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Theorem relexpaddg 15044
Description: Relation composition becomes addition under exponentiation except when the exponents total to one and the class isn't a relation. (Contributed by RP, 30-May-2020.)
Assertion
Ref Expression
relexpaddg ((𝑁 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))

Proof of Theorem relexpaddg
StepHypRef Expression
1 elnn0 12512 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 elnn0 12512 . . . . . . 7 (𝑀 ∈ ℕ0 ↔ (𝑀 ∈ ℕ ∨ 𝑀 = 0))
3 relexpaddnn 15042 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
43a1d 25 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅𝑉) → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
543exp 1116 . . . . . . . . 9 (𝑁 ∈ ℕ → (𝑀 ∈ ℕ → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
65com12 32 . . . . . . . 8 (𝑀 ∈ ℕ → (𝑁 ∈ ℕ → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
7 elnn1uz2 12947 . . . . . . . . . 10 (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ‘2)))
8 coires1 6270 . . . . . . . . . . . . . 14 ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅))
9 simpll 765 . . . . . . . . . . . . . . . . . . 19 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 = 1)
10 simplr 767 . . . . . . . . . . . . . . . . . . 19 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑀 = 0)
119, 10oveq12d 7437 . . . . . . . . . . . . . . . . . 18 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = (1 + 0))
12 1p0e1 12374 . . . . . . . . . . . . . . . . . 18 (1 + 0) = 1
1311, 12eqtrdi 2781 . . . . . . . . . . . . . . . . 17 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = 1)
14 simprr 771 . . . . . . . . . . . . . . . . 17 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑁 + 𝑀) = 1 → Rel 𝑅))
1513, 14mpd 15 . . . . . . . . . . . . . . . 16 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → Rel 𝑅)
169oveq2d 7435 . . . . . . . . . . . . . . . . . 18 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑁) = (𝑅𝑟1))
17 simprl 769 . . . . . . . . . . . . . . . . . . 19 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑅𝑉)
18 relexp1g 15017 . . . . . . . . . . . . . . . . . . 19 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
1917, 18syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟1) = 𝑅)
2016, 19eqtrd 2765 . . . . . . . . . . . . . . . . 17 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑁) = 𝑅)
2120releqd 5780 . . . . . . . . . . . . . . . 16 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (Rel (𝑅𝑟𝑁) ↔ Rel 𝑅))
2215, 21mpbird 256 . . . . . . . . . . . . . . 15 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → Rel (𝑅𝑟𝑁))
23 1nn 12261 . . . . . . . . . . . . . . . . . 18 1 ∈ ℕ
249, 23eqeltrdi 2833 . . . . . . . . . . . . . . . . 17 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 ∈ ℕ)
25 relexpnndm 15032 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)
2624, 17, 25syl2anc 582 . . . . . . . . . . . . . . . 16 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)
27 ssun1 4170 . . . . . . . . . . . . . . . 16 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2826, 27sstrdi 3989 . . . . . . . . . . . . . . 15 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → dom (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
29 relssres 6027 . . . . . . . . . . . . . . 15 ((Rel (𝑅𝑟𝑁) ∧ dom (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑁))
3022, 28, 29syl2anc 582 . . . . . . . . . . . . . 14 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑁))
318, 30eqtrid 2777 . . . . . . . . . . . . 13 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅𝑟𝑁))
3210oveq2d 7435 . . . . . . . . . . . . . . 15 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑀) = (𝑅𝑟0))
33 relexp0g 15013 . . . . . . . . . . . . . . . 16 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
3417, 33syl 17 . . . . . . . . . . . . . . 15 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
3532, 34eqtrd 2765 . . . . . . . . . . . . . 14 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
3635coeq2d 5865 . . . . . . . . . . . . 13 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
3710oveq2d 7435 . . . . . . . . . . . . . . 15 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = (𝑁 + 0))
38 ax-1cn 11203 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
399, 38eqeltrdi 2833 . . . . . . . . . . . . . . . 16 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 ∈ ℂ)
4039addridd 11451 . . . . . . . . . . . . . . 15 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 0) = 𝑁)
4137, 40eqtrd 2765 . . . . . . . . . . . . . 14 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = 𝑁)
4241oveq2d 7435 . . . . . . . . . . . . 13 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟𝑁))
4331, 36, 423eqtr4d 2775 . . . . . . . . . . . 12 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
4443exp43 435 . . . . . . . . . . 11 (𝑁 = 1 → (𝑀 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
45 simp1 1133 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ (ℤ‘2))
46 simp3 1135 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑅𝑉)
47 relexpuzrel 15043 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))
4845, 46, 47syl2anc 582 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))
49 eluz2nn 12906 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℕ)
5045, 49syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ ℕ)
5150, 46, 25syl2anc 582 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)
5251, 27sstrdi 3989 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
5348, 52, 29syl2anc 582 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑁))
548, 53eqtrid 2777 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅𝑟𝑁))
55 simp2 1134 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑀 = 0)
5655oveq2d 7435 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟0))
5746, 33syl 17 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
5856, 57eqtrd 2765 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
5958coeq2d 5865 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
6055oveq2d 7435 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (𝑁 + 0))
61 eluzelcn 12872 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℂ)
6245, 61syl 17 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ ℂ)
6362addridd 11451 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 0) = 𝑁)
6460, 63eqtrd 2765 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 𝑁)
6564oveq2d 7435 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟𝑁))
6654, 59, 653eqtr4d 2775 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
6766a1d 25 . . . . . . . . . . . 12 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
68673exp 1116 . . . . . . . . . . 11 (𝑁 ∈ (ℤ‘2) → (𝑀 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
6944, 68jaoi 855 . . . . . . . . . 10 ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ‘2)) → (𝑀 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
707, 69sylbi 216 . . . . . . . . 9 (𝑁 ∈ ℕ → (𝑀 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
7170com12 32 . . . . . . . 8 (𝑀 = 0 → (𝑁 ∈ ℕ → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
726, 71jaoi 855 . . . . . . 7 ((𝑀 ∈ ℕ ∨ 𝑀 = 0) → (𝑁 ∈ ℕ → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
732, 72sylbi 216 . . . . . 6 (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
7473com12 32 . . . . 5 (𝑁 ∈ ℕ → (𝑀 ∈ ℕ0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
75743impd 1345 . . . 4 (𝑁 ∈ ℕ → ((𝑀 ∈ ℕ0𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅)) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
76 elnn1uz2 12947 . . . . . . . . 9 (𝑀 ∈ ℕ ↔ (𝑀 = 1 ∨ 𝑀 ∈ (ℤ‘2)))
77 coires1 6270 . . . . . . . . . . . . . . 15 (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
78 relcnv 6109 . . . . . . . . . . . . . . . . 17 Rel 𝑅
79 ssun1 4170 . . . . . . . . . . . . . . . . 17 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
8078, 79pm3.2i 469 . . . . . . . . . . . . . . . 16 (Rel 𝑅 ∧ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅))
81 relssres 6027 . . . . . . . . . . . . . . . 16 ((Rel 𝑅 ∧ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = 𝑅)
8280, 81mp1i 13 . . . . . . . . . . . . . . 15 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = 𝑅)
8377, 82eqtrid 2777 . . . . . . . . . . . . . 14 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = 𝑅)
84 cnvco 5888 . . . . . . . . . . . . . . 15 ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑀) ∘ (𝑅𝑟𝑁))
85 simplr 767 . . . . . . . . . . . . . . . . . . 19 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑀 = 1)
86 1nn0 12526 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℕ0
8785, 86eqeltrdi 2833 . . . . . . . . . . . . . . . . . 18 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑀 ∈ ℕ0)
88 simprl 769 . . . . . . . . . . . . . . . . . 18 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑅𝑉)
89 relexpcnv 15026 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟𝑀))
9087, 88, 89syl2anc 582 . . . . . . . . . . . . . . . . 17 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑀) = (𝑅𝑟𝑀))
9185oveq2d 7435 . . . . . . . . . . . . . . . . 17 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑀) = (𝑅𝑟1))
92 cnvexg 7932 . . . . . . . . . . . . . . . . . . 19 (𝑅𝑉𝑅 ∈ V)
9388, 92syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑅 ∈ V)
94 relexp1g 15017 . . . . . . . . . . . . . . . . . 18 (𝑅 ∈ V → (𝑅𝑟1) = 𝑅)
9593, 94syl 17 . . . . . . . . . . . . . . . . 17 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟1) = 𝑅)
9690, 91, 953eqtrd 2769 . . . . . . . . . . . . . . . 16 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑀) = 𝑅)
97 simpll 765 . . . . . . . . . . . . . . . . . . 19 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 = 0)
98 0nn0 12525 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℕ0
9997, 98eqeltrdi 2833 . . . . . . . . . . . . . . . . . 18 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 ∈ ℕ0)
100 relexpcnv 15026 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))
10199, 88, 100syl2anc 582 . . . . . . . . . . . . . . . . 17 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))
10297oveq2d 7435 . . . . . . . . . . . . . . . . 17 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑁) = (𝑅𝑟0))
103 relexp0g 15013 . . . . . . . . . . . . . . . . . 18 (𝑅 ∈ V → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
10493, 103syl 17 . . . . . . . . . . . . . . . . 17 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
105101, 102, 1043eqtrd 2769 . . . . . . . . . . . . . . . 16 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
10696, 105coeq12d 5867 . . . . . . . . . . . . . . 15 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑀) ∘ (𝑅𝑟𝑁)) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
10784, 106eqtrid 2777 . . . . . . . . . . . . . 14 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
10899, 87nn0addcld 12574 . . . . . . . . . . . . . . . 16 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) ∈ ℕ0)
109 relexpcnv 15026 . . . . . . . . . . . . . . . 16 (((𝑁 + 𝑀) ∈ ℕ0𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
110108, 88, 109syl2anc 582 . . . . . . . . . . . . . . 15 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
11197, 85oveq12d 7437 . . . . . . . . . . . . . . . . 17 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = (0 + 1))
112 0p1e1 12372 . . . . . . . . . . . . . . . . 17 (0 + 1) = 1
113111, 112eqtrdi 2781 . . . . . . . . . . . . . . . 16 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = 1)
114113oveq2d 7435 . . . . . . . . . . . . . . 15 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟1))
115110, 114, 953eqtrd 2769 . . . . . . . . . . . . . 14 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟(𝑁 + 𝑀)) = 𝑅)
11683, 107, 1153eqtr4d 2775 . . . . . . . . . . . . 13 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
117 relco 6113 . . . . . . . . . . . . . 14 Rel ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀))
118 simprr 771 . . . . . . . . . . . . . . . 16 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑁 + 𝑀) = 1 → Rel 𝑅))
119113, 118mpd 15 . . . . . . . . . . . . . . 15 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → Rel 𝑅)
120113oveq2d 7435 . . . . . . . . . . . . . . . . 17 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟1))
12188, 18syl 17 . . . . . . . . . . . . . . . . 17 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟1) = 𝑅)
122120, 121eqtrd 2765 . . . . . . . . . . . . . . . 16 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟(𝑁 + 𝑀)) = 𝑅)
123122releqd 5780 . . . . . . . . . . . . . . 15 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (Rel (𝑅𝑟(𝑁 + 𝑀)) ↔ Rel 𝑅))
124119, 123mpbird 256 . . . . . . . . . . . . . 14 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → Rel (𝑅𝑟(𝑁 + 𝑀)))
125 cnveqb 6202 . . . . . . . . . . . . . 14 ((Rel ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ∧ Rel (𝑅𝑟(𝑁 + 𝑀))) → (((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)) ↔ ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
126117, 124, 125sylancr 585 . . . . . . . . . . . . 13 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)) ↔ ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
127116, 126mpbird 256 . . . . . . . . . . . 12 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
128127exp43 435 . . . . . . . . . . 11 (𝑁 = 0 → (𝑀 = 1 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
129128com12 32 . . . . . . . . . 10 (𝑀 = 1 → (𝑁 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
130 coires1 6270 . . . . . . . . . . . . . . . . 17 ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅))
131 simp2 1134 . . . . . . . . . . . . . . . . . . 19 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ (ℤ‘2))
132 simp3 1135 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑅𝑉)
133132, 92syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑅 ∈ V)
134 relexpuzrel 15043 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ (ℤ‘2) ∧ 𝑅 ∈ V) → Rel (𝑅𝑟𝑀))
135131, 133, 134syl2anc 582 . . . . . . . . . . . . . . . . . 18 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑀))
136 eluz2nn 12906 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ (ℤ‘2) → 𝑀 ∈ ℕ)
137131, 136syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℕ)
138 relexpnndm 15032 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 ∈ ℕ ∧ 𝑅 ∈ V) → dom (𝑅𝑟𝑀) ⊆ dom 𝑅)
139137, 133, 138syl2anc 582 . . . . . . . . . . . . . . . . . . 19 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → dom (𝑅𝑟𝑀) ⊆ dom 𝑅)
140139, 79sstrdi 3989 . . . . . . . . . . . . . . . . . 18 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → dom (𝑅𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅))
141 relssres 6027 . . . . . . . . . . . . . . . . . 18 ((Rel (𝑅𝑟𝑀) ∧ dom (𝑅𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑀))
142135, 140, 141syl2anc 582 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑀))
143130, 142eqtrid 2777 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅𝑟𝑀))
144 simp1 1133 . . . . . . . . . . . . . . . . . . 19 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑁 = 0)
145144oveq2d 7435 . . . . . . . . . . . . . . . . . 18 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
146133, 103syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
147145, 146eqtrd 2765 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
148147coeq2d 5865 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ∘ (𝑅𝑟𝑁)) = ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
149144oveq1d 7434 . . . . . . . . . . . . . . . . . 18 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (0 + 𝑀))
150 eluzelcn 12872 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (ℤ‘2) → 𝑀 ∈ ℂ)
151131, 150syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℂ)
152151addlidd 11452 . . . . . . . . . . . . . . . . . 18 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (0 + 𝑀) = 𝑀)
153149, 152eqtrd 2765 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 𝑀)
154153oveq2d 7435 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟𝑀))
155143, 148, 1543eqtr4d 2775 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ∘ (𝑅𝑟𝑁)) = (𝑅𝑟(𝑁 + 𝑀)))
156 nnnn0 12517 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
157131, 136, 1563syl 18 . . . . . . . . . . . . . . . . . 18 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℕ0)
158157, 132, 89syl2anc 582 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟𝑀))
159144, 98eqeltrdi 2833 . . . . . . . . . . . . . . . . . 18 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑁 ∈ ℕ0)
160159, 132, 100syl2anc 582 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))
161158, 160coeq12d 5867 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ∘ (𝑅𝑟𝑁)) = ((𝑅𝑟𝑀) ∘ (𝑅𝑟𝑁)))
16284, 161eqtrid 2777 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑀) ∘ (𝑅𝑟𝑁)))
163159, 157nn0addcld 12574 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑁 + 𝑀) ∈ ℕ0)
164163, 132, 109syl2anc 582 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
165155, 162, 1643eqtr4d 2775 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
166159nn0cnd 12572 . . . . . . . . . . . . . . . . . 18 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑁 ∈ ℂ)
167151, 166addcomd 11453 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑀 + 𝑁) = (𝑁 + 𝑀))
168 uzaddcl 12926 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ (ℤ‘2))
169131, 159, 168syl2anc 582 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑀 + 𝑁) ∈ (ℤ‘2))
170167, 169eqeltrrd 2826 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑁 + 𝑀) ∈ (ℤ‘2))
171 relexpuzrel 15043 . . . . . . . . . . . . . . . 16 (((𝑁 + 𝑀) ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟(𝑁 + 𝑀)))
172170, 132, 171syl2anc 582 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟(𝑁 + 𝑀)))
173117, 172, 125sylancr 585 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)) ↔ ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
174165, 173mpbird 256 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
175174a1d 25 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
1761753exp 1116 . . . . . . . . . . 11 (𝑁 = 0 → (𝑀 ∈ (ℤ‘2) → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
177176com12 32 . . . . . . . . . 10 (𝑀 ∈ (ℤ‘2) → (𝑁 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
178129, 177jaoi 855 . . . . . . . . 9 ((𝑀 = 1 ∨ 𝑀 ∈ (ℤ‘2)) → (𝑁 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
17976, 178sylbi 216 . . . . . . . 8 (𝑀 ∈ ℕ → (𝑁 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
180 coires1 6270 . . . . . . . . . . . . 13 ((𝑅𝑟0) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅𝑟0) ↾ (dom 𝑅 ∪ ran 𝑅))
181 simp3 1135 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑅𝑉)
182 relexp0rel 15028 . . . . . . . . . . . . . . 15 (𝑅𝑉 → Rel (𝑅𝑟0))
183181, 182syl 17 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → Rel (𝑅𝑟0))
184181, 33syl 17 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
185184dmeqd 5908 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟0) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
186 dmresi 6056 . . . . . . . . . . . . . . . 16 dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
187185, 186eqtrdi 2781 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟0) = (dom 𝑅 ∪ ran 𝑅))
188 eqimss 4035 . . . . . . . . . . . . . . 15 (dom (𝑅𝑟0) = (dom 𝑅 ∪ ran 𝑅) → dom (𝑅𝑟0) ⊆ (dom 𝑅 ∪ ran 𝑅))
189187, 188syl 17 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟0) ⊆ (dom 𝑅 ∪ ran 𝑅))
190 relssres 6027 . . . . . . . . . . . . . 14 ((Rel (𝑅𝑟0) ∧ dom (𝑅𝑟0) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅𝑟0) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟0))
191183, 189, 190syl2anc 582 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟0) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟0))
192180, 191eqtrid 2777 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟0) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅𝑟0))
193 simp1 1133 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 = 0)
194193oveq2d 7435 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
195 simp2 1134 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑀 = 0)
196195oveq2d 7435 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟0))
197196, 184eqtrd 2765 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
198194, 197coeq12d 5867 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟0) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
199193, 195oveq12d 7437 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (0 + 0))
200 00id 11426 . . . . . . . . . . . . . 14 (0 + 0) = 0
201199, 200eqtrdi 2781 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 0)
202201oveq2d 7435 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟0))
203192, 198, 2023eqtr4d 2775 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
204203a1d 25 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
2052043exp 1116 . . . . . . . . 9 (𝑁 = 0 → (𝑀 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
206205com12 32 . . . . . . . 8 (𝑀 = 0 → (𝑁 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
207179, 206jaoi 855 . . . . . . 7 ((𝑀 ∈ ℕ ∨ 𝑀 = 0) → (𝑁 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
2082, 207sylbi 216 . . . . . 6 (𝑀 ∈ ℕ0 → (𝑁 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
209208com12 32 . . . . 5 (𝑁 = 0 → (𝑀 ∈ ℕ0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
2102093impd 1345 . . . 4 (𝑁 = 0 → ((𝑀 ∈ ℕ0𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅)) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
21175, 210jaoi 855 . . 3 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝑀 ∈ ℕ0𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅)) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
2121, 211sylbi 216 . 2 (𝑁 ∈ ℕ0 → ((𝑀 ∈ ℕ0𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅)) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
213212imp 405 1 ((𝑁 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3461  cun 3942  wss 3944   I cid 5575  ccnv 5677  dom cdm 5678  ran crn 5679  cres 5680  ccom 5682  Rel wrel 5683  cfv 6549  (class class class)co 7419  cc 11143  0cc0 11145  1c1 11146   + caddc 11148  cn 12250  2c2 12305  0cn0 12510  cuz 12860  𝑟crelexp 15010
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-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222
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 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-nn 12251  df-2 12313  df-n0 12511  df-z 12597  df-uz 12861  df-seq 14008  df-relexp 15011
This theorem is referenced by:  relexpaddd  15045
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