MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relexpsucnnr Structured version   Visualization version   GIF version

Theorem relexpsucnnr 14969
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 2725 . . . 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 770 . . . . 5 (((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅𝑛 = (𝑁 + 1))) → 𝑛 = (𝑁 + 1))
3 dmeq 5893 . . . . . . . . . . 11 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
4 rneq 5925 . . . . . . . . . . 11 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
53, 4uneq12d 4156 . . . . . . . . . 10 (𝑟 = 𝑅 → (dom 𝑟 ∪ ran 𝑟) = (dom 𝑅 ∪ ran 𝑅))
65reseq2d 5971 . . . . . . . . 9 (𝑟 = 𝑅 → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7 eqidd 2725 . . . . . . . . . . 11 (𝑟 = 𝑅 → 1 = 1)
8 coeq2 5848 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (𝑥𝑟) = (𝑥𝑅))
98mpoeq3dv 7480 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)) = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)))
10 id 22 . . . . . . . . . . . 12 (𝑟 = 𝑅𝑟 = 𝑅)
1110mpteq2dv 5240 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑧 ∈ V ↦ 𝑟) = (𝑧 ∈ V ↦ 𝑅))
127, 9, 11seqeq123d 13972 . . . . . . . . . 10 (𝑟 = 𝑅 → seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟)) = seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅)))
1312fveq1d 6883 . . . . . . . . 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 4541 . . . . . . . 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 725 . . . . . . 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 2728 . . . . . . . 8 (𝑛 = (𝑁 + 1) → (𝑛 = (𝑁 + 1) ↔ (𝑁 + 1) = (𝑁 + 1)))
1817anbi2d 628 . . . . . . 7 (𝑛 = (𝑁 + 1) → ((𝑟 = 𝑅𝑛 = (𝑁 + 1)) ↔ (𝑟 = 𝑅 ∧ (𝑁 + 1) = (𝑁 + 1))))
1918anbi2d 628 . . . . . 6 (𝑛 = (𝑁 + 1) → (((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅𝑛 = (𝑁 + 1))) ↔ ((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅 ∧ (𝑁 + 1) = (𝑁 + 1)))))
20 eqeq1 2728 . . . . . . . 8 (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
21 fveq2 6881 . . . . . . . 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 4546 . . . . . . 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 2726 . . . . . 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 294 . . . . 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 3485 . . . . 5 (𝑅𝑉𝑅 ∈ V)
2726adantr 480 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑅 ∈ V)
28 simpr 484 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
2928peano2nnd 12226 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ)
3029nnnn0d 12529 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ0)
31 dmexg 7887 . . . . . . . 8 (𝑅𝑉 → dom 𝑅 ∈ V)
32 rnexg 7888 . . . . . . . 8 (𝑅𝑉 → ran 𝑅 ∈ V)
33 unexg 7729 . . . . . . . 8 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
3431, 32, 33syl2anc 583 . . . . . . 7 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
35 resiexg 7898 . . . . . . 7 ((dom 𝑅 ∪ ran 𝑅) ∈ V → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
3634, 35syl 17 . . . . . 6 (𝑅𝑉 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
3736adantr 480 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
38 fvexd 6896 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)) ∈ V)
3937, 38ifcld 4566 . . . 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, 39ovmpod 7552 . . 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 12243 . . . . . 6 ((𝑁 + 1) ∈ ℕ → (𝑁 + 1) ≠ 0)
4241neneqd 2937 . . . . 5 ((𝑁 + 1) ∈ ℕ → ¬ (𝑁 + 1) = 0)
4329, 42syl 17 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → ¬ (𝑁 + 1) = 0)
4443iffalsed 4531 . . 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 12863 . . . . . . 7 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ‘1))
4645biimpi 215 . . . . . 6 (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ‘1))
4746adantl 481 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ∈ (ℤ‘1))
48 seqp1 13978 . . . . 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 7434 . . . . . 6 (𝑁 + 1) ∈ V
51 simpl 482 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑅𝑉)
52 eqidd 2725 . . . . . . 7 (𝑧 = (𝑁 + 1) → 𝑅 = 𝑅)
53 eqid 2724 . . . . . . 7 (𝑧 ∈ V ↦ 𝑅) = (𝑧 ∈ V ↦ 𝑅)
5452, 53fvmptg 6986 . . . . . 6 (((𝑁 + 1) ∈ V ∧ 𝑅𝑉) → ((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1)) = 𝑅)
5550, 51, 54sylancr 586 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → ((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1)) = 𝑅)
5655oveq2d 7417 . . . 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 2895 . . . . . . 7 𝑎(𝑥𝑅)
58 nfcv 2895 . . . . . . 7 𝑏(𝑥𝑅)
59 nfcv 2895 . . . . . . 7 𝑥(𝑎𝑅)
60 nfcv 2895 . . . . . . 7 𝑦(𝑎𝑅)
61 simpl 482 . . . . . . . 8 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑥 = 𝑎)
6261coeq1d 5851 . . . . . . 7 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑥𝑅) = (𝑎𝑅))
6357, 58, 59, 60, 62cbvmpo 7495 . . . . . 6 (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎𝑅))
64 oveq 7407 . . . . . 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 2725 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎𝑅)))
67 simprl 768 . . . . . . 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 5851 . . . . . 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 6896 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V)
70 fvex 6894 . . . . . . 7 (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V
71 coexg 7913 . . . . . . 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 586 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∘ 𝑅) ∈ V)
7366, 68, 69, 27, 72ovmpod 7552 . . . . 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 484 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑛 = 𝑁) → 𝑛 = 𝑁)
7574eqeq1d 2726 . . . . . . . . . 10 ((𝑟 = 𝑅𝑛 = 𝑁) → (𝑛 = 0 ↔ 𝑁 = 0))
766adantr 480 . . . . . . . . . 10 ((𝑟 = 𝑅𝑛 = 𝑁) → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7712adantr 480 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑛 = 𝑁) → seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟)) = seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅)))
7877, 74fveq12d 6888 . . . . . . . . . 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 4548 . . . . . . . . 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 481 . . . . . . . 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 12529 . . . . . . . 8 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
8237, 69ifcld 4566 . . . . . . . 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, 82ovmpod 7552 . . . . . . 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 12243 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝑁 ≠ 0)
8584adantl 481 . . . . . . . . 9 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ≠ 0)
8685neneqd 2937 . . . . . . . 8 ((𝑅𝑉𝑁 ∈ ℕ) → ¬ 𝑁 = 0)
8786iffalsed 4531 . . . . . . 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 2765 . . . . . 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 5851 . . . . 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 2768 . . . 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 2768 . . 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 2768 . 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 14964 . . 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 7407 . . . . 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 7407 . . . . . 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 5851 . . . . 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 2740 . . . 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 340 . . 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 230 1 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wne 2932  Vcvv 3466  cun 3938  ifcif 4520  cmpt 5221   I cid 5563  dom cdm 5666  ran crn 5667  cres 5668  ccom 5670  cfv 6533  (class class class)co 7401  cmpo 7403  0cc0 11106  1c1 11107   + caddc 11109  cn 12209  0cn0 12469  cuz 12819  seqcseq 13963  𝑟crelexp 14963
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 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-n0 12470  df-z 12556  df-uz 12820  df-seq 13964  df-relexp 14964
This theorem is referenced by:  relexpsucnnl  14974  relexpsucr  14976  relexpcnv  14979  relexprelg  14982  relexpnndm  14985  relexp2  42917  relexpxpnnidm  42943  relexpss1d  42945  relexpmulnn  42949  trclrelexplem  42951  relexp0a  42956  trclfvcom  42963  cotrcltrcl  42965  trclfvdecomr  42968  cotrclrcl  42982
  Copyright terms: Public domain W3C validator