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Theorem relexpsucnnr 13562
Description: A reduction for relation exponentiation to the right. (Contributed by RP, 22-May-2020.)
Assertion
Ref Expression
relexpsucnnr ((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅))

Proof of Theorem relexpsucnnr
Dummy variables 𝑎 𝑏 𝑧 𝑛 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2610 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))))
2 simprr 791 . . . . 5 (((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅𝑛 = (𝑁 + 1))) → 𝑛 = (𝑁 + 1))
3 dmeq 5233 . . . . . . . . . . 11 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
4 rneq 5259 . . . . . . . . . . 11 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
53, 4uneq12d 3729 . . . . . . . . . 10 (𝑟 = 𝑅 → (dom 𝑟 ∪ ran 𝑟) = (dom 𝑅 ∪ ran 𝑅))
65reseq2d 5304 . . . . . . . . 9 (𝑟 = 𝑅 → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7 eqidd 2610 . . . . . . . . . . 11 (𝑟 = 𝑅 → 1 = 1)
8 coeq2 5190 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (𝑥𝑟) = (𝑥𝑅))
98mpt2eq3dv 6597 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)) = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)))
10 id 22 . . . . . . . . . . . 12 (𝑟 = 𝑅𝑟 = 𝑅)
1110mpteq2dv 4667 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑧 ∈ V ↦ 𝑟) = (𝑧 ∈ V ↦ 𝑅))
127, 9, 11seqeq123d 12630 . . . . . . . . . 10 (𝑟 = 𝑅 → seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟)) = seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅)))
1312fveq1d 6090 . . . . . . . . 9 (𝑟 = 𝑅 → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘(𝑁 + 1)) = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)))
146, 13ifeq12d 4055 . . . . . . . 8 (𝑟 = 𝑅 → if((𝑁 + 1) = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘(𝑁 + 1))) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1))))
1514ad2antrl 759 . . . . . . 7 (((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅 ∧ (𝑁 + 1) = (𝑁 + 1))) → if((𝑁 + 1) = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘(𝑁 + 1))) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1))))
1615a1i 11 . . . . . 6 (𝑛 = (𝑁 + 1) → (((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅 ∧ (𝑁 + 1) = (𝑁 + 1))) → if((𝑁 + 1) = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘(𝑁 + 1))) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)))))
17 eqeq1 2613 . . . . . . . 8 (𝑛 = (𝑁 + 1) → (𝑛 = (𝑁 + 1) ↔ (𝑁 + 1) = (𝑁 + 1)))
1817anbi2d 735 . . . . . . 7 (𝑛 = (𝑁 + 1) → ((𝑟 = 𝑅𝑛 = (𝑁 + 1)) ↔ (𝑟 = 𝑅 ∧ (𝑁 + 1) = (𝑁 + 1))))
1918anbi2d 735 . . . . . 6 (𝑛 = (𝑁 + 1) → (((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅𝑛 = (𝑁 + 1))) ↔ ((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅 ∧ (𝑁 + 1) = (𝑁 + 1)))))
20 eqeq1 2613 . . . . . . . 8 (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
21 fveq2 6088 . . . . . . . 8 (𝑛 = (𝑁 + 1) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛) = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘(𝑁 + 1)))
2220, 21ifbieq2d 4060 . . . . . . 7 (𝑛 = (𝑁 + 1) → if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘(𝑁 + 1))))
2322eqeq1d 2611 . . . . . 6 (𝑛 = (𝑁 + 1) → (if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1))) ↔ if((𝑁 + 1) = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘(𝑁 + 1))) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)))))
2416, 19, 233imtr4d 281 . . . . 5 (𝑛 = (𝑁 + 1) → (((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅𝑛 = (𝑁 + 1))) → if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)))))
252, 24mpcom 37 . . . 4 (((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅𝑛 = (𝑁 + 1))) → if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1))))
26 elex 3184 . . . . 5 (𝑅𝑉𝑅 ∈ V)
2726adantr 479 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑅 ∈ V)
28 simpr 475 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
2928peano2nnd 10887 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ)
3029nnnn0d 11201 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ0)
31 dmexg 6967 . . . . . . . 8 (𝑅𝑉 → dom 𝑅 ∈ V)
32 rnexg 6968 . . . . . . . 8 (𝑅𝑉 → ran 𝑅 ∈ V)
33 unexg 6835 . . . . . . . 8 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
3431, 32, 33syl2anc 690 . . . . . . 7 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
35 resiexg 6972 . . . . . . 7 ((dom 𝑅 ∪ ran 𝑅) ∈ V → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
3634, 35syl 17 . . . . . 6 (𝑅𝑉 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
3736adantr 479 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
38 fvex 6098 . . . . . 6 (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)) ∈ V
3938a1i 11 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)) ∈ V)
4037, 39ifcld 4080 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1))) ∈ V)
411, 25, 27, 30, 40ovmpt2d 6664 . . 3 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))(𝑁 + 1)) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1))))
42 nnne0 10903 . . . . . 6 ((𝑁 + 1) ∈ ℕ → (𝑁 + 1) ≠ 0)
4342neneqd 2786 . . . . 5 ((𝑁 + 1) ∈ ℕ → ¬ (𝑁 + 1) = 0)
4429, 43syl 17 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → ¬ (𝑁 + 1) = 0)
4544iffalsed 4046 . . 3 ((𝑅𝑉𝑁 ∈ ℕ) → if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1))) = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)))
46 elnnuz 11559 . . . . . . 7 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ‘1))
4746biimpi 204 . . . . . 6 (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ‘1))
4847adantl 480 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ∈ (ℤ‘1))
49 seqp1 12636 . . . . 5 (𝑁 ∈ (ℤ‘1) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)) = ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅))((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1))))
5048, 49syl 17 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)) = ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅))((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1))))
51 ovex 6555 . . . . . 6 (𝑁 + 1) ∈ V
52 simpl 471 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑅𝑉)
53 eqidd 2610 . . . . . . 7 (𝑧 = (𝑁 + 1) → 𝑅 = 𝑅)
54 eqid 2609 . . . . . . 7 (𝑧 ∈ V ↦ 𝑅) = (𝑧 ∈ V ↦ 𝑅)
5553, 54fvmptg 6174 . . . . . 6 (((𝑁 + 1) ∈ V ∧ 𝑅𝑉) → ((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1)) = 𝑅)
5651, 52, 55sylancr 693 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → ((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1)) = 𝑅)
5756oveq2d 6543 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅))((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1))) = ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅))𝑅))
58 nfcv 2750 . . . . . . 7 𝑎(𝑥𝑅)
59 nfcv 2750 . . . . . . 7 𝑏(𝑥𝑅)
60 nfcv 2750 . . . . . . 7 𝑥(𝑎𝑅)
61 nfcv 2750 . . . . . . 7 𝑦(𝑎𝑅)
62 simpl 471 . . . . . . . 8 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑥 = 𝑎)
6362coeq1d 5193 . . . . . . 7 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑥𝑅) = (𝑎𝑅))
6458, 59, 60, 61, 63cbvmpt2 6610 . . . . . 6 (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎𝑅))
65 oveq 6533 . . . . . 6 ((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎𝑅)) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅))𝑅) = ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎𝑅))𝑅))
6664, 65mp1i 13 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅))𝑅) = ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎𝑅))𝑅))
67 eqidd 2610 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎𝑅)))
68 simprl 789 . . . . . . 7 (((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑎 = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∧ 𝑏 = 𝑅)) → 𝑎 = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁))
6968coeq1d 5193 . . . . . 6 (((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑎 = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∧ 𝑏 = 𝑅)) → (𝑎𝑅) = ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∘ 𝑅))
70 fvex 6098 . . . . . . 7 (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V
7170a1i 11 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V)
72 coexg 6988 . . . . . . 7 (((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V ∧ 𝑅𝑉) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∘ 𝑅) ∈ V)
7370, 52, 72sylancr 693 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∘ 𝑅) ∈ V)
7467, 69, 71, 27, 73ovmpt2d 6664 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎𝑅))𝑅) = ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∘ 𝑅))
75 simpr 475 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑛 = 𝑁) → 𝑛 = 𝑁)
7675eqeq1d 2611 . . . . . . . . . 10 ((𝑟 = 𝑅𝑛 = 𝑁) → (𝑛 = 0 ↔ 𝑁 = 0))
776adantr 479 . . . . . . . . . 10 ((𝑟 = 𝑅𝑛 = 𝑁) → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7812adantr 479 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑛 = 𝑁) → seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟)) = seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅)))
7978, 75fveq12d 6094 . . . . . . . . . 10 ((𝑟 = 𝑅𝑛 = 𝑁) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛) = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁))
8076, 77, 79ifbieq12d 4062 . . . . . . . . 9 ((𝑟 = 𝑅𝑛 = 𝑁) → if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)) = if(𝑁 = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)))
8180adantl 480 . . . . . . . 8 (((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅𝑛 = 𝑁)) → if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)) = if(𝑁 = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)))
8228nnnn0d 11201 . . . . . . . 8 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
8337, 71ifcld 4080 . . . . . . . 8 ((𝑅𝑉𝑁 ∈ ℕ) → if(𝑁 = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)) ∈ V)
841, 81, 27, 82, 83ovmpt2d 6664 . . . . . . 7 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) = if(𝑁 = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)))
85 nnne0 10903 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝑁 ≠ 0)
8685adantl 480 . . . . . . . . 9 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ≠ 0)
8786neneqd 2786 . . . . . . . 8 ((𝑅𝑉𝑁 ∈ ℕ) → ¬ 𝑁 = 0)
8887iffalsed 4046 . . . . . . 7 ((𝑅𝑉𝑁 ∈ ℕ) → if(𝑁 = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)) = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁))
8984, 88eqtr2d 2644 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) = (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁))
9089coeq1d 5193 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∘ 𝑅) = ((𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) ∘ 𝑅))
9166, 74, 903eqtrd 2647 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅))𝑅) = ((𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) ∘ 𝑅))
9250, 57, 913eqtrd 2647 . . 3 ((𝑅𝑉𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)) = ((𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) ∘ 𝑅))
9341, 45, 923eqtrd 2647 . 2 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))(𝑁 + 1)) = ((𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) ∘ 𝑅))
94 df-relexp 13558 . . 3 𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))
95 oveq 6533 . . . . 5 (↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) → (𝑅𝑟(𝑁 + 1)) = (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))(𝑁 + 1)))
96 oveq 6533 . . . . . 6 (↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) → (𝑅𝑟𝑁) = (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁))
9796coeq1d 5193 . . . . 5 (↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) → ((𝑅𝑟𝑁) ∘ 𝑅) = ((𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) ∘ 𝑅))
9895, 97eqeq12d 2624 . . . 4 (↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) → ((𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅) ↔ (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))(𝑁 + 1)) = ((𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) ∘ 𝑅)))
9998imbi2d 328 . . 3 (↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) → (((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅)) ↔ ((𝑅𝑉𝑁 ∈ ℕ) → (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))(𝑁 + 1)) = ((𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) ∘ 𝑅))))
10094, 99ax-mp 5 . 2 (((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅)) ↔ ((𝑅𝑉𝑁 ∈ ℕ) → (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))(𝑁 + 1)) = ((𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) ∘ 𝑅)))
10193, 100mpbir 219 1 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779  Vcvv 3172  cun 3537  ifcif 4035  cmpt 4637   I cid 4938  dom cdm 5028  ran crn 5029  cres 5030  ccom 5032  cfv 5790  (class class class)co 6527  cmpt2 6529  0cc0 9793  1c1 9794   + caddc 9796  cn 10870  0cn0 11142  cuz 11522  seqcseq 12621  𝑟crelexp 13557
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-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
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 6936  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-er 7607  df-en 7820  df-dom 7821  df-sdom 7822  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-nn 10871  df-n0 11143  df-z 11214  df-uz 11523  df-seq 12622  df-relexp 13558
This theorem is referenced by:  relexpsucr  13566  relexpsucnnl  13569  relexpcnv  13572  relexprelg  13575  relexpnndm  13578  relexp2  36812  relexpxpnnidm  36838  relexpss1d  36840  relexpmulnn  36844  trclrelexplem  36846  relexp0a  36851  trclfvcom  36858  cotrcltrcl  36860  trclfvdecomr  36863  cotrclrcl  36877
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