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Theorem relexp01min 40585
 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 (((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) ∧ (𝐽 ∈ {0, 1} ∧ 𝐾 ∈ {0, 1})) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))

Proof of Theorem relexp01min
StepHypRef Expression
1 elpri 4550 . . 3 (𝐽 ∈ {0, 1} → (𝐽 = 0 ∨ 𝐽 = 1))
2 elpri 4550 . . 3 (𝐾 ∈ {0, 1} → (𝐾 = 0 ∨ 𝐾 = 1))
3 dmresi 5892 . . . . . . . . . . 11 dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
4 rnresi 5914 . . . . . . . . . . 11 ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
53, 4uneq12i 4091 . . . . . . . . . 10 (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅))
6 unidm 4082 . . . . . . . . . 10 ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
75, 6eqtri 2821 . . . . . . . . 9 (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (dom 𝑅 ∪ ran 𝑅)
87reseq2i 5819 . . . . . . . 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 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 0)
109oveq2d 7161 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟0))
11 simp3l 1198 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅𝑉)
12 relexp0g 14393 . . . . . . . . . . . 12 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
1311, 12syl 17 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
1410, 13eqtrd 2833 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
15 simp2 1134 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 0)
1614, 15oveq12d 7163 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟0))
17 dmexg 7607 . . . . . . . . . . . 12 (𝑅𝑉 → dom 𝑅 ∈ V)
18 rnexg 7608 . . . . . . . . . . . 12 (𝑅𝑉 → ran 𝑅 ∈ V)
19 unexg 7465 . . . . . . . . . . . 12 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
2017, 18, 19syl2anc 587 . . . . . . . . . . 11 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
2120resiexd 6966 . . . . . . . . . 10 (𝑅𝑉 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
22 relexp0g 14393 . . . . . . . . . 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)))))
2311, 21, 223syl 18 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟0) = ( I ↾ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))))
2416, 23eqtrd 2833 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = ( I ↾ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))))
25 simp3r 1199 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
26 0re 10650 . . . . . . . . . . . . . 14 0 ∈ ℝ
2726ltnri 10756 . . . . . . . . . . . . 13 ¬ 0 < 0
289, 15breq12d 5047 . . . . . . . . . . . . 13 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 0 < 0))
2927, 28mtbiri 330 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
3029iffalsed 4439 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
3125, 30, 153eqtrd 2837 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
3231oveq2d 7161 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
3332, 13eqtrd 2833 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
348, 24, 333eqtr4a 2859 . . . . . . 7 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
35343exp 1116 . . . . . 6 (𝐽 = 0 → (𝐾 = 0 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
36 simp1 1133 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 1)
3736oveq2d 7161 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟1))
38 simp3l 1198 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅𝑉)
39 relexp1g 14397 . . . . . . . . . . 11 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
4038, 39syl 17 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟1) = 𝑅)
4137, 40eqtrd 2833 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = 𝑅)
42 simp2 1134 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 0)
4341, 42oveq12d 7163 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟0))
44 simp3r 1199 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
45 0lt1 11169 . . . . . . . . . . . . 13 0 < 1
46 1re 10648 . . . . . . . . . . . . . 14 1 ∈ ℝ
4726, 46ltnsymi 10766 . . . . . . . . . . . . 13 (0 < 1 → ¬ 1 < 0)
4845, 47mp1i 13 . . . . . . . . . . . 12 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 1 < 0)
4936, 42breq12d 5047 . . . . . . . . . . . 12 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 1 < 0))
5048, 49mtbird 328 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
5150iffalsed 4439 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
5244, 51, 423eqtrd 2837 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
5352oveq2d 7161 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
5443, 53eqtr4d 2836 . . . . . . 7 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
55543exp 1116 . . . . . 6 (𝐽 = 1 → (𝐾 = 0 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
5635, 55jaoi 854 . . . . 5 ((𝐽 = 0 ∨ 𝐽 = 1) → (𝐾 = 0 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
57 ovex 7178 . . . . . . . . 9 (𝑅𝑟0) ∈ V
58 relexp1g 14397 . . . . . . . . 9 ((𝑅𝑟0) ∈ V → ((𝑅𝑟0)↑𝑟1) = (𝑅𝑟0))
5957, 58mp1i 13 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟0)↑𝑟1) = (𝑅𝑟0))
60 simp1 1133 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 0)
6160oveq2d 7161 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟0))
62 simp2 1134 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 1)
6361, 62oveq12d 7163 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = ((𝑅𝑟0)↑𝑟1))
64 simp3r 1199 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
6560, 62breq12d 5047 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 0 < 1))
6645, 65mpbiri 261 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 < 𝐾)
6766iftrued 4436 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽)
6864, 67, 603eqtrd 2837 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
6968oveq2d 7161 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
7059, 63, 693eqtr4d 2843 . . . . . . 7 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
71703exp 1116 . . . . . 6 (𝐽 = 0 → (𝐾 = 1 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
72 simp1 1133 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 1)
7372oveq2d 7161 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟1))
74 simp3l 1198 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅𝑉)
7574, 39syl 17 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟1) = 𝑅)
7673, 75eqtrd 2833 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = 𝑅)
77 simp2 1134 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 1)
7876, 77oveq12d 7163 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟1))
79 simp3r 1199 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
8046ltnri 10756 . . . . . . . . . . . 12 ¬ 1 < 1
8172, 77breq12d 5047 . . . . . . . . . . . 12 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 1 < 1))
8280, 81mtbiri 330 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
8382iffalsed 4439 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
8479, 83, 773eqtrd 2837 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 1)
8584oveq2d 7161 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟1))
8678, 85eqtr4d 2836 . . . . . . 7 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
87863exp 1116 . . . . . 6 (𝐽 = 1 → (𝐾 = 1 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
8871, 87jaoi 854 . . . . 5 ((𝐽 = 0 ∨ 𝐽 = 1) → (𝐾 = 1 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
8956, 88jaod 856 . . . 4 ((𝐽 = 0 ∨ 𝐽 = 1) → ((𝐾 = 0 ∨ 𝐾 = 1) → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
9089imp 410 . . 3 (((𝐽 = 0 ∨ 𝐽 = 1) ∧ (𝐾 = 0 ∨ 𝐾 = 1)) → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
911, 2, 90syl2an 598 . 2 ((𝐽 ∈ {0, 1} ∧ 𝐾 ∈ {0, 1}) → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
9291impcom 411 1 (((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) ∧ (𝐽 ∈ {0, 1} ∧ 𝐾 ∈ {0, 1})) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3442   ∪ cun 3881  ifcif 4428  {cpr 4530   class class class wbr 5034   I cid 5428  dom cdm 5523  ran crn 5524   ↾ cres 5525  (class class class)co 7145  0cc0 10544  1c1 10545   < clt 10682  ↑𝑟crelexp 14390 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-er 8290  df-en 8511  df-dom 8512  df-sdom 8513  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-nn 11644  df-n0 11904  df-z 11990  df-uz 12252  df-seq 13385  df-relexp 14391 This theorem is referenced by:  relexp1idm  40586  relexp0idm  40587
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