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

Theorem relexpsucnnr 15061
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 2736 . . . 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 773 . . . . 5 (((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅𝑛 = (𝑁 + 1))) → 𝑛 = (𝑁 + 1))
3 dmeq 5917 . . . . . . . . . . 11 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
4 rneq 5950 . . . . . . . . . . 11 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
53, 4uneq12d 4179 . . . . . . . . . 10 (𝑟 = 𝑅 → (dom 𝑟 ∪ ran 𝑟) = (dom 𝑅 ∪ ran 𝑅))
65reseq2d 6000 . . . . . . . . 9 (𝑟 = 𝑅 → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7 eqidd 2736 . . . . . . . . . . 11 (𝑟 = 𝑅 → 1 = 1)
8 coeq2 5872 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (𝑥𝑟) = (𝑥𝑅))
98mpoeq3dv 7512 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)) = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)))
10 id 22 . . . . . . . . . . . 12 (𝑟 = 𝑅𝑟 = 𝑅)
1110mpteq2dv 5250 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑧 ∈ V ↦ 𝑟) = (𝑧 ∈ V ↦ 𝑅))
127, 9, 11seqeq123d 14048 . . . . . . . . . 10 (𝑟 = 𝑅 → seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟)) = seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅)))
1312fveq1d 6909 . . . . . . . . 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 4552 . . . . . . . 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 728 . . . . . . 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 2739 . . . . . . . 8 (𝑛 = (𝑁 + 1) → (𝑛 = (𝑁 + 1) ↔ (𝑁 + 1) = (𝑁 + 1)))
1817anbi2d 630 . . . . . . 7 (𝑛 = (𝑁 + 1) → ((𝑟 = 𝑅𝑛 = (𝑁 + 1)) ↔ (𝑟 = 𝑅 ∧ (𝑁 + 1) = (𝑁 + 1))))
1918anbi2d 630 . . . . . 6 (𝑛 = (𝑁 + 1) → (((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅𝑛 = (𝑁 + 1))) ↔ ((𝑅𝑉𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅 ∧ (𝑁 + 1) = (𝑁 + 1)))))
20 eqeq1 2739 . . . . . . . 8 (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
21 fveq2 6907 . . . . . . . 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 4557 . . . . . . 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 2737 . . . . . 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 3499 . . . . 5 (𝑅𝑉𝑅 ∈ V)
2726adantr 480 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑅 ∈ V)
28 simpr 484 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
2928peano2nnd 12281 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ)
3029nnnn0d 12585 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ0)
31 dmexg 7924 . . . . . . . 8 (𝑅𝑉 → dom 𝑅 ∈ V)
32 rnexg 7925 . . . . . . . 8 (𝑅𝑉 → ran 𝑅 ∈ V)
33 unexg 7762 . . . . . . . 8 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
3431, 32, 33syl2anc 584 . . . . . . 7 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
35 resiexg 7935 . . . . . . 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 6922 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)) ∈ V)
3937, 38ifcld 4577 . . . 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 7585 . . 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 12298 . . . . . 6 ((𝑁 + 1) ∈ ℕ → (𝑁 + 1) ≠ 0)
4241neneqd 2943 . . . . 5 ((𝑁 + 1) ∈ ℕ → ¬ (𝑁 + 1) = 0)
4329, 42syl 17 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → ¬ (𝑁 + 1) = 0)
4443iffalsed 4542 . . 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 12920 . . . . . . 7 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ‘1))
4645biimpi 216 . . . . . 6 (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ‘1))
4746adantl 481 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ∈ (ℤ‘1))
48 seqp1 14054 . . . . 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 7464 . . . . . 6 (𝑁 + 1) ∈ V
51 simpl 482 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑅𝑉)
52 eqidd 2736 . . . . . . 7 (𝑧 = (𝑁 + 1) → 𝑅 = 𝑅)
53 eqid 2735 . . . . . . 7 (𝑧 ∈ V ↦ 𝑅) = (𝑧 ∈ V ↦ 𝑅)
5452, 53fvmptg 7014 . . . . . 6 (((𝑁 + 1) ∈ V ∧ 𝑅𝑉) → ((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1)) = 𝑅)
5550, 51, 54sylancr 587 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → ((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1)) = 𝑅)
5655oveq2d 7447 . . . 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 2903 . . . . . . 7 𝑎(𝑥𝑅)
58 nfcv 2903 . . . . . . 7 𝑏(𝑥𝑅)
59 nfcv 2903 . . . . . . 7 𝑥(𝑎𝑅)
60 nfcv 2903 . . . . . . 7 𝑦(𝑎𝑅)
61 simpl 482 . . . . . . . 8 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑥 = 𝑎)
6261coeq1d 5875 . . . . . . 7 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑥𝑅) = (𝑎𝑅))
6357, 58, 59, 60, 62cbvmpo 7527 . . . . . 6 (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎𝑅))
64 oveq 7437 . . . . . 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 2736 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎𝑅)))
67 simprl 771 . . . . . . 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 5875 . . . . . 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 6922 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V)
70 fvex 6920 . . . . . . 7 (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V
71 coexg 7952 . . . . . . 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 587 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∘ 𝑅) ∈ V)
7366, 68, 69, 27, 72ovmpod 7585 . . . . 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 2737 . . . . . . . . . 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 6914 . . . . . . . . . 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 4559 . . . . . . . . 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 12585 . . . . . . . 8 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
8237, 69ifcld 4577 . . . . . . . 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 7585 . . . . . . 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 12298 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝑁 ≠ 0)
8584adantl 481 . . . . . . . . 9 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ≠ 0)
8685neneqd 2943 . . . . . . . 8 ((𝑅𝑉𝑁 ∈ ℕ) → ¬ 𝑁 = 0)
8786iffalsed 4542 . . . . . . 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 2776 . . . . . 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 5875 . . . . 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 2779 . . . 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 2779 . . 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 2779 . 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 15056 . . 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 7437 . . . . 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 7437 . . . . . 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 5875 . . . . 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 2751 . . . 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 231 1 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478  cun 3961  ifcif 4531  cmpt 5231   I cid 5582  dom cdm 5689  ran crn 5690  cres 5691  ccom 5693  cfv 6563  (class class class)co 7431  cmpo 7433  0cc0 11153  1c1 11154   + caddc 11156  cn 12264  0cn0 12524  cuz 12876  seqcseq 14039  𝑟crelexp 15055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-seq 14040  df-relexp 15056
This theorem is referenced by:  relexpsucnnl  15066  relexpsucr  15068  relexpcnv  15071  relexprelg  15074  relexpnndm  15077  relexp2  43667  relexpxpnnidm  43693  relexpss1d  43695  relexpmulnn  43699  trclrelexplem  43701  relexp0a  43706  trclfvcom  43713  cotrcltrcl  43715  trclfvdecomr  43718  cotrclrcl  43732
  Copyright terms: Public domain W3C validator