Theorem relexpsucnnl 15003

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 7397	. . . . . 6 ⊢ (𝑛 = 1 → (𝑛 + 1) = (1 + 1))
2	1	oveq2d 7406	. . . . 5 ⊢ (𝑛 = 1 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅↑𝑟(1 + 1)))
3		oveq2 7398	. . . . . 6 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
4	3	coeq2d 5829	. . . . 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 340	. . 3 ⊢ (𝑛 = 1 → ((𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛))) ↔ (𝑅 ∈ 𝑉 → (𝑅↑𝑟(1 + 1)) = (𝑅 ∘ (𝑅↑𝑟1)))))
7		oveq1 7397	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑛 + 1) = (𝑚 + 1))
8	7	oveq2d 7406	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅↑𝑟(𝑚 + 1)))
9		oveq2 7398	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑚))
10	9	coeq2d 5829	. . . . 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 340	. . 3 ⊢ (𝑛 = 𝑚 → ((𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛))) ↔ (𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚)))))
13		oveq1 7397	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 + 1) = ((𝑚 + 1) + 1))
14	13	oveq2d 7406	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟(𝑛 + 1)) = (𝑅↑𝑟((𝑚 + 1) + 1)))
15		oveq2 7398	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
16	15	coeq2d 5829	. . . . 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 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 7397	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
20	19	oveq2d 7406	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅↑𝑟(𝑁 + 1)))
21		oveq2 7398	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
22	21	coeq2d 5829	. . . . 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 340	. . 3 ⊢ (𝑛 = 𝑁 → ((𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛))) ↔ (𝑅 ∈ 𝑉 → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁)))))
25		relexp1g 14999	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
26	25	coeq1d 5828	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟1) ∘ 𝑅) = (𝑅 ∘ 𝑅))
27		1nn 12204	. . . . 5 ⊢ 1 ∈ ℕ
28		relexpsucnnr 14998	. . . . 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 5829	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∘ (𝑅↑𝑟1)) = (𝑅 ∘ 𝑅))
31	26, 29, 30	3eqtr4d 2775	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟(1 + 1)) = (𝑅 ∘ (𝑅↑𝑟1)))
32		coeq1 5824	. . . . . . . . 9 ⊢ ((𝑅↑𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑚)) → ((𝑅↑𝑟(𝑚 + 1)) ∘ 𝑅) = ((𝑅 ∘ (𝑅↑𝑟𝑚)) ∘ 𝑅))
33		coass 6241	. . . . . . . . 9 ⊢ ((𝑅 ∘ (𝑅↑𝑟𝑚)) ∘ 𝑅) = (𝑅 ∘ ((𝑅↑𝑟𝑚) ∘ 𝑅))
34	32, 33	eqtrdi 2781	. . . . . . . 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 12205	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
38	37	anim2i 617	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅 ∈ 𝑉 ∧ (𝑚 + 1) ∈ ℕ))
39		relexpsucnnr 14998	. . . . . . . 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 14998	. . . . . . . . 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 5829	. . . . . . 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 2775	. . . . . 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 12211	. 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 5645  (class class class)co 7390  1c1 11076   + caddc 11078  ℕcn 12193  ↑_𝑟crelexp 14992

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-n0 12450  df-z 12537  df-uz 12801  df-seq 13974  df-relexp 14993

This theorem is referenced by:  relexpsucl  15004  relexpcnv  15008  relexpaddnn  15024  trclfvcom  43719  trclimalb2  43722