Theorem relexpsucnnl 15069

Description: A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.)

Assertion

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

Proof of Theorem relexpsucnnl

Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7438	. . . . . 6 ⊢ (𝑛 = 1 → (𝑛 + 1) = (1 + 1))
2	1	oveq2d 7447	. . . . 5 ⊢ (𝑛 = 1 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅↑𝑟(1 + 1)))
3		oveq2 7439	. . . . . 6 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
4	3	coeq2d 5873	. . . . 5 ⊢ (𝑛 = 1 → (𝑅 ∘ (𝑅↑𝑟𝑛)) = (𝑅 ∘ (𝑅↑𝑟1)))
5	2, 4	eqeq12d 2753	. . . 4 ⊢ (𝑛 = 1 → ((𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛)) ↔ (𝑅↑𝑟(1 + 1)) = (𝑅 ∘ (𝑅↑𝑟1))))
6	5	imbi2d 340	. . 3 ⊢ (𝑛 = 1 → ((𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛))) ↔ (𝑅 ∈ 𝑉 → (𝑅↑𝑟(1 + 1)) = (𝑅 ∘ (𝑅↑𝑟1)))))
7		oveq1 7438	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑛 + 1) = (𝑚 + 1))
8	7	oveq2d 7447	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅↑𝑟(𝑚 + 1)))
9		oveq2 7439	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑚))
10	9	coeq2d 5873	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑅 ∘ (𝑅↑𝑟𝑛)) = (𝑅 ∘ (𝑅↑𝑟𝑚)))
11	8, 10	eqeq12d 2753	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛)) ↔ (𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚))))
12	11	imbi2d 340	. . 3 ⊢ (𝑛 = 𝑚 → ((𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛))) ↔ (𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚)))))
13		oveq1 7438	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 + 1) = ((𝑚 + 1) + 1))
14	13	oveq2d 7447	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟(𝑛 + 1)) = (𝑅↑𝑟((𝑚 + 1) + 1)))
15		oveq2 7439	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
16	15	coeq2d 5873	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑅 ∘ (𝑅↑𝑟𝑛)) = (𝑅 ∘ (𝑅↑𝑟(𝑚 + 1))))
17	14, 16	eqeq12d 2753	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → ((𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛)) ↔ (𝑅↑𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅↑𝑟(𝑚 + 1)))))
18	17	imbi2d 340	. . 3 ⊢ (𝑛 = (𝑚 + 1) → ((𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛))) ↔ (𝑅 ∈ 𝑉 → (𝑅↑𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅↑𝑟(𝑚 + 1))))))
19		oveq1 7438	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
20	19	oveq2d 7447	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅↑𝑟(𝑁 + 1)))
21		oveq2 7439	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
22	21	coeq2d 5873	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑅 ∘ (𝑅↑𝑟𝑛)) = (𝑅 ∘ (𝑅↑𝑟𝑁)))
23	20, 22	eqeq12d 2753	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛)) ↔ (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))))
24	23	imbi2d 340	. . 3 ⊢ (𝑛 = 𝑁 → ((𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛))) ↔ (𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁)))))
25		relexp1g 15065	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
26	25	coeq1d 5872	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟1) ∘ 𝑅) = (𝑅 ∘ 𝑅))
27		1nn 12277	. . . . 5 ⊢ 1 ∈ ℕ
28		relexpsucnnr 15064	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 1 ∈ ℕ) → (𝑅↑𝑟(1 + 1)) = ((𝑅↑𝑟1) ∘ 𝑅))
29	27, 28	mpan2 691	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟(1 + 1)) = ((𝑅↑𝑟1) ∘ 𝑅))
30	25	coeq2d 5873	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∘ (𝑅↑𝑟1)) = (𝑅 ∘ 𝑅))
31	26, 29, 30	3eqtr4d 2787	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟(1 + 1)) = (𝑅 ∘ (𝑅↑𝑟1)))
32		coeq1 5868	. . . . . . . . 9 ⊢ ((𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚)) → ((𝑅↑𝑟(𝑚 + 1)) ∘ 𝑅) = ((𝑅 ∘ (𝑅↑𝑟𝑚)) ∘ 𝑅))
33		coass 6285	. . . . . . . . 9 ⊢ ((𝑅 ∘ (𝑅↑𝑟𝑚)) ∘ 𝑅) = (𝑅 ∘ ((𝑅↑𝑟𝑚) ∘ 𝑅))
34	32, 33	eqtrdi 2793	. . . . . . . 8 ⊢ ((𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚)) → ((𝑅↑𝑟(𝑚 + 1)) ∘ 𝑅) = (𝑅 ∘ ((𝑅↑𝑟𝑚) ∘ 𝑅)))
35	34	adantl 481	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) ∧ (𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚))) → ((𝑅↑𝑟(𝑚 + 1)) ∘ 𝑅) = (𝑅 ∘ ((𝑅↑𝑟𝑚) ∘ 𝑅)))
36		simpl 482	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) ∧ (𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚))) → (𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ))
37		peano2nn 12278	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
38	37	anim2i 617	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅 ∈ 𝑉 ∧ (𝑚 + 1) ∈ ℕ))
39		relexpsucnnr 15064	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑚 + 1) ∈ ℕ) → (𝑅↑𝑟((𝑚 + 1) + 1)) = ((𝑅↑𝑟(𝑚 + 1)) ∘ 𝑅))
40	36, 38, 39	3syl 18	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) ∧ (𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚))) → (𝑅↑𝑟((𝑚 + 1) + 1)) = ((𝑅↑𝑟(𝑚 + 1)) ∘ 𝑅))
41		relexpsucnnr 15064	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
42	41	adantr 480	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) ∧ (𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚))) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
43	42	coeq2d 5873	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) ∧ (𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚))) → (𝑅 ∘ (𝑅↑𝑟(𝑚 + 1))) = (𝑅 ∘ ((𝑅↑𝑟𝑚) ∘ 𝑅)))
44	35, 40, 43	3eqtr4d 2787	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) ∧ (𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚))) → (𝑅↑𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅↑𝑟(𝑚 + 1))))
45	44	ex 412	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → ((𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚)) → (𝑅↑𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅↑𝑟(𝑚 + 1)))))
46	45	expcom 413	. . . 4 ⊢ (𝑚 ∈ ℕ → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚)) → (𝑅↑𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅↑𝑟(𝑚 + 1))))))
47	46	a2d 29	. . 3 ⊢ (𝑚 ∈ ℕ → ((𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚))) → (𝑅 ∈ 𝑉 → (𝑅↑𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅↑𝑟(𝑚 + 1))))))
48	6, 12, 18, 24, 31, 47	nnind 12284	. 2 ⊢ (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))))
49	48	impcom 407	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ∘ ccom 5689  (class class class)co 7431  1c1 11156   + caddc 11158  ℕcn 12266  ↑_𝑟crelexp 15058

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-seq 14043  df-relexp 15059

This theorem is referenced by:  relexpsucl  15070  relexpcnv  15074  relexpaddnn  15090  trclfvcom  43736  trclimalb2  43739