Theorem relexp01min 40048

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 4581	. . 3 ⊢ (𝐽 ∈ {0, 1} → (𝐽 = 0 ∨ 𝐽 = 1))
2		elpri 4581	. . 3 ⊢ (𝐾 ∈ {0, 1} → (𝐾 = 0 ∨ 𝐾 = 1))
3		dmresi 5914	. . . . . . . . . . 11 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
4		rnresi 5936	. . . . . . . . . . 11 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
5	3, 4	uneq12i 4135	. . . . . . . . . 10 ⊢ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅))
6		unidm 4126	. . . . . . . . . 10 ⊢ ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
7	5, 6	eqtri 2842	. . . . . . . . 9 ⊢ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (dom 𝑅 ∪ ran 𝑅)
8	7	reseq2i 5843	. . . . . . . 8 ⊢ ( I ↾ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
9		simp1 1131	. . . . . . . . . . . 12 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 0)
10	9	oveq2d 7164	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = (𝑅↑𝑟0))
11		simp3l 1196	. . . . . . . . . . . 12 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅 ∈ 𝑉)
12		relexp0g 14373	. . . . . . . . . . . 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 2854	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
15		simp2 1132	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 0)
16	14, 15	oveq12d 7166	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟0))
17		dmexg 7605	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
18		rnexg 7606	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
19		unexg 7464	. . . . . . . . . . . 12 ⊢ ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
20	17, 18, 19	syl2anc 586	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
21	20	resiexd 6971	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
22		relexp0g 14373	. . . . . . . . . 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)))))
23	11, 21, 22	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)))))
24	16, 23	eqtrd 2854	. . . . . . . 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)))))
25		simp3r 1197	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
26		0re 10635	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
27	26	ltnri 10741	. . . . . . . . . . . . 13 ⊢ ¬ 0 < 0
28	9, 15	breq12d 5070	. . . . . . . . . . . . 13 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 0 < 0))
29	27, 28	mtbiri 329	. . . . . . . . . . . 12 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
30	29	iffalsed 4476	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
31	25, 30, 15	3eqtrd 2858	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
32	31	oveq2d 7164	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = (𝑅↑𝑟0))
33	32, 13	eqtrd 2854	. . . . . . . 8 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
34	8, 24, 33	3eqtr4a 2880	. . . . . . 7 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
35	34	3exp 1114	. . . . . 6 ⊢ (𝐽 = 0 → (𝐾 = 0 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
36		simp1 1131	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 1)
37	36	oveq2d 7164	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = (𝑅↑𝑟1))
38		simp3l 1196	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅 ∈ 𝑉)
39		relexp1g 14377	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
40	38, 39	syl 17	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟1) = 𝑅)
41	37, 40	eqtrd 2854	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = 𝑅)
42		simp2 1132	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 0)
43	41, 42	oveq12d 7166	. . . . . . . 8 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟0))
44		simp3r 1197	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
45		0lt1 11154	. . . . . . . . . . . . 13 ⊢ 0 < 1
46		1re 10633	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
47	26, 46	ltnsymi 10751	. . . . . . . . . . . . 13 ⊢ (0 < 1 → ¬ 1 < 0)
48	45, 47	mp1i 13	. . . . . . . . . . . 12 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 1 < 0)
49	36, 42	breq12d 5070	. . . . . . . . . . . 12 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 1 < 0))
50	48, 49	mtbird 327	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
51	50	iffalsed 4476	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
52	44, 51, 42	3eqtrd 2858	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
53	52	oveq2d 7164	. . . . . . . 8 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = (𝑅↑𝑟0))
54	43, 53	eqtr4d 2857	. . . . . . 7 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
55	54	3exp 1114	. . . . . 6 ⊢ (𝐽 = 1 → (𝐾 = 0 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
56	35, 55	jaoi 853	. . . . 5 ⊢ ((𝐽 = 0 ∨ 𝐽 = 1) → (𝐾 = 0 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
57		ovex 7181	. . . . . . . . 9 ⊢ (𝑅↑𝑟0) ∈ V
58		relexp1g 14377	. . . . . . . . 9 ⊢ ((𝑅↑𝑟0) ∈ V → ((𝑅↑𝑟0)↑𝑟1) = (𝑅↑𝑟0))
59	57, 58	mp1i 13	. . . . . . . 8 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟0)↑𝑟1) = (𝑅↑𝑟0))
60		simp1 1131	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 0)
61	60	oveq2d 7164	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = (𝑅↑𝑟0))
62		simp2 1132	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 1)
63	61, 62	oveq12d 7166	. . . . . . . 8 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = ((𝑅↑𝑟0)↑𝑟1))
64		simp3r 1197	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
65	60, 62	breq12d 5070	. . . . . . . . . . . 12 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 0 < 1))
66	45, 65	mpbiri 260	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 < 𝐾)
67	66	iftrued 4473	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽)
68	64, 67, 60	3eqtrd 2858	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
69	68	oveq2d 7164	. . . . . . . 8 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = (𝑅↑𝑟0))
70	59, 63, 69	3eqtr4d 2864	. . . . . . 7 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
71	70	3exp 1114	. . . . . 6 ⊢ (𝐽 = 0 → (𝐾 = 1 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
72		simp1 1131	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 1)
73	72	oveq2d 7164	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = (𝑅↑𝑟1))
74		simp3l 1196	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅 ∈ 𝑉)
75	74, 39	syl 17	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟1) = 𝑅)
76	73, 75	eqtrd 2854	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = 𝑅)
77		simp2 1132	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 1)
78	76, 77	oveq12d 7166	. . . . . . . 8 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟1))
79		simp3r 1197	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
80	46	ltnri 10741	. . . . . . . . . . . 12 ⊢ ¬ 1 < 1
81	72, 77	breq12d 5070	. . . . . . . . . . . 12 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 1 < 1))
82	80, 81	mtbiri 329	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
83	82	iffalsed 4476	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
84	79, 83, 77	3eqtrd 2858	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 1)
85	84	oveq2d 7164	. . . . . . . 8 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = (𝑅↑𝑟1))
86	78, 85	eqtr4d 2857	. . . . . . 7 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
87	86	3exp 1114	. . . . . 6 ⊢ (𝐽 = 1 → (𝐾 = 1 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
88	71, 87	jaoi 853	. . . . 5 ⊢ ((𝐽 = 0 ∨ 𝐽 = 1) → (𝐾 = 1 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
89	56, 88	jaod 855	. . . 4 ⊢ ((𝐽 = 0 ∨ 𝐽 = 1) → ((𝐾 = 0 ∨ 𝐾 = 1) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
90	89	imp 409	. . 3 ⊢ (((𝐽 = 0 ∨ 𝐽 = 1) ∧ (𝐾 = 0 ∨ 𝐾 = 1)) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼)))
91	1, 2, 90	syl2an 597	. 2 ⊢ ((𝐽 ∈ {0, 1} ∧ 𝐾 ∈ {0, 1}) → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼)))
92	91	impcom 410	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 398   ∨ wo 843   ∧ w3a 1082   = wceq 1531   ∈ wcel 2108  Vcvv 3493   ∪ cun 3932  ifcif 4465  {cpr 4561   class class class wbr 5057   I cid 5452  dom cdm 5548  ran crn 5549   ↾ cres 5550  (class class class)co 7148  0cc0 10529  1c1 10530   < clt 10667  ↑_𝑟crelexp 14371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11890  df-z 11974  df-uz 12236  df-seq 13362  df-relexp 14372

This theorem is referenced by:  relexp1idm  40049  relexp0idm  40050