Theorem relexpsucnnr 15049

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 2737	. . . 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 5888	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
4		rneq 5921	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
5	3, 4	uneq12d 4149	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (dom 𝑟 ∪ ran 𝑟) = (dom 𝑅 ∪ ran 𝑅))
6	5	reseq2d 5971	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7		eqidd 2737	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → 1 = 1)
8		coeq2 5843	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (𝑥 ∘ 𝑟) = (𝑥 ∘ 𝑅))
9	8	mpoeq3dv 7491	. . . . . . . . . . 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 5220	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (𝑧 ∈ V ↦ 𝑟) = (𝑧 ∈ V ↦ 𝑅))
12	7, 9, 11	seqeq123d 14033	. . . . . . . . . 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 6883	. . . . . . . . 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 4527	. . . . . . . 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 2740	. . . . . . . 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 2740	. . . . . . . 8 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
21		fveq2 6881	. . . . . . . 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 4532	. . . . . . 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 2738	. . . . . 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 3485	. . . . 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 12262	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ)
30	29	nnnn0d 12567	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ0)
31		dmexg 7902	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
32		rnexg 7903	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V)
33		unexg 7742	. . . . . . . 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 7913	. . . . . . 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 6896	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘(𝑁 + 1)) ∈ V)
39	37, 38	ifcld 4552	. . . 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 7564	. . 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 12279	. . . . . 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 4516	. . 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 12901	. . . . . . 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 14039	. . . . 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 7443	. . . . . 6 ⊢ (𝑁 + 1) ∈ V
51		simpl 482	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ 𝑉)
52		eqidd 2737	. . . . . . 7 ⊢ (𝑧 = (𝑁 + 1) → 𝑅 = 𝑅)
53		eqid 2736	. . . . . . 7 ⊢ (𝑧 ∈ V ↦ 𝑅) = (𝑧 ∈ V ↦ 𝑅)
54	52, 53	fvmptg 6989	. . . . . 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 7426	. . . 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 5846	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥 ∘ 𝑅) = (𝑎 ∘ 𝑅))
63	57, 58, 59, 60, 62	cbvmpo 7506	. . . . . 6 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∘ 𝑅))
64		oveq 7416	. . . . . 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 2737	. . . . . 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 5846	. . . . . 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 6896	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V)
70		fvex 6894	. . . . . . 7 ⊢ (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑅)), (𝑧 ∈ V ↦ 𝑅))‘𝑁) ∈ V
71		coexg 7930	. . . . . . 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 7564	. . . . 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 2738	. . . . . . . . . 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 6888	. . . . . . . . . 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 4534	. . . . . . . . 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 12567	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
82	37, 69	ifcld 4552	. . . . . . . 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 7564	. . . . . . 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 12279	. . . . . . . . . 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 4516	. . . . . . 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 2772	. . . . . 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 5846	. . . . 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 2775	. . . 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 2775	. . 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 2775	. 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 15044	. . 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 7416	. . . . 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 7416	. . . . . 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 5846	. . . . 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 2752	. . . 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 2933  Vcvv 3464   ∪ cun 3929  ifcif 4505   ↦ cmpt 5206   I cid 5552  dom cdm 5659  ran crn 5660   ↾ cres 5661   ∘ ccom 5663  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  0cc0 11134  1c1 11135   + caddc 11137  ℕcn 12245  ℕ₀cn0 12506  ℤ_≥cuz 12857  seqcseq 14024  ↑_𝑟crelexp 15043

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 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-n0 12507  df-z 12594  df-uz 12858  df-seq 14025  df-relexp 15044

This theorem is referenced by:  relexpsucnnl  15054  relexpsucr  15056  relexpcnv  15059  relexprelg  15062  relexpnndm  15065  relexp2  43668  relexpxpnnidm  43694  relexpss1d  43696  relexpmulnn  43700  trclrelexplem  43702  relexp0a  43707  trclfvcom  43714  cotrcltrcl  43716  trclfvdecomr  43719  cotrclrcl  43733