Theorem relexp01min 43204

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 4648	. . 3 ⊢ (𝐽 ∈ {0, 1} → (𝐽 = 0 ∨ 𝐽 = 1))
2		elpri 4648	. . 3 ⊢ (𝐾 ∈ {0, 1} → (𝐾 = 0 ∨ 𝐾 = 1))
3		dmresi 6051	. . . . . . . . . . 11 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
4		rnresi 6074	. . . . . . . . . . 11 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
5	3, 4	uneq12i 4155	. . . . . . . . . 10 ⊢ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅))
6		unidm 4146	. . . . . . . . . 10 ⊢ ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
7	5, 6	eqtri 2753	. . . . . . . . 9 ⊢ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (dom 𝑅 ∪ ran 𝑅)
8	7	reseq2i 5977	. . . . . . . 8 ⊢ ( I ↾ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
9		simp1 1133	. . . . . . . . . . . 12 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 0)
10	9	oveq2d 7429	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = (𝑅↑𝑟0))
11		simp3l 1198	. . . . . . . . . . . 12 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅 ∈ 𝑉)
12		relexp0g 14996	. . . . . . . . . . . 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 2765	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
15		simp2 1134	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 0)
16	14, 15	oveq12d 7431	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟0))
17		dmexg 7903	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
18		rnexg 7904	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
19	17, 18	unexd 7751	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
20	19	resiexd 7222	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
21		relexp0g 14996	. . . . . . . . . 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 2765	. . . . . . . 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 1199	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
25		0re 11241	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
26	25	ltnri 11348	. . . . . . . . . . . . 13 ⊢ ¬ 0 < 0
27	9, 15	breq12d 5157	. . . . . . . . . . . . 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 4536	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
30	24, 29, 15	3eqtrd 2769	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
31	30	oveq2d 7429	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = (𝑅↑𝑟0))
32	31, 13	eqtrd 2765	. . . . . . . 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 2791	. . . . . . 7 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
34	33	3exp 1116	. . . . . 6 ⊢ (𝐽 = 0 → (𝐾 = 0 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
35		simp1 1133	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 1)
36	35	oveq2d 7429	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = (𝑅↑𝑟1))
37		simp3l 1198	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅 ∈ 𝑉)
38	37	relexp1d 15003	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟1) = 𝑅)
39	36, 38	eqtrd 2765	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = 𝑅)
40		simp2 1134	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 0)
41	39, 40	oveq12d 7431	. . . . . . . 8 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟0))
42		simp3r 1199	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
43		0lt1 11761	. . . . . . . . . . . . 13 ⊢ 0 < 1
44		1re 11239	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
45	25, 44	ltnsymi 11358	. . . . . . . . . . . . 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 5157	. . . . . . . . . . . 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 4536	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
50	42, 49, 40	3eqtrd 2769	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
51	50	oveq2d 7429	. . . . . . . 8 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = (𝑅↑𝑟0))
52	41, 51	eqtr4d 2768	. . . . . . 7 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
53	52	3exp 1116	. . . . . 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 7446	. . . . . . . . 9 ⊢ (𝑅↑𝑟0) ∈ V
56		relexp1g 15000	. . . . . . . . 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 1133	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 0)
59	58	oveq2d 7429	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = (𝑅↑𝑟0))
60		simp2 1134	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 1)
61	59, 60	oveq12d 7431	. . . . . . . 8 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = ((𝑅↑𝑟0)↑𝑟1))
62		simp3r 1199	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
63	58, 60	breq12d 5157	. . . . . . . . . . . 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 4533	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽)
66	62, 65, 58	3eqtrd 2769	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
67	66	oveq2d 7429	. . . . . . . 8 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = (𝑅↑𝑟0))
68	57, 61, 67	3eqtr4d 2775	. . . . . . 7 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
69	68	3exp 1116	. . . . . 6 ⊢ (𝐽 = 0 → (𝐾 = 1 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
70		simp1 1133	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 1)
71	70	oveq2d 7429	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = (𝑅↑𝑟1))
72		simp3l 1198	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅 ∈ 𝑉)
73	72	relexp1d 15003	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟1) = 𝑅)
74	71, 73	eqtrd 2765	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = 𝑅)
75		simp2 1134	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 1)
76	74, 75	oveq12d 7431	. . . . . . . 8 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟1))
77		simp3r 1199	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
78	44	ltnri 11348	. . . . . . . . . . . 12 ⊢ ¬ 1 < 1
79	70, 75	breq12d 5157	. . . . . . . . . . . 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 4536	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
82	77, 81, 75	3eqtrd 2769	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 1)
83	82	oveq2d 7429	. . . . . . . 8 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = (𝑅↑𝑟1))
84	76, 83	eqtr4d 2768	. . . . . . 7 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
85	84	3exp 1116	. . . . . 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 405	. . 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 594	. 2 ⊢ ((𝐽 ∈ {0, 1} ∧ 𝐾 ∈ {0, 1}) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼)))
90	89	impcom 406	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 394   ∨ wo 845   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3463   ∪ cun 3939  ifcif 4525  {cpr 4627   class class class wbr 5144   I cid 5570  dom cdm 5673  ran crn 5674   ↾ cres 5675  (class class class)co 7413  0cc0 11133  1c1 11134   < clt 11273  ↑_𝑟crelexp 14993

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 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-n0 12498  df-z 12584  df-uz 12848  df-seq 13994  df-relexp 14994

This theorem is referenced by:  relexp1idm  43205  relexp0idm  43206