Theorem relexp01min 43702

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 4613	. . 3 ⊢ (𝐽 ∈ {0, 1} → (𝐽 = 0 ∨ 𝐽 = 1))
2		elpri 4613	. . 3 ⊢ (𝐾 ∈ {0, 1} → (𝐾 = 0 ∨ 𝐾 = 1))
3		dmresi 6023	. . . . . . . . . . 11 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
4		rnresi 6046	. . . . . . . . . . 11 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
5	3, 4	uneq12i 4129	. . . . . . . . . 10 ⊢ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅))
6		unidm 4120	. . . . . . . . . 10 ⊢ ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
7	5, 6	eqtri 2752	. . . . . . . . 9 ⊢ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (dom 𝑅 ∪ ran 𝑅)
8	7	reseq2i 5947	. . . . . . . 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 7403	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = (𝑅↑𝑟0))
11		simp3l 1202	. . . . . . . . . . . 12 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅 ∈ 𝑉)
12		relexp0g 14988	. . . . . . . . . . . 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 2764	. . . . . . . . . 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 7405	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟0))
17		dmexg 7877	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
18		rnexg 7878	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
19	17, 18	unexd 7730	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
20	19	resiexd 7190	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
21		relexp0g 14988	. . . . . . . . . 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 2764	. . . . . . . 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 1203	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
25		0re 11176	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
26	25	ltnri 11283	. . . . . . . . . . . . 13 ⊢ ¬ 0 < 0
27	9, 15	breq12d 5120	. . . . . . . . . . . . 13 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 0 < 0))
28	26, 27	mtbiri 327	. . . . . . . . . . . 12 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
29	28	iffalsed 4499	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
30	24, 29, 15	3eqtrd 2768	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
31	30	oveq2d 7403	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = (𝑅↑𝑟0))
32	31, 13	eqtrd 2764	. . . . . . . 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 2790	. . . . . . 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 7403	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = (𝑅↑𝑟1))
37		simp3l 1202	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅 ∈ 𝑉)
38	37	relexp1d 14995	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟1) = 𝑅)
39	36, 38	eqtrd 2764	. . . . . . . . 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 7405	. . . . . . . 8 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟0))
42		simp3r 1203	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
43		0lt1 11700	. . . . . . . . . . . . 13 ⊢ 0 < 1
44		1re 11174	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
45	25, 44	ltnsymi 11293	. . . . . . . . . . . . 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 5120	. . . . . . . . . . . 12 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 1 < 0))
48	46, 47	mtbird 325	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
49	48	iffalsed 4499	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
50	42, 49, 40	3eqtrd 2768	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
51	50	oveq2d 7403	. . . . . . . 8 ⊢ ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = (𝑅↑𝑟0))
52	41, 51	eqtr4d 2767	. . . . . . 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 857	. . . . 5 ⊢ ((𝐽 = 0 ∨ 𝐽 = 1) → (𝐾 = 0 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
55		ovex 7420	. . . . . . . . 9 ⊢ (𝑅↑𝑟0) ∈ V
56		relexp1g 14992	. . . . . . . . 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 7403	. . . . . . . . 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 7405	. . . . . . . 8 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = ((𝑅↑𝑟0)↑𝑟1))
62		simp3r 1203	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
63	58, 60	breq12d 5120	. . . . . . . . . . . 12 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 0 < 1))
64	43, 63	mpbiri 258	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 < 𝐾)
65	64	iftrued 4496	. . . . . . . . . 10 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽)
66	62, 65, 58	3eqtrd 2768	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
67	66	oveq2d 7403	. . . . . . . 8 ⊢ ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = (𝑅↑𝑟0))
68	57, 61, 67	3eqtr4d 2774	. . . . . . 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 7403	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐽) = (𝑅↑𝑟1))
72		simp3l 1202	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅 ∈ 𝑉)
73	72	relexp1d 14995	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟1) = 𝑅)
74	71, 73	eqtrd 2764	. . . . . . . . 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 7405	. . . . . . . 8 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟1))
77		simp3r 1203	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
78	44	ltnri 11283	. . . . . . . . . . . 12 ⊢ ¬ 1 < 1
79	70, 75	breq12d 5120	. . . . . . . . . . . 12 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 1 < 1))
80	78, 79	mtbiri 327	. . . . . . . . . . 11 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
81	80	iffalsed 4499	. . . . . . . . . 10 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
82	77, 81, 75	3eqtrd 2768	. . . . . . . . 9 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 1)
83	82	oveq2d 7403	. . . . . . . 8 ⊢ ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅↑𝑟𝐼) = (𝑅↑𝑟1))
84	76, 83	eqtr4d 2767	. . . . . . 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 857	. . . . 5 ⊢ ((𝐽 = 0 ∨ 𝐽 = 1) → (𝐾 = 1 → ((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))))
87	54, 86	jaod 859	. . . 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 406	. . 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 407	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 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ∪ cun 3912  ifcif 4488  {cpr 4591   class class class wbr 5107   I cid 5532  dom cdm 5638  ran crn 5639   ↾ cres 5640  (class class class)co 7387  0cc0 11068  1c1 11069   < clt 11208  ↑_𝑟crelexp 14985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-n0 12443  df-z 12530  df-uz 12794  df-seq 13967  df-relexp 14986

This theorem is referenced by:  relexp1idm  43703  relexp0idm  43704