Theorem relexpcnv 13623

Description: Commutation of converse and relation exponentiation. (Contributed by RP, 23-May-2020.)

Assertion

Ref	Expression
relexpcnv	⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))

Proof of Theorem relexpcnv

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

Step	Hyp	Ref	Expression
1		elnn0 11171	. . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		oveq2 6557	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
3	2	cnveqd 5220	. . . . . . 7 ⊢ (𝑛 = 1 → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟1))
4		oveq2 6557	. . . . . . 7 ⊢ (𝑛 = 1 → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟1))
5	3, 4	eqeq12d 2625	. . . . . 6 ⊢ (𝑛 = 1 → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟1) = (◡𝑅↑𝑟1)))
6	5	imbi2d 329	. . . . 5 ⊢ (𝑛 = 1 → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟1) = (◡𝑅↑𝑟1))))
7		oveq2 6557	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑚))
8	7	cnveqd 5220	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟𝑚))
9		oveq2 6557	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑚))
10	8, 9	eqeq12d 2625	. . . . . 6 ⊢ (𝑛 = 𝑚 → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)))
11	10	imbi2d 329	. . . . 5 ⊢ (𝑛 = 𝑚 → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚))))
12		oveq2 6557	. . . . . . . 8 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
13	12	cnveqd 5220	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟(𝑚 + 1)))
14		oveq2 6557	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟(𝑚 + 1)))
15	13, 14	eqeq12d 2625	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟(𝑚 + 1)) = (◡𝑅↑𝑟(𝑚 + 1))))
16	15	imbi2d 329	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟(𝑚 + 1)) = (◡𝑅↑𝑟(𝑚 + 1)))))
17		oveq2 6557	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
18	17	cnveqd 5220	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟𝑁))
19		oveq2 6557	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑁))
20	18, 19	eqeq12d 2625	. . . . . 6 ⊢ (𝑛 = 𝑁 → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
21	20	imbi2d 329	. . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))))
22		relexp1g 13614	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
23	22	cnveqd 5220	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟1) = ◡𝑅)
24		cnvexg 7005	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ◡𝑅 ∈ V)
25		relexp1g 13614	. . . . . . 7 ⊢ (◡𝑅 ∈ V → (◡𝑅↑𝑟1) = ◡𝑅)
26	24, 25	syl 17	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (◡𝑅↑𝑟1) = ◡𝑅)
27	23, 26	eqtr4d 2647	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟1) = (◡𝑅↑𝑟1))
28		cnvco 5230	. . . . . . . . 9 ⊢ ◡((𝑅↑𝑟𝑚) ∘ 𝑅) = (◡𝑅 ∘ ◡(𝑅↑𝑟𝑚))
29		simp3 1056	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚))
30	29	coeq2d 5206	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → (◡𝑅 ∘ ◡(𝑅↑𝑟𝑚)) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
31	28, 30	syl5eq 2656	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → ◡((𝑅↑𝑟𝑚) ∘ 𝑅) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
32		simp2 1055	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → 𝑅 ∈ 𝑉)
33		simp1 1054	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → 𝑚 ∈ ℕ)
34		relexpsucnnr 13613	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
35	32, 33, 34	syl2anc 691	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
36	35	cnveqd 5220	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → ◡(𝑅↑𝑟(𝑚 + 1)) = ◡((𝑅↑𝑟𝑚) ∘ 𝑅))
37	32, 24	syl 17	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → ◡𝑅 ∈ V)
38		relexpsucnnl 13620	. . . . . . . . 9 ⊢ ((◡𝑅 ∈ V ∧ 𝑚 ∈ ℕ) → (◡𝑅↑𝑟(𝑚 + 1)) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
39	37, 33, 38	syl2anc 691	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → (◡𝑅↑𝑟(𝑚 + 1)) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
40	31, 36, 39	3eqtr4d 2654	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → ◡(𝑅↑𝑟(𝑚 + 1)) = (◡𝑅↑𝑟(𝑚 + 1)))
41	40	3exp 1256	. . . . . 6 ⊢ (𝑚 ∈ ℕ → (𝑅 ∈ 𝑉 → (◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚) → ◡(𝑅↑𝑟(𝑚 + 1)) = (◡𝑅↑𝑟(𝑚 + 1)))))
42	41	a2d 29	. . . . 5 ⊢ (𝑚 ∈ ℕ → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟(𝑚 + 1)) = (◡𝑅↑𝑟(𝑚 + 1)))))
43	6, 11, 16, 21, 27, 42	nnind 10915	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
44		cnvresid 5882	. . . . . . 7 ⊢ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
45		uncom 3719	. . . . . . . . 9 ⊢ (dom 𝑅 ∪ ran 𝑅) = (ran 𝑅 ∪ dom 𝑅)
46		df-rn 5049	. . . . . . . . . 10 ⊢ ran 𝑅 = dom ◡𝑅
47		dfdm4 5238	. . . . . . . . . 10 ⊢ dom 𝑅 = ran ◡𝑅
48	46, 47	uneq12i 3727	. . . . . . . . 9 ⊢ (ran 𝑅 ∪ dom 𝑅) = (dom ◡𝑅 ∪ ran ◡𝑅)
49	45, 48	eqtri 2632	. . . . . . . 8 ⊢ (dom 𝑅 ∪ ran 𝑅) = (dom ◡𝑅 ∪ ran ◡𝑅)
50	49	reseq2i 5314	. . . . . . 7 ⊢ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))
51	44, 50	eqtri 2632	. . . . . 6 ⊢ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))
52		oveq2 6557	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
53		relexp0g 13610	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
54	52, 53	sylan9eq 2664	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
55	54	cnveqd 5220	. . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)))
56		oveq2 6557	. . . . . . . 8 ⊢ (𝑁 = 0 → (◡𝑅↑𝑟𝑁) = (◡𝑅↑𝑟0))
57	56	adantr 480	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟𝑁) = (◡𝑅↑𝑟0))
58		simpr 476	. . . . . . . 8 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
59		relexp0g 13610	. . . . . . . 8 ⊢ (◡𝑅 ∈ V → (◡𝑅↑𝑟0) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅)))
60	58, 24, 59	3syl 18	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟0) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅)))
61	57, 60	eqtrd 2644	. . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟𝑁) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅)))
62	51, 55, 61	3eqtr4a 2670	. . . . 5 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))
63	62	ex 449	. . . 4 ⊢ (𝑁 = 0 → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
64	43, 63	jaoi 393	. . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
65	1, 64	sylbi 206	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
66	65	imp 444	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   I cid 4948  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   ∘ ccom 5042  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  ℕcn 10897  ℕ₀cn0 11169  ↑_𝑟crelexp 13608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-seq 12664  df-relexp 13609

This theorem is referenced by:  relexpcnvd  13624  relexpnnrn  13633  relexpaddg  13641  relexpaddss  37029  cnvtrclfv  37035