Theorem relexpsucnnr 14967

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 2730	. . . 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 5857	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
4		rneq 5889	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
5	3, 4	uneq12d 4128	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (dom 𝑟 ∪ ran 𝑟) = (dom 𝑅 ∪ ran 𝑅))
6	5	reseq2d 5939	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7		eqidd 2730	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → 1 = 1)
8		coeq2 5812	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (𝑥 ∘ 𝑟) = (𝑥 ∘ 𝑅))
9	8	mpoeq3dv 7448	. . . . . . . . . . 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 5196	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (𝑧 ∈ V ↦ 𝑟) = (𝑧 ∈ V ↦ 𝑅))
12	7, 9, 11	seqeq123d 13951	. . . . . . . . . 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 6842	. . . . . . . . 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 4506	. . . . . . . 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 2733	. . . . . . . 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 2733	. . . . . . . 8 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
21		fveq2 6840	. . . . . . . 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 4511	. . . . . . 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 2731	. . . . . 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 3465	. . . . 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 12179	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ)
30	29	nnnn0d 12479	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ0)
31		dmexg 7857	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
32		rnexg 7858	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
33		unexg 7699	. . . . . . . 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 7868	. . . . . . 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 6855	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)) ∈ V)
39	37, 38	ifcld 4531	. . . 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 7521	. . 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 12196	. . . . . 6 ⊢ ((𝑁 + 1) ∈ ℕ → (𝑁 + 1) ≠ 0)
42	41	neneqd 2930	. . . . 5 ⊢ ((𝑁 + 1) ∈ ℕ → ¬ (𝑁 + 1) = 0)
43	29, 42	syl 17	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ¬ (𝑁 + 1) = 0)
44	43	iffalsed 4495	. . 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 12813	. . . . . . 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 13957	. . . . 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 7402	. . . . . 6 ⊢ (𝑁 + 1) ∈ V
51		simpl 482	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ 𝑉)
52		eqidd 2730	. . . . . . 7 ⊢ (𝑧 = (𝑁 + 1) → 𝑅 = 𝑅)
53		eqid 2729	. . . . . . 7 ⊢ (𝑧 ∈ V ↦ 𝑅) = (𝑧 ∈ V ↦ 𝑅)
54	52, 53	fvmptg 6948	. . . . . 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 7385	. . . 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 2891	. . . . . . 7 ⊢ Ⅎ𝑎(𝑥 ∘ 𝑅)
58		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑏(𝑥 ∘ 𝑅)
59		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎 ∘ 𝑅)
60		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑦(𝑎 ∘ 𝑅)
61		simpl 482	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑥 = 𝑎)
62	61	coeq1d 5815	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥 ∘ 𝑅) = (𝑎 ∘ 𝑅))
63	57, 58, 59, 60, 62	cbvmpo 7463	. . . . . 6 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∘ 𝑅))
64		oveq 7375	. . . . . 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 2730	. . . . . 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 5815	. . . . . 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 6855	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V)
70		fvex 6853	. . . . . . 7 ⊢ (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V
71		coexg 7885	. . . . . . 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 7521	. . . . 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 2731	. . . . . . . . . 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 6847	. . . . . . . . . 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 4513	. . . . . . . . 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 12479	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
82	37, 69	ifcld 4531	. . . . . . . 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 7521	. . . . . . 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 12196	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
85	84	adantl 481	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ≠ 0)
86	85	neneqd 2930	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ¬ 𝑁 = 0)
87	86	iffalsed 4495	. . . . . . 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 2765	. . . . . 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 5815	. . . . 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 2768	. . . 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 2768	. . 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 2768	. 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 14962	. . 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 7375	. . . . 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 7375	. . . . . 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 5815	. . . . 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 2745	. . . 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 2925  Vcvv 3444   ∪ cun 3909  ifcif 4484   ↦ cmpt 5183   I cid 5525  dom cdm 5631  ran crn 5632   ↾ cres 5633   ∘ ccom 5635  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  0cc0 11044  1c1 11045   + caddc 11047  ℕcn 12162  ℕ₀cn0 12418  ℤ_≥cuz 12769  seqcseq 13942  ↑_𝑟crelexp 14961

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 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-n0 12419  df-z 12506  df-uz 12770  df-seq 13943  df-relexp 14962

This theorem is referenced by:  relexpsucnnl  14972  relexpsucr  14974  relexpcnv  14977  relexprelg  14980  relexpnndm  14983  relexp2  43639  relexpxpnnidm  43665  relexpss1d  43667  relexpmulnn  43671  trclrelexplem  43673  relexp0a  43678  trclfvcom  43685  cotrcltrcl  43687  trclfvdecomr  43690  cotrclrcl  43704