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Theorem relexp01min 36820
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 4144 . . 3 (𝐽 ∈ {0, 1} → (𝐽 = 0 ∨ 𝐽 = 1))
2 elpri 4144 . . 3 (𝐾 ∈ {0, 1} → (𝐾 = 0 ∨ 𝐾 = 1))
3 dmresi 5363 . . . . . . . . . . 11 dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
4 rnresi 5385 . . . . . . . . . . 11 ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
53, 4uneq12i 3726 . . . . . . . . . 10 (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅))
6 unidm 3717 . . . . . . . . . 10 ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
75, 6eqtri 2631 . . . . . . . . 9 (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (dom 𝑅 ∪ ran 𝑅)
87reseq2i 5301 . . . . . . . 8 ( I ↾ (dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∪ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
9 simp1 1053 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 0)
109oveq2d 6543 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟0))
11 simp3l 1081 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅𝑉)
12 relexp0g 13556 . . . . . . . . . . . 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 2643 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
15 simp2 1054 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 0)
1614, 15oveq12d 6545 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟0))
17 dmexg 6966 . . . . . . . . . . . 12 (𝑅𝑉 → dom 𝑅 ∈ V)
18 rnexg 6967 . . . . . . . . . . . 12 (𝑅𝑉 → ran 𝑅 ∈ V)
19 unexg 6834 . . . . . . . . . . . 12 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
2017, 18, 19syl2anc 690 . . . . . . . . . . 11 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
2120resiexd 6363 . . . . . . . . . 10 (𝑅𝑉 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
22 relexp0g 13556 . . . . . . . . . 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 2643 . . . . . . . 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 1082 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
26 0re 9896 . . . . . . . . . . . . . 14 0 ∈ ℝ
2726ltnri 9997 . . . . . . . . . . . . 13 ¬ 0 < 0
289, 15breq12d 4590 . . . . . . . . . . . . 13 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 0 < 0))
2927, 28mtbiri 315 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
3029iffalsed 4046 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
3125, 30, 153eqtrd 2647 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
3231oveq2d 6543 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
3332, 13eqtrd 2643 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
348, 24, 333eqtr4a 2669 . . . . . . 7 ((𝐽 = 0 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
35343exp 1255 . . . . . 6 (𝐽 = 0 → (𝐾 = 0 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
36 simp1 1053 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 1)
3736oveq2d 6543 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟1))
38 simp3l 1081 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝑅𝑉)
39 relexp1g 13560 . . . . . . . . . . 11 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
4038, 39syl 17 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟1) = 𝑅)
4137, 40eqtrd 2643 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = 𝑅)
42 simp2 1054 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 0)
4341, 42oveq12d 6545 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟0))
44 simp3r 1082 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
45 0lt1 10399 . . . . . . . . . . . . 13 0 < 1
46 1re 9895 . . . . . . . . . . . . . 14 1 ∈ ℝ
4726, 46ltnsymi 10007 . . . . . . . . . . . . 13 (0 < 1 → ¬ 1 < 0)
4845, 47mp1i 13 . . . . . . . . . . . 12 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 1 < 0)
4936, 42breq12d 4590 . . . . . . . . . . . 12 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 1 < 0))
5048, 49mtbird 313 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
5150iffalsed 4046 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
5244, 51, 423eqtrd 2647 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
5352oveq2d 6543 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
5443, 53eqtr4d 2646 . . . . . . 7 ((𝐽 = 1 ∧ 𝐾 = 0 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
55543exp 1255 . . . . . 6 (𝐽 = 1 → (𝐾 = 0 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
5635, 55jaoi 392 . . . . 5 ((𝐽 = 0 ∨ 𝐽 = 1) → (𝐾 = 0 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
57 ovex 6555 . . . . . . . . 9 (𝑅𝑟0) ∈ V
58 relexp1g 13560 . . . . . . . . 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 1053 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 0)
6160oveq2d 6543 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟0))
62 simp2 1054 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 1)
6361, 62oveq12d 6545 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = ((𝑅𝑟0)↑𝑟1))
64 simp3r 1082 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
6560, 62breq12d 4590 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 0 < 1))
6645, 65mpbiri 246 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 < 𝐾)
6766iftrued 4043 . . . . . . . . . 10 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐽)
6864, 67, 603eqtrd 2647 . . . . . . . . 9 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 0)
6968oveq2d 6543 . . . . . . . 8 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
7059, 63, 693eqtr4d 2653 . . . . . . 7 ((𝐽 = 0 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
71703exp 1255 . . . . . 6 (𝐽 = 0 → (𝐾 = 1 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
72 simp1 1053 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐽 = 1)
7372oveq2d 6543 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟1))
74 simp3l 1081 . . . . . . . . . . 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 2643 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐽) = 𝑅)
77 simp2 1054 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐾 = 1)
7876, 77oveq12d 6545 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟1))
79 simp3r 1082 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))
8046ltnri 9997 . . . . . . . . . . . 12 ¬ 1 < 1
8172, 77breq12d 4590 . . . . . . . . . . . 12 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝐽 < 𝐾 ↔ 1 < 1))
8280, 81mtbiri 315 . . . . . . . . . . 11 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ¬ 𝐽 < 𝐾)
8382iffalsed 4046 . . . . . . . . . 10 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → if(𝐽 < 𝐾, 𝐽, 𝐾) = 𝐾)
8479, 83, 773eqtrd 2647 . . . . . . . . 9 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → 𝐼 = 1)
8584oveq2d 6543 . . . . . . . 8 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟1))
8678, 85eqtr4d 2646 . . . . . . 7 ((𝐽 = 1 ∧ 𝐾 = 1 ∧ (𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
87863exp 1255 . . . . . 6 (𝐽 = 1 → (𝐾 = 1 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
8871, 87jaoi 392 . . . . 5 ((𝐽 = 0 ∨ 𝐽 = 1) → (𝐾 = 1 → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
8956, 88jaod 393 . . . 4 ((𝐽 = 0 ∨ 𝐽 = 1) → ((𝐾 = 0 ∨ 𝐾 = 1) → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
9089imp 443 . . 3 (((𝐽 = 0 ∨ 𝐽 = 1) ∧ (𝐾 = 0 ∨ 𝐾 = 1)) → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
911, 2, 90syl2an 492 . 2 ((𝐽 ∈ {0, 1} ∧ 𝐾 ∈ {0, 1}) → ((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
9291impcom 444 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  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  Vcvv 3172  cun 3537  ifcif 4035  {cpr 4126   class class class wbr 4577   I cid 4938  dom cdm 5028  ran crn 5029  cres 5030  (class class class)co 6527  0cc0 9792  1c1 9793   < clt 9930  𝑟crelexp 13554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-n0 11140  df-z 11211  df-uz 11520  df-seq 12619  df-relexp 13555
This theorem is referenced by:  relexp1idm  36821  relexp0idm  36822
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