Theorem relexpsucnnr 14248

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 2779	. . . 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 760	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑟 = 𝑅 ∧ 𝑛 = (𝑁 + 1))) → 𝑛 = (𝑁 + 1))
3		dmeq 5623	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
4		rneq 5650	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
5	3, 4	uneq12d 4031	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (dom 𝑟 ∪ ran 𝑟) = (dom 𝑅 ∪ ran 𝑅))
6	5	reseq2d 5696	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7		eqidd 2779	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → 1 = 1)
8		coeq2 5580	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (𝑥 ∘ 𝑟) = (𝑥 ∘ 𝑅))
9	8	mpoeq3dv 7053	. . . . . . . . . . 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 5024	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (𝑧 ∈ V ↦ 𝑟) = (𝑧 ∈ V ↦ 𝑅))
12	7, 9, 11	seqeq123d 13196	. . . . . . . . . 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 6503	. . . . . . . . 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 4371	. . . . . . . 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 715	. . . . . . 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 2782	. . . . . . . 8 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 = (𝑁 + 1) ↔ (𝑁 + 1) = (𝑁 + 1)))
18	17	anbi2d 619	. . . . . . 7 ⊢ (𝑛 = (𝑁 + 1) → ((𝑟 = 𝑅 ∧ 𝑛 = (𝑁 + 1)) ↔ (𝑟 = 𝑅 ∧ (𝑁 + 1) = (𝑁 + 1))))
19	18	anbi2d 619	. . . . . 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 2782	. . . . . . . 8 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
21		fveq2 6501	. . . . . . . 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 4376	. . . . . . 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 2780	. . . . . 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 286	. . . . 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 3433	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
27	26	adantr 473	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ V)
28		simpr 477	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
29	28	peano2nnd 11460	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ)
30	29	nnnn0d 11770	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ0)
31		dmexg 7430	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
32		rnexg 7431	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
33		unexg 7291	. . . . . . . 8 ⊢ ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
34	31, 32, 33	syl2anc 576	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
35		resiexg 7436	. . . . . . 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 473	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∈ V)
38		fvexd 6516	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)) ∈ V)
39	37, 38	ifcld 4396	. . . 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 7120	. . 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 11477	. . . . . 6 ⊢ ((𝑁 + 1) ∈ ℕ → (𝑁 + 1) ≠ 0)
42	41	neneqd 2972	. . . . 5 ⊢ ((𝑁 + 1) ∈ ℕ → ¬ (𝑁 + 1) = 0)
43	29, 42	syl 17	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ¬ (𝑁 + 1) = 0)
44	43	iffalsed 4362	. . 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 12099	. . . . . . 7 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1))
46	45	biimpi 208	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘1))
47	46	adantl 474	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (ℤ≥‘1))
48		seqp1 13202	. . . . 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 7010	. . . . . 6 ⊢ (𝑁 + 1) ∈ V
51		simpl 475	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ 𝑉)
52		eqidd 2779	. . . . . . 7 ⊢ (𝑧 = (𝑁 + 1) → 𝑅 = 𝑅)
53		eqid 2778	. . . . . . 7 ⊢ (𝑧 ∈ V ↦ 𝑅) = (𝑧 ∈ V ↦ 𝑅)
54	52, 53	fvmptg 6595	. . . . . 6 ⊢ (((𝑁 + 1) ∈ V ∧ 𝑅 ∈ 𝑉) → ((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1)) = 𝑅)
55	50, 51, 54	sylancr 578	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑧 ∈ V ↦ 𝑅)‘(𝑁 + 1)) = 𝑅)
56	55	oveq2d 6994	. . . 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 2932	. . . . . . 7 ⊢ Ⅎ𝑎(𝑥 ∘ 𝑅)
58		nfcv 2932	. . . . . . 7 ⊢ Ⅎ𝑏(𝑥 ∘ 𝑅)
59		nfcv 2932	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎 ∘ 𝑅)
60		nfcv 2932	. . . . . . 7 ⊢ Ⅎ𝑦(𝑎 ∘ 𝑅)
61		simpl 475	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑥 = 𝑎)
62	61	coeq1d 5583	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥 ∘ 𝑅) = (𝑎 ∘ 𝑅))
63	57, 58, 59, 60, 62	cbvmpo 7066	. . . . . 6 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∘ 𝑅))
64		oveq 6984	. . . . . 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 2779	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∘ 𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∘ 𝑅)))
67		simprl 758	. . . . . . 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 5583	. . . . . 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 6516	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V)
70		fvex 6514	. . . . . . 7 ⊢ (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V
71		coexg 7451	. . . . . . 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 578	. . . . . 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 7120	. . . . 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 477	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑛 = 𝑁) → 𝑛 = 𝑁)
75	74	eqeq1d 2780	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑛 = 𝑁) → (𝑛 = 0 ↔ 𝑁 = 0))
76	6	adantr 473	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑛 = 𝑁) → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
77	12	adantr 473	. . . . . . . . . . 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 6508	. . . . . . . . . 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 4378	. . . . . . . . 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 474	. . . . . . . 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 11770	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
82	37, 69	ifcld 4396	. . . . . . . 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 7120	. . . . . . 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 11477	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
85	84	adantl 474	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ≠ 0)
86	85	neneqd 2972	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ¬ 𝑁 = 0)
87	86	iffalsed 4362	. . . . . . 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 2815	. . . . . 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 5583	. . . . 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 2818	. . . 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 2818	. . 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 2818	. 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 14244	. . 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 6984	. . . . 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 6984	. . . . . 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 5583	. . . . 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 2793	. . . 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 333	. . 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 223	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050   ≠ wne 2967  Vcvv 3415   ∪ cun 3829  ifcif 4351   ↦ cmpt 5009   I cid 5312  dom cdm 5408  ran crn 5409   ↾ cres 5410   ∘ ccom 5412  ‘cfv 6190  (class class class)co 6978   ∈ cmpo 6980  0cc0 10337  1c1 10338   + caddc 10340  ℕcn 11441  ℕ₀cn0 11710  ℤ_≥cuz 12061  seqcseq 13187  ↑_𝑟crelexp 14243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-cnex 10393  ax-resscn 10394  ax-1cn 10395  ax-icn 10396  ax-addcl 10397  ax-addrcl 10398  ax-mulcl 10399  ax-mulrcl 10400  ax-mulcom 10401  ax-addass 10402  ax-mulass 10403  ax-distr 10404  ax-i2m1 10405  ax-1ne0 10406  ax-1rid 10407  ax-rnegex 10408  ax-rrecex 10409  ax-cnre 10410  ax-pre-lttri 10411  ax-pre-lttrn 10412  ax-pre-ltadd 10413  ax-pre-mulgt0 10414

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-riota 6939  df-ov 6981  df-oprab 6982  df-mpo 6983  df-om 7399  df-2nd 7504  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-er 8091  df-en 8309  df-dom 8310  df-sdom 8311  df-pnf 10478  df-mnf 10479  df-xr 10480  df-ltxr 10481  df-le 10482  df-sub 10674  df-neg 10675  df-nn 11442  df-n0 11711  df-z 11797  df-uz 12062  df-seq 13188  df-relexp 14244

This theorem is referenced by:  relexpsucr  14252  relexpsucnnl  14255  relexpcnv  14258  relexprelg  14261  relexpnndm  14264  relexp2  39385  relexpxpnnidm  39411  relexpss1d  39413  relexpmulnn  39417  trclrelexplem  39419  relexp0a  39424  trclfvcom  39431  cotrcltrcl  39433  trclfvdecomr  39436  cotrclrcl  39450