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Theorem relexpsucnnr 15064
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 2738 . . . 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 5914 . . . . . . . . . . 11 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
4 rneq 5947 . . . . . . . . . . 11 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
53, 4uneq12d 4169 . . . . . . . . . 10 (𝑟 = 𝑅 → (dom 𝑟 ∪ ran 𝑟) = (dom 𝑅 ∪ ran 𝑅))
65reseq2d 5997 . . . . . . . . 9 (𝑟 = 𝑅 → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7 eqidd 2738 . . . . . . . . . . 11 (𝑟 = 𝑅 → 1 = 1)
8 coeq2 5869 . . . . . . . . . . . 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 5244 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑧 ∈ V ↦ 𝑟) = (𝑧 ∈ V ↦ 𝑅))
127, 9, 11seqeq123d 14051 . . . . . . . . . 10 (𝑟 = 𝑅 → seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟)) = seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅)))
1312fveq1d 6908 . . . . . . . . 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 4547 . . . . . . . 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 2741 . . . . . . . 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 2741 . . . . . . . 8 (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
21 fveq2 6906 . . . . . . . 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 4552 . . . . . . 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 2739 . . . . . 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 3501 . . . . 5 (𝑅𝑉𝑅 ∈ V)
2726adantr 480 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑅 ∈ V)
28 simpr 484 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
2928peano2nnd 12283 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ)
3029nnnn0d 12587 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ0)
31 dmexg 7923 . . . . . . . 8 (𝑅𝑉 → dom 𝑅 ∈ V)
32 rnexg 7924 . . . . . . . 8 (𝑅𝑉 → ran 𝑅 ∈ V)
33 unexg 7763 . . . . . . . 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 7934 . . . . . . 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 6921 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)) ∈ V)
3937, 38ifcld 4572 . . . 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 12300 . . . . . 6 ((𝑁 + 1) ∈ ℕ → (𝑁 + 1) ≠ 0)
4241neneqd 2945 . . . . 5 ((𝑁 + 1) ∈ ℕ → ¬ (𝑁 + 1) = 0)
4329, 42syl 17 . . . 4 ((𝑅𝑉𝑁 ∈ ℕ) → ¬ (𝑁 + 1) = 0)
4443iffalsed 4536 . . 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 12922 . . . . . . 7 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ‘1))
4645biimpi 216 . . . . . 6 (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ‘1))
4746adantl 481 . . . . 5 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ∈ (ℤ‘1))
48 seqp1 14057 . . . . 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 2738 . . . . . . 7 (𝑧 = (𝑁 + 1) → 𝑅 = 𝑅)
53 eqid 2737 . . . . . . 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 2905 . . . . . . 7 𝑎(𝑥𝑅)
58 nfcv 2905 . . . . . . 7 𝑏(𝑥𝑅)
59 nfcv 2905 . . . . . . 7 𝑥(𝑎𝑅)
60 nfcv 2905 . . . . . . 7 𝑦(𝑎𝑅)
61 simpl 482 . . . . . . . 8 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑥 = 𝑎)
6261coeq1d 5872 . . . . . . 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 2738 . . . . . 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 5872 . . . . . 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 6921 . . . . . 6 ((𝑅𝑉𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V)
70 fvex 6919 . . . . . . 7 (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V
71 coexg 7951 . . . . . . 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 2739 . . . . . . . . . 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 6913 . . . . . . . . . 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 4554 . . . . . . . . 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 12587 . . . . . . . 8 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
8237, 69ifcld 4572 . . . . . . . 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 12300 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝑁 ≠ 0)
8584adantl 481 . . . . . . . . 9 ((𝑅𝑉𝑁 ∈ ℕ) → 𝑁 ≠ 0)
8685neneqd 2945 . . . . . . . 8 ((𝑅𝑉𝑁 ∈ ℕ) → ¬ 𝑁 = 0)
8786iffalsed 4536 . . . . . . 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 2778 . . . . . 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 5872 . . . . 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 2781 . . . 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 2781 . . 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 2781 . 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 15059 . . 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 5872 . . . . 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 2753 . . . 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 1540  wcel 2108  wne 2940  Vcvv 3480  cun 3949  ifcif 4525  cmpt 5225   I cid 5577  dom cdm 5685  ran crn 5686  cres 5687  ccom 5689  cfv 6561  (class class class)co 7431  cmpo 7433  0cc0 11155  1c1 11156   + caddc 11158  cn 12266  0cn0 12526  cuz 12878  seqcseq 14042  𝑟crelexp 15058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-seq 14043  df-relexp 15059
This theorem is referenced by:  relexpsucnnl  15069  relexpsucr  15071  relexpcnv  15074  relexprelg  15077  relexpnndm  15080  relexp2  43690  relexpxpnnidm  43716  relexpss1d  43718  relexpmulnn  43722  trclrelexplem  43724  relexp0a  43729  trclfvcom  43736  cotrcltrcl  43738  trclfvdecomr  43741  cotrclrcl  43755
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