Theorem relexpsucnnl 14981

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 7418	. . . . . 6 ⊢ (𝑛 = 1 → (𝑛 + 1) = (1 + 1))
2	1	oveq2d 7427	. . . . 5 ⊢ (𝑛 = 1 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅↑𝑟(1 + 1)))
3		oveq2 7419	. . . . . 6 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
4	3	coeq2d 5861	. . . . 5 ⊢ (𝑛 = 1 → (𝑅 ∘ (𝑅↑𝑟𝑛)) = (𝑅 ∘ (𝑅↑𝑟1)))
5	2, 4	eqeq12d 2746	. . . 4 ⊢ (𝑛 = 1 → ((𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛)) ↔ (𝑅↑𝑟(1 + 1)) = (𝑅 ∘ (𝑅↑𝑟1))))
6	5	imbi2d 339	. . 3 ⊢ (𝑛 = 1 → ((𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛))) ↔ (𝑅 ∈ 𝑉 → (𝑅↑𝑟(1 + 1)) = (𝑅 ∘ (𝑅↑𝑟1)))))
7		oveq1 7418	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑛 + 1) = (𝑚 + 1))
8	7	oveq2d 7427	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅↑𝑟(𝑚 + 1)))
9		oveq2 7419	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑚))
10	9	coeq2d 5861	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑅 ∘ (𝑅↑𝑟𝑛)) = (𝑅 ∘ (𝑅↑𝑟𝑚)))
11	8, 10	eqeq12d 2746	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛)) ↔ (𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚))))
12	11	imbi2d 339	. . 3 ⊢ (𝑛 = 𝑚 → ((𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛))) ↔ (𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚)))))
13		oveq1 7418	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 + 1) = ((𝑚 + 1) + 1))
14	13	oveq2d 7427	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟(𝑛 + 1)) = (𝑅↑𝑟((𝑚 + 1) + 1)))
15		oveq2 7419	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
16	15	coeq2d 5861	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑅 ∘ (𝑅↑𝑟𝑛)) = (𝑅 ∘ (𝑅↑𝑟(𝑚 + 1))))
17	14, 16	eqeq12d 2746	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → ((𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛)) ↔ (𝑅↑𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅↑𝑟(𝑚 + 1)))))
18	17	imbi2d 339	. . 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 7418	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
20	19	oveq2d 7427	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅↑𝑟(𝑁 + 1)))
21		oveq2 7419	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
22	21	coeq2d 5861	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑅 ∘ (𝑅↑𝑟𝑛)) = (𝑅 ∘ (𝑅↑𝑟𝑁)))
23	20, 22	eqeq12d 2746	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛)) ↔ (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))))
24	23	imbi2d 339	. . 3 ⊢ (𝑛 = 𝑁 → ((𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛))) ↔ (𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁)))))
25		relexp1g 14977	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
26	25	coeq1d 5860	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟1) ∘ 𝑅) = (𝑅 ∘ 𝑅))
27		1nn 12227	. . . . 5 ⊢ 1 ∈ ℕ
28		relexpsucnnr 14976	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 1 ∈ ℕ) → (𝑅↑𝑟(1 + 1)) = ((𝑅↑𝑟1) ∘ 𝑅))
29	27, 28	mpan2 687	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟(1 + 1)) = ((𝑅↑𝑟1) ∘ 𝑅))
30	25	coeq2d 5861	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∘ (𝑅↑𝑟1)) = (𝑅 ∘ 𝑅))
31	26, 29, 30	3eqtr4d 2780	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟(1 + 1)) = (𝑅 ∘ (𝑅↑𝑟1)))
32		coeq1 5856	. . . . . . . . 9 ⊢ ((𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚)) → ((𝑅↑𝑟(𝑚 + 1)) ∘ 𝑅) = ((𝑅 ∘ (𝑅↑𝑟𝑚)) ∘ 𝑅))
33		coass 6263	. . . . . . . . 9 ⊢ ((𝑅 ∘ (𝑅↑𝑟𝑚)) ∘ 𝑅) = (𝑅 ∘ ((𝑅↑𝑟𝑚) ∘ 𝑅))
34	32, 33	eqtrdi 2786	. . . . . . . 8 ⊢ ((𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚)) → ((𝑅↑𝑟(𝑚 + 1)) ∘ 𝑅) = (𝑅 ∘ ((𝑅↑𝑟𝑚) ∘ 𝑅)))
35	34	adantl 480	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) ∧ (𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚))) → ((𝑅↑𝑟(𝑚 + 1)) ∘ 𝑅) = (𝑅 ∘ ((𝑅↑𝑟𝑚) ∘ 𝑅)))
36		simpl 481	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) ∧ (𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚))) → (𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ))
37		peano2nn 12228	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
38	37	anim2i 615	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅 ∈ 𝑉 ∧ (𝑚 + 1) ∈ ℕ))
39		relexpsucnnr 14976	. . . . . . . 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 14976	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
42	41	adantr 479	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) ∧ (𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚))) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
43	42	coeq2d 5861	. . . . . . 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 2780	. . . . . 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 411	. . . . 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 412	. . . 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 12234	. 2 ⊢ (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))))
49	48	impcom 406	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁)))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104   ∘ ccom 5679  (class class class)co 7411  1c1 11113   + caddc 11115  ℕcn 12216  ↑_𝑟crelexp 14970

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-seq 13971  df-relexp 14971

This theorem is referenced by:  relexpsucl  14982  relexpcnv  14986  relexpaddnn  15002  trclfvcom  42776  trclimalb2  42779