Theorem relexpsucnnr 14978

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 2738	. . . 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 773	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅 ∧ 𝑛 = (𝑁 + 1))) → 𝑛 = (𝑁 + 1))
3		dmeq 5852	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
4		rneq 5885	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
5	3, 4	uneq12d 4110	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (dom 𝑟 ∪ ran 𝑟) = (dom 𝑅 ∪ ran 𝑅))
6	5	reseq2d 5938	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7		eqidd 2738	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → 1 = 1)
8		coeq2 5807	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (𝑥 ∘ 𝑟) = (𝑥 ∘ 𝑅))
9	8	mpoeq3dv 7439	. . . . . . . . . . 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 5180	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (𝑧 ∈ V ↦ 𝑟) = (𝑧 ∈ V ↦ 𝑅))
12	7, 9, 11	seqeq123d 13963	. . . . . . . . . 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 6836	. . . . . . . . 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 4489	. . . . . . . 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 729	. . . . . . 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 2741	. . . . . . . 8 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 = (𝑁 + 1) ↔ (𝑁 + 1) = (𝑁 + 1)))
18	17	anbi2d 631	. . . . . . 7 ⊢ (𝑛 = (𝑁 + 1) → ((𝑟 = 𝑅 ∧ 𝑛 = (𝑁 + 1)) ↔ (𝑟 = 𝑅 ∧ (𝑁 + 1) = (𝑁 + 1))))
19	18	anbi2d 631	. . . . . 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 2741	. . . . . . . 8 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
21		fveq2 6834	. . . . . . . 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 4494	. . . . . . 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 2739	. . . . . 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 3451	. . . . 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 12182	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ)
30	29	nnnn0d 12489	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ0)
31		dmexg 7845	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
32		rnexg 7846	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
33		unexg 7690	. . . . . . . 8 ⊢ ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
34	31, 32, 33	syl2anc 585	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
35		resiexg 7856	. . . . . . 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 6849	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)) ∈ V)
39	37, 38	ifcld 4514	. . . 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 7512	. . 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 12202	. . . . . 6 ⊢ ((𝑁 + 1) ∈ ℕ → (𝑁 + 1) ≠ 0)
42	41	neneqd 2938	. . . . 5 ⊢ ((𝑁 + 1) ∈ ℕ → ¬ (𝑁 + 1) = 0)
43	29, 42	syl 17	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ¬ (𝑁 + 1) = 0)
44	43	iffalsed 4478	. . 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 12819	. . . . . . 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 13969	. . . . 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 7393	. . . . . 6 ⊢ (𝑁 + 1) ∈ V
51		simpl 482	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ 𝑉)
52		eqidd 2738	. . . . . . 7 ⊢ (𝑧 = (𝑁 + 1) → 𝑅 = 𝑅)
53		eqid 2737	. . . . . . 7 ⊢ (𝑧 ∈ V ↦ 𝑅) = (𝑧 ∈ V ↦ 𝑅)
54	52, 53	fvmptg 6939	. . . . . 6 ⊢ (((𝑁 + 1) ∈ V ∧ 𝑅 ∈ 𝑉) → ((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1)) = 𝑅)
55	50, 51, 54	sylancr 588	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1)) = 𝑅)
56	55	oveq2d 7376	. . . 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 2899	. . . . . . 7 ⊢ Ⅎ𝑎(𝑥 ∘ 𝑅)
58		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑏(𝑥 ∘ 𝑅)
59		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎 ∘ 𝑅)
60		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑦(𝑎 ∘ 𝑅)
61		simpl 482	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑥 = 𝑎)
62	61	coeq1d 5810	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥 ∘ 𝑅) = (𝑎 ∘ 𝑅))
63	57, 58, 59, 60, 62	cbvmpo 7454	. . . . . 6 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∘ 𝑅))
64		oveq 7366	. . . . . 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 2738	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∘ 𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∘ 𝑅)))
67		simprl 771	. . . . . . 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 5810	. . . . . 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 6849	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V)
70		fvex 6847	. . . . . . 7 ⊢ (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V
71		coexg 7873	. . . . . . 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 588	. . . . . 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 7512	. . . . 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 2739	. . . . . . . . . 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 6841	. . . . . . . . . 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 4496	. . . . . . . . 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 12489	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
82	37, 69	ifcld 4514	. . . . . . . 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 7512	. . . . . . 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 12202	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
85	84	adantl 481	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ≠ 0)
86	85	neneqd 2938	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ¬ 𝑁 = 0)
87	86	iffalsed 4478	. . . . . . 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 2773	. . . . . 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 5810	. . . . 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 2776	. . . 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 2776	. . 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 2776	. 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 14973	. . 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 7366	. . . . 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 7366	. . . . . 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 5810	. . . . 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 2753	. . . 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 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3430   ∪ cun 3888  ifcif 4467   ↦ cmpt 5167   I cid 5518  dom cdm 5624  ran crn 5625   ↾ cres 5626   ∘ ccom 5628  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  0cc0 11029  1c1 11030   + caddc 11032  ℕcn 12165  ℕ₀cn0 12428  ℤ_≥cuz 12779  seqcseq 13954  ↑_𝑟crelexp 14972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-seq 13955  df-relexp 14973

This theorem is referenced by:  relexpsucnnl  14983  relexpsucr  14985  relexpcnv  14988  relexprelg  14991  relexpnndm  14994  relexp2  44122  relexpxpnnidm  44148  relexpss1d  44150  relexpmulnn  44154  trclrelexplem  44156  relexp0a  44161  trclfvcom  44168  cotrcltrcl  44170  trclfvdecomr  44173  cotrclrcl  44187