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Theorem relexpaddg 13722
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 11239 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 elnn0 11239 . . . . . . 7 (𝑀 ∈ ℕ0 ↔ (𝑀 ∈ ℕ ∨ 𝑀 = 0))
3 relexpaddnn 13720 . . . . . . . . . . 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 1261 . . . . . . . . 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 11709 . . . . . . . . . 10 (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ‘2)))
8 coires1 5615 . . . . . . . . . . . . . 14 ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅))
9 simpll 789 . . . . . . . . . . . . . . . . . . 19 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 = 1)
10 simplr 791 . . . . . . . . . . . . . . . . . . 19 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑀 = 0)
119, 10oveq12d 6623 . . . . . . . . . . . . . . . . . 18 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = (1 + 0))
12 1p0e1 11078 . . . . . . . . . . . . . . . . . 18 (1 + 0) = 1
1311, 12syl6eq 2676 . . . . . . . . . . . . . . . . 17 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = 1)
14 simprr 795 . . . . . . . . . . . . . . . . 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 6621 . . . . . . . . . . . . . . . . . 18 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑁) = (𝑅𝑟1))
17 simprl 793 . . . . . . . . . . . . . . . . . . 19 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑅𝑉)
18 relexp1g 13695 . . . . . . . . . . . . . . . . . . 19 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
1917, 18syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟1) = 𝑅)
2016, 19eqtrd 2660 . . . . . . . . . . . . . . . . 17 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑁) = 𝑅)
2120releqd 5169 . . . . . . . . . . . . . . . 16 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (Rel (𝑅𝑟𝑁) ↔ Rel 𝑅))
2215, 21mpbird 247 . . . . . . . . . . . . . . 15 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → Rel (𝑅𝑟𝑁))
23 1nn 10976 . . . . . . . . . . . . . . . . . 18 1 ∈ ℕ
249, 23syl6eqel 2712 . . . . . . . . . . . . . . . . 17 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 ∈ ℕ)
25 relexpnndm 13710 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)
2624, 17, 25syl2anc 692 . . . . . . . . . . . . . . . 16 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)
27 ssun1 3759 . . . . . . . . . . . . . . . 16 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2826, 27syl6ss 3600 . . . . . . . . . . . . . . 15 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → dom (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
29 relssres 5400 . . . . . . . . . . . . . . 15 ((Rel (𝑅𝑟𝑁) ∧ dom (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑁))
3022, 28, 29syl2anc 692 . . . . . . . . . . . . . 14 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑁))
318, 30syl5eq 2672 . . . . . . . . . . . . 13 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅𝑟𝑁))
3210oveq2d 6621 . . . . . . . . . . . . . . 15 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑀) = (𝑅𝑟0))
33 relexp0g 13691 . . . . . . . . . . . . . . . 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 2660 . . . . . . . . . . . . . 14 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
3635coeq2d 5249 . . . . . . . . . . . . 13 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
3710oveq2d 6621 . . . . . . . . . . . . . . 15 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = (𝑁 + 0))
38 ax-1cn 9939 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
399, 38syl6eqel 2712 . . . . . . . . . . . . . . . 16 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 ∈ ℂ)
4039addid1d 10181 . . . . . . . . . . . . . . 15 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 0) = 𝑁)
4137, 40eqtrd 2660 . . . . . . . . . . . . . 14 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = 𝑁)
4241oveq2d 6621 . . . . . . . . . . . . 13 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟𝑁))
4331, 36, 423eqtr4d 2670 . . . . . . . . . . . 12 (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
4443exp43 639 . . . . . . . . . . 11 (𝑁 = 1 → (𝑀 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
45 simp1 1059 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ (ℤ‘2))
46 simp3 1061 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑅𝑉)
47 relexpuzrel 13721 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))
4845, 46, 47syl2anc 692 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))
49 eluz2nn 11670 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℕ)
5045, 49syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ ℕ)
5150, 46, 25syl2anc 692 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)
5251, 27syl6ss 3600 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
5348, 52, 29syl2anc 692 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑁))
548, 53syl5eq 2672 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅𝑟𝑁))
55 simp2 1060 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑀 = 0)
5655oveq2d 6621 . . . . . . . . . . . . . . . 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 2660 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
5958coeq2d 5249 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
6055oveq2d 6621 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (𝑁 + 0))
61 eluzelcn 11643 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℂ)
6245, 61syl 17 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ ℂ)
6362addid1d 10181 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 0) = 𝑁)
6460, 63eqtrd 2660 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 𝑁)
6564oveq2d 6621 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟𝑁))
6654, 59, 653eqtr4d 2670 . . . . . . . . . . . . 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 1261 . . . . . . . . . . 11 (𝑁 ∈ (ℤ‘2) → (𝑀 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
6944, 68jaoi 394 . . . . . . . . . 10 ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ‘2)) → (𝑀 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
707, 69sylbi 207 . . . . . . . . 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 394 . . . . . . 7 ((𝑀 ∈ ℕ ∨ 𝑀 = 0) → (𝑁 ∈ ℕ → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
732, 72sylbi 207 . . . . . 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 1278 . . . 4 (𝑁 ∈ ℕ → ((𝑀 ∈ ℕ0𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅)) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
76 elnn1uz2 11709 . . . . . . . . 9 (𝑀 ∈ ℕ ↔ (𝑀 = 1 ∨ 𝑀 ∈ (ℤ‘2)))
77 coires1 5615 . . . . . . . . . . . . . . 15 (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
78 relcnv 5466 . . . . . . . . . . . . . . . . 17 Rel 𝑅
79 ssun1 3759 . . . . . . . . . . . . . . . . 17 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
8078, 79pm3.2i 471 . . . . . . . . . . . . . . . 16 (Rel 𝑅 ∧ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅))
81 relssres 5400 . . . . . . . . . . . . . . . 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, 82syl5eq 2672 . . . . . . . . . . . . . 14 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = 𝑅)
84 cnvco 5273 . . . . . . . . . . . . . . 15 ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑀) ∘ (𝑅𝑟𝑁))
85 simplr 791 . . . . . . . . . . . . . . . . . . 19 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑀 = 1)
86 1nn0 11253 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℕ0
8785, 86syl6eqel 2712 . . . . . . . . . . . . . . . . . 18 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑀 ∈ ℕ0)
88 simprl 793 . . . . . . . . . . . . . . . . . 18 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑅𝑉)
89 relexpcnv 13704 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟𝑀))
9087, 88, 89syl2anc 692 . . . . . . . . . . . . . . . . 17 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑀) = (𝑅𝑟𝑀))
9185oveq2d 6621 . . . . . . . . . . . . . . . . 17 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑀) = (𝑅𝑟1))
92 cnvexg 7062 . . . . . . . . . . . . . . . . . . 19 (𝑅𝑉𝑅 ∈ V)
9388, 92syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑅 ∈ V)
94 relexp1g 13695 . . . . . . . . . . . . . . . . . 18 (𝑅 ∈ V → (𝑅𝑟1) = 𝑅)
9593, 94syl 17 . . . . . . . . . . . . . . . . 17 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟1) = 𝑅)
9690, 91, 953eqtrd 2664 . . . . . . . . . . . . . . . 16 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑀) = 𝑅)
97 simpll 789 . . . . . . . . . . . . . . . . . . 19 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 = 0)
98 0nn0 11252 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℕ0
9997, 98syl6eqel 2712 . . . . . . . . . . . . . . . . . 18 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 ∈ ℕ0)
100 relexpcnv 13704 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))
10199, 88, 100syl2anc 692 . . . . . . . . . . . . . . . . 17 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))
10297oveq2d 6621 . . . . . . . . . . . . . . . . 17 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑁) = (𝑅𝑟0))
103 relexp0g 13691 . . . . . . . . . . . . . . . . . 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 2664 . . . . . . . . . . . . . . . 16 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
10696, 105coeq12d 5251 . . . . . . . . . . . . . . 15 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑀) ∘ (𝑅𝑟𝑁)) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
10784, 106syl5eq 2672 . . . . . . . . . . . . . 14 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
10899, 87nn0addcld 11300 . . . . . . . . . . . . . . . 16 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) ∈ ℕ0)
109 relexpcnv 13704 . . . . . . . . . . . . . . . 16 (((𝑁 + 𝑀) ∈ ℕ0𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
110108, 88, 109syl2anc 692 . . . . . . . . . . . . . . 15 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
11197, 85oveq12d 6623 . . . . . . . . . . . . . . . . 17 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = (0 + 1))
112 0p1e1 11077 . . . . . . . . . . . . . . . . 17 (0 + 1) = 1
113111, 112syl6eq 2676 . . . . . . . . . . . . . . . 16 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = 1)
114113oveq2d 6621 . . . . . . . . . . . . . . 15 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟1))
115110, 114, 953eqtrd 2664 . . . . . . . . . . . . . 14 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟(𝑁 + 𝑀)) = 𝑅)
11683, 107, 1153eqtr4d 2670 . . . . . . . . . . . . 13 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
117 relco 5595 . . . . . . . . . . . . . 14 Rel ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀))
118 simprr 795 . . . . . . . . . . . . . . . 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 6621 . . . . . . . . . . . . . . . . 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 2660 . . . . . . . . . . . . . . . 16 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅𝑟(𝑁 + 𝑀)) = 𝑅)
123122releqd 5169 . . . . . . . . . . . . . . 15 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (Rel (𝑅𝑟(𝑁 + 𝑀)) ↔ Rel 𝑅))
124119, 123mpbird 247 . . . . . . . . . . . . . 14 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → Rel (𝑅𝑟(𝑁 + 𝑀)))
125 cnveqb 5552 . . . . . . . . . . . . . 14 ((Rel ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ∧ Rel (𝑅𝑟(𝑁 + 𝑀))) → (((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)) ↔ ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
126117, 124, 125sylancr 694 . . . . . . . . . . . . 13 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)) ↔ ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
127116, 126mpbird 247 . . . . . . . . . . . 12 (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
128127exp43 639 . . . . . . . . . . 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 5615 . . . . . . . . . . . . . . . . 17 ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅))
131 simp2 1060 . . . . . . . . . . . . . . . . . . 19 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ (ℤ‘2))
132 simp3 1061 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑅𝑉)
133132, 92syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑅 ∈ V)
134 relexpuzrel 13721 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ (ℤ‘2) ∧ 𝑅 ∈ V) → Rel (𝑅𝑟𝑀))
135131, 133, 134syl2anc 692 . . . . . . . . . . . . . . . . . 18 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑀))
136 eluz2nn 11670 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ (ℤ‘2) → 𝑀 ∈ ℕ)
137131, 136syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℕ)
138 relexpnndm 13710 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 ∈ ℕ ∧ 𝑅 ∈ V) → dom (𝑅𝑟𝑀) ⊆ dom 𝑅)
139137, 133, 138syl2anc 692 . . . . . . . . . . . . . . . . . . 19 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → dom (𝑅𝑟𝑀) ⊆ dom 𝑅)
140139, 79syl6ss 3600 . . . . . . . . . . . . . . . . . 18 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → dom (𝑅𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅))
141 relssres 5400 . . . . . . . . . . . . . . . . . 18 ((Rel (𝑅𝑟𝑀) ∧ dom (𝑅𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑀))
142135, 140, 141syl2anc 692 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑀))
143130, 142syl5eq 2672 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅𝑟𝑀))
144 simp1 1059 . . . . . . . . . . . . . . . . . . 19 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑁 = 0)
145144oveq2d 6621 . . . . . . . . . . . . . . . . . 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 2660 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
148147coeq2d 5249 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ∘ (𝑅𝑟𝑁)) = ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
149144oveq1d 6620 . . . . . . . . . . . . . . . . . 18 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (0 + 𝑀))
150 eluzelcn 11643 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (ℤ‘2) → 𝑀 ∈ ℂ)
151131, 150syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℂ)
152151addid2d 10182 . . . . . . . . . . . . . . . . . 18 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (0 + 𝑀) = 𝑀)
153149, 152eqtrd 2660 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 𝑀)
154153oveq2d 6621 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟𝑀))
155143, 148, 1543eqtr4d 2670 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ∘ (𝑅𝑟𝑁)) = (𝑅𝑟(𝑁 + 𝑀)))
156 nnnn0 11244 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
157131, 136, 1563syl 18 . . . . . . . . . . . . . . . . . 18 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℕ0)
158157, 132, 89syl2anc 692 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟𝑀))
159144, 98syl6eqel 2712 . . . . . . . . . . . . . . . . . 18 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑁 ∈ ℕ0)
160159, 132, 100syl2anc 692 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))
161158, 160coeq12d 5251 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ∘ (𝑅𝑟𝑁)) = ((𝑅𝑟𝑀) ∘ (𝑅𝑟𝑁)))
16284, 161syl5eq 2672 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑀) ∘ (𝑅𝑟𝑁)))
163159, 157nn0addcld 11300 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑁 + 𝑀) ∈ ℕ0)
164163, 132, 109syl2anc 692 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
165155, 162, 1643eqtr4d 2670 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
166159nn0cnd 11298 . . . . . . . . . . . . . . . . . 18 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑁 ∈ ℂ)
167151, 166addcomd 10183 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑀 + 𝑁) = (𝑁 + 𝑀))
168 uzaddcl 11688 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ (ℤ‘2))
169131, 159, 168syl2anc 692 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑀 + 𝑁) ∈ (ℤ‘2))
170167, 169eqeltrrd 2705 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑁 + 𝑀) ∈ (ℤ‘2))
171 relexpuzrel 13721 . . . . . . . . . . . . . . . 16 (((𝑁 + 𝑀) ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟(𝑁 + 𝑀)))
172170, 132, 171syl2anc 692 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟(𝑁 + 𝑀)))
173117, 172, 125sylancr 694 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)) ↔ ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
174165, 173mpbird 247 . . . . . . . . . . . . 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 1261 . . . . . . . . . . 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 394 . . . . . . . . 9 ((𝑀 = 1 ∨ 𝑀 ∈ (ℤ‘2)) → (𝑁 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
17976, 178sylbi 207 . . . . . . . 8 (𝑀 ∈ ℕ → (𝑁 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
180 coires1 5615 . . . . . . . . . . . . 13 ((𝑅𝑟0) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅𝑟0) ↾ (dom 𝑅 ∪ ran 𝑅))
181 simp3 1061 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑅𝑉)
182 relexp0rel 13706 . . . . . . . . . . . . . . 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 5291 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟0) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
186 dmresi 5420 . . . . . . . . . . . . . . . 16 dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
187185, 186syl6eq 2676 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟0) = (dom 𝑅 ∪ ran 𝑅))
188 eqimss 3641 . . . . . . . . . . . . . . 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 5400 . . . . . . . . . . . . . 14 ((Rel (𝑅𝑟0) ∧ dom (𝑅𝑟0) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅𝑟0) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟0))
191183, 189, 190syl2anc 692 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟0) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟0))
192180, 191syl5eq 2672 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟0) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅𝑟0))
193 simp1 1059 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 = 0)
194193oveq2d 6621 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
195 simp2 1060 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑀 = 0)
196195oveq2d 6621 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟0))
197196, 184eqtrd 2660 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
198194, 197coeq12d 5251 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟0) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
199193, 195oveq12d 6623 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (0 + 0))
200 00id 10156 . . . . . . . . . . . . . 14 (0 + 0) = 0
201199, 200syl6eq 2676 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 0)
202201oveq2d 6621 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟0))
203192, 198, 2023eqtr4d 2670 . . . . . . . . . . 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 1261 . . . . . . . . 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 394 . . . . . . 7 ((𝑀 ∈ ℕ ∨ 𝑀 = 0) → (𝑁 = 0 → (𝑅𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))))
2082, 207sylbi 207 . . . . . 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 1278 . . . 4 (𝑁 = 0 → ((𝑀 ∈ ℕ0𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅)) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
21175, 210jaoi 394 . . 3 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝑀 ∈ ℕ0𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅)) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
2121, 211sylbi 207 . 2 (𝑁 ∈ ℕ0 → ((𝑀 ∈ ℕ0𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅)) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
213212imp 445 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 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1992  Vcvv 3191  cun 3558  wss 3560   I cid 4989  ccnv 5078  dom cdm 5079  ran crn 5080  cres 5081  ccom 5083  Rel wrel 5084  cfv 5850  (class class class)co 6605  cc 9879  0cc0 9881  1c1 9882   + caddc 9884  cn 10965  2c2 11015  0cn0 11237  cuz 11631  𝑟crelexp 13689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-n0 11238  df-z 11323  df-uz 11632  df-seq 12739  df-relexp 13690
This theorem is referenced by:  relexpaddd  13723  relexpnul  37437
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