝑟0))
52	41, 51	eqtr4d 2775	. . . . . . 7 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
53	52	3exp 1119	. . . . . 6 ⊢ (𝐽 = 1 → (𝐾 = 0 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
54	34, 53	jaoi 855	. . . . 5 ⊢ ((𝐽 = 0 ∨ 𝐽 = 1) → (𝐾 = 0 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
55		ovex 7438	. . . . . . . . 9 ⊢ (𝑅↑𝑟0) ∈ V
56		relexp1g 14969	. . . . . . . . 9 ⊢ ((𝑅↑𝑟0) ∈ V → ((𝑅↑𝑟0)↑𝑟1) = (𝑅↑𝑟0))
57	55, 56	mp1i 13	. . . . . . . 8 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟0)↑𝑟1) = (𝑅↑𝑟0))
58		simp1 1136	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 0)
59	58	oveq2d 7421	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = (𝑅↑𝑟0))
60		simp2 1137	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 1)
61	59, 60	oveq12d 7423	. . . . . . . 8 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = ((𝑅↑𝑟0)↑𝑟1))
62		simp3r 1202	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
63	58, 60	breq12d 5160	. . . . . . . . . . . 12 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 0 < 1))
64	43, 63	mpbiri 257	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 < 𝐾)
65	64	iftrued 4535	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽)
66	62, 65, 58	3eqtrd 2776	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
67	66	oveq2d 7421	. . . . . . . 8 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = (𝑅↑𝑟0))
68	57, 61, 67	3eqtr4d 2782	. . . . . . 7 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
69	68	3exp 1119	. . . . . 6 ⊢ (𝐽 = 0 → (𝐾 = 1 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
70		simp1 1136	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 1)
71	70	oveq2d 7421	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = (𝑅↑𝑟1))
72		simp3l 1201	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅 ∈ 𝑉)
73	72	relexp1d 14972	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟1) = 𝑅)
74	71, 73	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = 𝑅)
75		simp2 1137	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 1)
76	74, 75	oveq12d 7423	. . . . . . . 8 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟1))
77		simp3r 1202	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
78	44	ltnri 11319	. . . . . . . . . . . 12 ⊢ ¬ 1 < 1
79	70, 75	breq12d 5160	. . . . . . . . . . . 12 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 1 < 1))
80	78, 79	mtbiri 326	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
81	80	iffalsed 4538	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
82	77, 81, 75	3eqtrd 2776	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 1)
83	82	oveq2d 7421	. . . . . . . 8 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = (𝑅↑𝑟1))
84	76, 83	eqtr4d 2775	. . . . . . 7 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
85	84	3exp 1119	. . . . . 6 ⊢ (𝐽 = 1 → (𝐾 = 1 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
86	69, 85	jaoi 855	. . . . 5 ⊢ ((𝐽 = 0 ∨ 𝐽 = 1) → (𝐾 = 1 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
87	54, 86	jaod 857	. . . 4 ⊢ ((𝐽 = 0 ∨ 𝐽 = 1) → ((𝐾 = 0 ∨ 𝐾 = 1) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
88	87	imp 407	. . 3 ⊢ (((𝐽 = 0 ∨ 𝐽 = 1) ∧ (𝐾 = 0 ∨ 𝐾 = 1)) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼)))
89	1, 2, 88	syl2an 596	. 2 ⊢ ((𝐽 ∈ {0, 1} ∧ 𝐾 ∈ {0, 1}) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼)))
90	89	impcom 408	1 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) ∧ (𝐽 ∈ {0, 1} ∧ 𝐾 ∈ {0, 1})) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∪ cun 3945  ifcif 4527  {cpr 4629   class class class wbr 5147   I cid 5572  dom cdm 5675  ran crn 5676   ↾ cres 5677  (class class class)co 7405  0cc0 11106  1c1 11107   < clt 11244  ↑_𝑟crelexp 14962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-seq 13963  df-relexp 14963

This theorem is referenced by:  relexp1idm  42450  relexp0idm  42451