Theorem relexpsucnnr 14998

Description: A reduction for relation exponentiation to the right. (Contributed by RP, 22-May-2020.)

Assertion

Ref	Expression
relexpsucnnr	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅))

Proof of Theorem relexpsucnnr

Dummy variables 𝑎 𝑏 𝑧 𝑛 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2731	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))))
2		simprr 772	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅 ∧ 𝑛 = (𝑁 + 1))) → 𝑛 = (𝑁 + 1))
3		dmeq 5870	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
4		rneq 5903	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
5	3, 4	uneq12d 4135	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (dom 𝑟 ∪ ran 𝑟) = (dom 𝑅 ∪ ran 𝑅))
6	5	reseq2d 5953	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7		eqidd 2731	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → 1 = 1)
8		coeq2 5825	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (𝑥 ∘ 𝑟) = (𝑥 ∘ 𝑅))
9	8	mpoeq3dv 7471	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)) = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)))
10		id 22	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
11	10	mpteq2dv 5204	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (𝑧 ∈ V ↦ 𝑟) = (𝑧 ∈ V ↦ 𝑅))
12	7, 9, 11	seqeq123d 13982	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟)) = seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅)))
13	12	fveq1d 6863	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘(𝑁 + 1)) = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)))
14	6, 13	ifeq12d 4513	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → if((𝑁 + 1) = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘(𝑁 + 1))) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1))))
15	14	ad2antrl 728	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅 ∧ (𝑁 + 1) = (𝑁 + 1))) → if((𝑁 + 1) = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘(𝑁 + 1))) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1))))
16	15	a1i 11	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → (((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅 ∧ (𝑁 + 1) = (𝑁 + 1))) → if((𝑁 + 1) = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘(𝑁 + 1))) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)))))
17		eqeq1 2734	. . . . . . . 8 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 = (𝑁 + 1) ↔ (𝑁 + 1) = (𝑁 + 1)))
18	17	anbi2d 630	. . . . . . 7 ⊢ (𝑛 = (𝑁 + 1) → ((𝑟 = 𝑅 ∧ 𝑛 = (𝑁 + 1)) ↔ (𝑟 = 𝑅 ∧ (𝑁 + 1) = (𝑁 + 1))))
19	18	anbi2d 630	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → (((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅 ∧ 𝑛 = (𝑁 + 1))) ↔ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅 ∧ (𝑁 + 1) = (𝑁 + 1)))))
20		eqeq1 2734	. . . . . . . 8 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
21		fveq2 6861	. . . . . . . 8 ⊢ (𝑛 = (𝑁 + 1) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛) = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘(𝑁 + 1)))
22	20, 21	ifbieq2d 4518	. . . . . . 7 ⊢ (𝑛 = (𝑁 + 1) → if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘(𝑁 + 1))))
23	22	eqeq1d 2732	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → (if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1))) ↔ if((𝑁 + 1) = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘(𝑁 + 1))) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)))))
24	16, 19, 23	3imtr4d 294	. . . . 5 ⊢ (𝑛 = (𝑁 + 1) → (((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅 ∧ 𝑛 = (𝑁 + 1))) → if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)))))
25	2, 24	mpcom 38	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅 ∧ 𝑛 = (𝑁 + 1))) → if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1))))
26		elex 3471	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
27	26	adantr 480	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ V)
28		simpr 484	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
29	28	peano2nnd 12210	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ)
30	29	nnnn0d 12510	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ0)
31		dmexg 7880	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
32		rnexg 7881	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
33		unexg 7722	. . . . . . . 8 ⊢ ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
34	31, 32, 33	syl2anc 584	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
35		resiexg 7891	. . . . . . 7 ⊢ ((dom 𝑅 ∪ ran 𝑅) ∈ V → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
36	34, 35	syl 17	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
37	36	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
38		fvexd 6876	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)) ∈ V)
39	37, 38	ifcld 4538	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1))) ∈ V)
40	1, 25, 27, 30, 39	ovmpod 7544	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))(𝑁 + 1)) = if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1))))
41		nnne0 12227	. . . . . 6 ⊢ ((𝑁 + 1) ∈ ℕ → (𝑁 + 1) ≠ 0)
42	41	neneqd 2931	. . . . 5 ⊢ ((𝑁 + 1) ∈ ℕ → ¬ (𝑁 + 1) = 0)
43	29, 42	syl 17	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ¬ (𝑁 + 1) = 0)
44	43	iffalsed 4502	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → if((𝑁 + 1) = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1))) = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)))
45		elnnuz 12844	. . . . . . 7 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1))
46	45	biimpi 216	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘1))
47	46	adantl 481	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (ℤ≥‘1))
48		seqp1 13988	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘1) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)) = ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅))((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1))))
49	47, 48	syl 17	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)) = ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅))((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1))))
50		ovex 7423	. . . . . 6 ⊢ (𝑁 + 1) ∈ V
51		simpl 482	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ 𝑉)
52		eqidd 2731	. . . . . . 7 ⊢ (𝑧 = (𝑁 + 1) → 𝑅 = 𝑅)
53		eqid 2730	. . . . . . 7 ⊢ (𝑧 ∈ V ↦ 𝑅) = (𝑧 ∈ V ↦ 𝑅)
54	52, 53	fvmptg 6969	. . . . . 6 ⊢ (((𝑁 + 1) ∈ V ∧ 𝑅 ∈ 𝑉) → ((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1)) = 𝑅)
55	50, 51, 54	sylancr 587	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1)) = 𝑅)
56	55	oveq2d 7406	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅))((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1))) = ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅))𝑅))
57		nfcv 2892	. . . . . . 7 ⊢ Ⅎ𝑎(𝑥 ∘ 𝑅)
58		nfcv 2892	. . . . . . 7 ⊢ Ⅎ𝑏(𝑥 ∘ 𝑅)
59		nfcv 2892	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎 ∘ 𝑅)
60		nfcv 2892	. . . . . . 7 ⊢ Ⅎ𝑦(𝑎 ∘ 𝑅)
61		simpl 482	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑥 = 𝑎)
62	61	coeq1d 5828	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥 ∘ 𝑅) = (𝑎 ∘ 𝑅))
63	57, 58, 59, 60, 62	cbvmpo 7486	. . . . . 6 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∘ 𝑅))
64		oveq 7396	. . . . . 6 ⊢ ((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∘ 𝑅)) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅))𝑅) = ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∘ 𝑅))𝑅))
65	63, 64	mp1i 13	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅))𝑅) = ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∘ 𝑅))𝑅))
66		eqidd 2731	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∘ 𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∘ 𝑅)))
67		simprl 770	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑎 = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∧ 𝑏 = 𝑅)) → 𝑎 = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁))
68	67	coeq1d 5828	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑎 = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∧ 𝑏 = 𝑅)) → (𝑎 ∘ 𝑅) = ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∘ 𝑅))
69		fvexd 6876	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V)
70		fvex 6874	. . . . . . 7 ⊢ (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V
71		coexg 7908	. . . . . . 7 ⊢ (((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V ∧ 𝑅 ∈ 𝑉) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∘ 𝑅) ∈ V)
72	70, 51, 71	sylancr 587	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∘ 𝑅) ∈ V)
73	66, 68, 69, 27, 72	ovmpod 7544	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∘ 𝑅))𝑅) = ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∘ 𝑅))
74		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑛 = 𝑁) → 𝑛 = 𝑁)
75	74	eqeq1d 2732	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑛 = 𝑁) → (𝑛 = 0 ↔ 𝑁 = 0))
76	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑛 = 𝑁) → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
77	12	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑛 = 𝑁) → seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟)) = seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅)))
78	77, 74	fveq12d 6868	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑛 = 𝑁) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛) = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁))
79	75, 76, 78	ifbieq12d 4520	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑛 = 𝑁) → if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)) = if(𝑁 = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)))
80	79	adantl 481	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅 ∧ 𝑛 = 𝑁)) → if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)) = if(𝑁 = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)))
81	28	nnnn0d 12510	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
82	37, 69	ifcld 4538	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → if(𝑁 = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)) ∈ V)
83	1, 80, 27, 81, 82	ovmpod 7544	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) = if(𝑁 = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)))
84		nnne0 12227	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
85	84	adantl 481	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ≠ 0)
86	85	neneqd 2931	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ¬ 𝑁 = 0)
87	86	iffalsed 4502	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → if(𝑁 = 0, ( I ↾ (dom 𝑅 ∪ ran 𝑅)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)) = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁))
88	83, 87	eqtr2d 2766	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) = (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁))
89	88	coeq1d 5828	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∘ 𝑅) = ((𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) ∘ 𝑅))
90	65, 73, 89	3eqtrd 2769	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁)(𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅))𝑅) = ((𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) ∘ 𝑅))
91	49, 56, 90	3eqtrd 2769	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)) = ((𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) ∘ 𝑅))
92	40, 44, 91	3eqtrd 2769	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))(𝑁 + 1)) = ((𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) ∘ 𝑅))
93		df-relexp 14993	. . 3 ⊢ ↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))
94		oveq 7396	. . . . 5 ⊢ (↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))(𝑁 + 1)))
95		oveq 7396	. . . . . 6 ⊢ (↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) → (𝑅↑𝑟𝑁) = (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁))
96	95	coeq1d 5828	. . . . 5 ⊢ (↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) → ((𝑅↑𝑟𝑁) ∘ 𝑅) = ((𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) ∘ 𝑅))
97	94, 96	eqeq12d 2746	. . . 4 ⊢ (↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) → ((𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅) ↔ (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))(𝑁 + 1)) = ((𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) ∘ 𝑅)))
98	97	imbi2d 340	. . 3 ⊢ (↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) → (((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) ↔ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))(𝑁 + 1)) = ((𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) ∘ 𝑅))))
99	93, 98	ax-mp 5	. 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) ↔ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))(𝑁 + 1)) = ((𝑅(𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))𝑁) ∘ 𝑅)))
100	92, 99	mpbir 231	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  Vcvv 3450   ∪ cun 3915  ifcif 4491   ↦ cmpt 5191   I cid 5535  dom cdm 5641  ran crn 5642   ↾ cres 5643   ∘ ccom 5645  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  0cc0 11075  1c1 11076   + caddc 11078  ℕcn 12193  ℕ₀cn0 12449  ℤ_≥cuz 12800  seqcseq 13973  ↑_𝑟crelexp 14992

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-n0 12450  df-z 12537  df-uz 12801  df-seq 13974  df-relexp 14993

This theorem is referenced by:  relexpsucnnl  15003  relexpsucr  15005  relexpcnv  15008  relexprelg  15011  relexpnndm  15014  relexp2  43673  relexpxpnnidm  43699  relexpss1d  43701  relexpmulnn  43705  trclrelexplem  43707  relexp0a  43712  trclfvcom  43719  cotrcltrcl  43721  trclfvdecomr  43724  cotrclrcl  43738