Theorem relexpsucnnl 14955

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 7360	. . . . . 6 ⊢ (𝑛 = 1 → (𝑛 + 1) = (1 + 1))
2	1	oveq2d 7369	. . . . 5 ⊢ (𝑛 = 1 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅↑𝑟(1 + 1)))
3		oveq2 7361	. . . . . 6 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
4	3	coeq2d 5809	. . . . 5 ⊢ (𝑛 = 1 → (𝑅 ∘ (𝑅↑𝑟𝑛)) = (𝑅 ∘ (𝑅↑𝑟1)))
5	2, 4	eqeq12d 2745	. . . 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 7360	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑛 + 1) = (𝑚 + 1))
8	7	oveq2d 7369	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅↑𝑟(𝑚 + 1)))
9		oveq2 7361	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑚))
10	9	coeq2d 5809	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑅 ∘ (𝑅↑𝑟𝑛)) = (𝑅 ∘ (𝑅↑𝑟𝑚)))
11	8, 10	eqeq12d 2745	. . . 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 7360	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 + 1) = ((𝑚 + 1) + 1))
14	13	oveq2d 7369	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟(𝑛 + 1)) = (𝑅↑𝑟((𝑚 + 1) + 1)))
15		oveq2 7361	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
16	15	coeq2d 5809	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑅 ∘ (𝑅↑𝑟𝑛)) = (𝑅 ∘ (𝑅↑𝑟(𝑚 + 1))))
17	14, 16	eqeq12d 2745	. . . 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 7360	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
20	19	oveq2d 7369	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅↑𝑟(𝑁 + 1)))
21		oveq2 7361	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
22	21	coeq2d 5809	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑅 ∘ (𝑅↑𝑟𝑛)) = (𝑅 ∘ (𝑅↑𝑟𝑁)))
23	20, 22	eqeq12d 2745	. . . 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 14951	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
26	25	coeq1d 5808	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟1) ∘ 𝑅) = (𝑅 ∘ 𝑅))
27		1nn 12157	. . . . 5 ⊢ 1 ∈ ℕ
28		relexpsucnnr 14950	. . . . 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 5809	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∘ (𝑅↑𝑟1)) = (𝑅 ∘ 𝑅))
31	26, 29, 30	3eqtr4d 2774	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟(1 + 1)) = (𝑅 ∘ (𝑅↑𝑟1)))
32		coeq1 5804	. . . . . . . . 9 ⊢ ((𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚)) → ((𝑅↑𝑟(𝑚 + 1)) ∘ 𝑅) = ((𝑅 ∘ (𝑅↑𝑟𝑚)) ∘ 𝑅))
33		coass 6218	. . . . . . . . 9 ⊢ ((𝑅 ∘ (𝑅↑𝑟𝑚)) ∘ 𝑅) = (𝑅 ∘ ((𝑅↑𝑟𝑚) ∘ 𝑅))
34	32, 33	eqtrdi 2780	. . . . . . . 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 12158	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
38	37	anim2i 617	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅 ∈ 𝑉 ∧ (𝑚 + 1) ∈ ℕ))
39		relexpsucnnr 14950	. . . . . . . 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 14950	. . . . . . . . 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 5809	. . . . . . 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 2774	. . . . . 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 12164	. 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 2109   ∘ ccom 5627  (class class class)co 7353  1c1 11029   + caddc 11031  ℕcn 12146  ↑_𝑟crelexp 14944

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 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-n0 12403  df-z 12490  df-uz 12754  df-seq 13927  df-relexp 14945

This theorem is referenced by:  relexpsucl  14956  relexpcnv  14960  relexpaddnn  14976  trclfvcom  43699  trclimalb2  43702