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Theorem relexpsucnnr 13982
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 2803 . . . 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 780 . . . . 5 (((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅𝑛 = (𝑁 + 1))) → 𝑛 = (𝑁 + 1))
3 dmeq 5519 . . . . . . . . . . 11 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
4 rneq 5546 . . . . . . . . . . 11 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
53, 4uneq12d 3961 . . . . . . . . . 10 (𝑟 = 𝑅 → (dom 𝑟 ∪ ran 𝑟) = (dom 𝑅 ∪ ran 𝑅))
65reseq2d 5591 . . . . . . . . 9 (𝑟 = 𝑅 → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7 eqidd 2803 . . . . . . . . . . 11 (𝑟 = 𝑅 → 1 = 1)
8 coeq2 5476 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (𝑥𝑟) = (𝑥𝑅))
98mpt2eq3dv 6945 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)) = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)))
10 id 22 . . . . . . . . . . . 12 (𝑟 = 𝑅𝑟 = 𝑅)
1110mpteq2dv 4932 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑧 ∈ V ↦ 𝑟) = (𝑧 ∈ V ↦ 𝑅))
127, 9, 11seqeq123d 13027 . . . . . . . . . 10 (𝑟 = 𝑅 → seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟)) = seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅)))
1312fveq1d 6404 . . . . . . . . 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 4293 . . . . . . . 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 710 . . . . . . 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 2806 . . . . . . . 8 (𝑛 = (𝑁 + 1) → (𝑛 = (𝑁 + 1) ↔ (𝑁 + 1) = (𝑁 + 1)))
1817anbi2d 616 . . . . . . 7 (𝑛 = (𝑁 + 1) → ((𝑟 = 𝑅𝑛 = (𝑁 + 1)) ↔ (𝑟 = 𝑅 ∧ (𝑁 + 1) = (𝑁 + 1))))
1918anbi2d 616 . . . . . 6 (𝑛 = (𝑁 + 1) → (((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅𝑛 = (𝑁 + 1))) ↔ ((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅 ∧ (𝑁 + 1) = (𝑁 + 1)))))
20 eqeq1 2806 . . . . . . . 8 (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
21 fveq2 6402 . . . . . . . 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 4298 . . . . . . 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 2804 . . . . . 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 285 . . . . 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 38 . . . 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 3402 . . . . 5 (𝑅𝑉𝑅 ∈ V)
2726adantr 468 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑅 ∈ V)
28 simpr 473 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
2928peano2nnd 11316 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ)
3029nnnn0d 11611 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ0)
31 dmexg 7321 . . . . . . . 8 (𝑅𝑉 → dom 𝑅 ∈ V)
32 rnexg 7322 . . . . . . . 8 (𝑅𝑉 → ran 𝑅 ∈ V)
33 unexg 7183 . . . . . . . 8 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
3431, 32, 33syl2anc 575 . . . . . . 7 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
35 resiexg 7326 . . . . . . 7 ((dom 𝑅 ∪ ran 𝑅) ∈ V → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
3634, 35syl 17 . . . . . 6 (𝑅𝑉 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
3736adantr 468 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
38 fvexd 6417 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)) ∈ V)
3937, 38ifcld 4318 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1))) ∈ V)
401, 25, 27, 30, 39ovmpt2d 7012 . . 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))))
41 nnne0 11333 . . . . . 6 ((𝑁 + 1) ∈ ℕ → (𝑁 + 1) ≠ 0)
4241neneqd 2979 . . . . 5 ((𝑁 + 1) ∈ ℕ → ¬ (𝑁 + 1) = 0)
4329, 42syl 17 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → ¬ (𝑁 + 1) = 0)
4443iffalsed 4284 . . 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)))
45 elnnuz 11936 . . . . . . 7 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ‘1))
4645biimpi 207 . . . . . 6 (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ‘1))
4746adantl 469 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ∈ (ℤ‘1))
48 seqp1 13033 . . . . 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))))
4947, 48syl 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))))
50 ovex 6900 . . . . . 6 (𝑁 + 1) ∈ V
51 simpl 470 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑅𝑉)
52 eqidd 2803 . . . . . . 7 (𝑧 = (𝑁 + 1) → 𝑅 = 𝑅)
53 eqid 2802 . . . . . . 7 (𝑧 ∈ V ↦ 𝑅) = (𝑧 ∈ V ↦ 𝑅)
5452, 53fvmptg 6495 . . . . . 6 (((𝑁 + 1) ∈ V ∧ 𝑅𝑉) → ((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1)) = 𝑅)
5550, 51, 54sylancr 577 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → ((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1)) = 𝑅)
5655oveq2d 6884 . . . 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))
57 nfcv 2944 . . . . . . 7 𝑎(𝑥𝑅)
58 nfcv 2944 . . . . . . 7 𝑏(𝑥𝑅)
59 nfcv 2944 . . . . . . 7 𝑥(𝑎𝑅)
60 nfcv 2944 . . . . . . 7 𝑦(𝑎𝑅)
61 simpl 470 . . . . . . . 8 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑥 = 𝑎)
6261coeq1d 5479 . . . . . . 7 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑥𝑅) = (𝑎𝑅))
6357, 58, 59, 60, 62cbvmpt2 6958 . . . . . 6 (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎𝑅))
64 oveq 6874 . . . . . 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))
6563, 64mp1i 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))
66 eqidd 2803 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎𝑅)))
67 simprl 778 . . . . . . 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))
6867coeq1d 5479 . . . . . 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))
69 fvexd 6417 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V)
70 fvex 6415 . . . . . . 7 (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V
71 coexg 7341 . . . . . . 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)
7270, 51, 71sylancr 577 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∘ 𝑅) ∈ V)
7366, 68, 69, 27, 72ovmpt2d 7012 . . . . 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))
74 simpr 473 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑛 = 𝑁) → 𝑛 = 𝑁)
7574eqeq1d 2804 . . . . . . . . . 10 ((𝑟 = 𝑅𝑛 = 𝑁) → (𝑛 = 0 ↔ 𝑁 = 0))
766adantr 468 . . . . . . . . . 10 ((𝑟 = 𝑅𝑛 = 𝑁) → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7712adantr 468 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑛 = 𝑁) → seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟)) = seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅)))
7877, 74fveq12d 6409 . . . . . . . . . 10 ((𝑟 = 𝑅𝑛 = 𝑁) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛) = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁))
7975, 76, 78ifbieq12d 4300 . . . . . . . . 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)))
8079adantl 469 . . . . . . . 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)))
8128nnnn0d 11611 . . . . . . . 8 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
8237, 69ifcld 4318 . . . . . . . 8 ((𝑅𝑉𝑁 ∈ ℕ) → if(𝑁 = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)) ∈ V)
831, 80, 27, 81, 82ovmpt2d 7012 . . . . . . 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)))
84 nnne0 11333 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝑁 ≠ 0)
8584adantl 469 . . . . . . . . 9 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ≠ 0)
8685neneqd 2979 . . . . . . . 8 ((𝑅𝑉𝑁 ∈ ℕ) → ¬ 𝑁 = 0)
8786iffalsed 4284 . . . . . . 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))
8883, 87eqtr2d 2837 . . . . . 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))
8988coeq1d 5479 . . . . 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))
9065, 73, 893eqtrd 2840 . . . 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))
9149, 56, 903eqtrd 2840 . . 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))
9240, 44, 913eqtrd 2840 . 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))
93 df-relexp 13978 . . 3 𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))
94 oveq 6874 . . . . 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)))
95 oveq 6874 . . . . . 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))
9695coeq1d 5479 . . . . 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))
9794, 96eqeq12d 2817 . . . 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)))
9897imbi2d 331 . . 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))))
9993, 98ax-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)))
10092, 99mpbir 222 1 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1637  wcel 2155  wne 2974  Vcvv 3387  cun 3761  ifcif 4273  cmpt 4916   I cid 5212  dom cdm 5305  ran crn 5306  cres 5307  ccom 5309  cfv 6095  (class class class)co 6868  cmpt2 6870  0cc0 10215  1c1 10216   + caddc 10218  cn 11299  0cn0 11553  cuz 11898  seqcseq 13018  𝑟crelexp 13977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173  ax-cnex 10271  ax-resscn 10272  ax-1cn 10273  ax-icn 10274  ax-addcl 10275  ax-addrcl 10276  ax-mulcl 10277  ax-mulrcl 10278  ax-mulcom 10279  ax-addass 10280  ax-mulass 10281  ax-distr 10282  ax-i2m1 10283  ax-1ne0 10284  ax-1rid 10285  ax-rnegex 10286  ax-rrecex 10287  ax-cnre 10288  ax-pre-lttri 10289  ax-pre-lttrn 10290  ax-pre-ltadd 10291  ax-pre-mulgt0 10292
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-nel 3078  df-ral 3097  df-rex 3098  df-reu 3099  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-ord 5933  df-on 5934  df-lim 5935  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-riota 6829  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-om 7290  df-2nd 7393  df-wrecs 7636  df-recs 7698  df-rdg 7736  df-er 7973  df-en 8187  df-dom 8188  df-sdom 8189  df-pnf 10355  df-mnf 10356  df-xr 10357  df-ltxr 10358  df-le 10359  df-sub 10547  df-neg 10548  df-nn 11300  df-n0 11554  df-z 11638  df-uz 11899  df-seq 13019  df-relexp 13978
This theorem is referenced by:  relexpsucr  13986  relexpsucnnl  13989  relexpcnv  13992  relexprelg  13995  relexpnndm  13998  relexp2  38463  relexpxpnnidm  38489  relexpss1d  38491  relexpmulnn  38495  trclrelexplem  38497  relexp0a  38502  trclfvcom  38509  cotrcltrcl  38511  trclfvdecomr  38514  cotrclrcl  38528
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