Theorem relexpcnv 14386

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 11887	. . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		oveq2 7143	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
3	2	cnveqd 5710	. . . . . . 7 ⊢ (𝑛 = 1 → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟1))
4		oveq2 7143	. . . . . . 7 ⊢ (𝑛 = 1 → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟1))
5	3, 4	eqeq12d 2814	. . . . . 6 ⊢ (𝑛 = 1 → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟1) = (◡𝑅↑𝑟1)))
6	5	imbi2d 344	. . . . 5 ⊢ (𝑛 = 1 → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟1) = (◡𝑅↑𝑟1))))
7		oveq2 7143	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑚))
8	7	cnveqd 5710	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟𝑚))
9		oveq2 7143	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑚))
10	8, 9	eqeq12d 2814	. . . . . 6 ⊢ (𝑛 = 𝑚 → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)))
11	10	imbi2d 344	. . . . 5 ⊢ (𝑛 = 𝑚 → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚))))
12		oveq2 7143	. . . . . . . 8 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
13	12	cnveqd 5710	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟(𝑚 + 1)))
14		oveq2 7143	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟(𝑚 + 1)))
15	13, 14	eqeq12d 2814	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟(𝑚 + 1)) = (◡𝑅↑𝑟(𝑚 + 1))))
16	15	imbi2d 344	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟(𝑚 + 1)) = (◡𝑅↑𝑟(𝑚 + 1)))))
17		oveq2 7143	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
18	17	cnveqd 5710	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟𝑁))
19		oveq2 7143	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑁))
20	18, 19	eqeq12d 2814	. . . . . 6 ⊢ (𝑛 = 𝑁 → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
21	20	imbi2d 344	. . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))))
22		relexp1g 14377	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
23	22	cnveqd 5710	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟1) = ◡𝑅)
24		cnvexg 7611	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ◡𝑅 ∈ V)
25		relexp1g 14377	. . . . . . 7 ⊢ (◡𝑅 ∈ V → (◡𝑅↑𝑟1) = ◡𝑅)
26	24, 25	syl 17	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (◡𝑅↑𝑟1) = ◡𝑅)
27	23, 26	eqtr4d 2836	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟1) = (◡𝑅↑𝑟1))
28		cnvco 5720	. . . . . . . . 9 ⊢ ◡((𝑅↑𝑟𝑚) ∘ 𝑅) = (◡𝑅 ∘ ◡(𝑅↑𝑟𝑚))
29		simp3 1135	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚))
30	29	coeq2d 5697	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → (◡𝑅 ∘ ◡(𝑅↑𝑟𝑚)) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
31	28, 30	syl5eq 2845	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → ◡((𝑅↑𝑟𝑚) ∘ 𝑅) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
32		simp2 1134	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → 𝑅 ∈ 𝑉)
33		simp1 1133	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → 𝑚 ∈ ℕ)
34		relexpsucnnr 14376	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
35	32, 33, 34	syl2anc 587	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
36	35	cnveqd 5710	. . . . . . . 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 14381	. . . . . . . . 9 ⊢ ((◡𝑅 ∈ V ∧ 𝑚 ∈ ℕ) → (◡𝑅↑𝑟(𝑚 + 1)) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
39	37, 33, 38	syl2anc 587	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → (◡𝑅↑𝑟(𝑚 + 1)) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
40	31, 36, 39	3eqtr4d 2843	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → ◡(𝑅↑𝑟(𝑚 + 1)) = (◡𝑅↑𝑟(𝑚 + 1)))
41	40	3exp 1116	. . . . . 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 11643	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
44		cnvresid 6403	. . . . . . 7 ⊢ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
45		uncom 4080	. . . . . . . . 9 ⊢ (dom 𝑅 ∪ ran 𝑅) = (ran 𝑅 ∪ dom 𝑅)
46		df-rn 5530	. . . . . . . . . 10 ⊢ ran 𝑅 = dom ◡𝑅
47		dfdm4 5728	. . . . . . . . . 10 ⊢ dom 𝑅 = ran ◡𝑅
48	46, 47	uneq12i 4088	. . . . . . . . 9 ⊢ (ran 𝑅 ∪ dom 𝑅) = (dom ◡𝑅 ∪ ran ◡𝑅)
49	45, 48	eqtri 2821	. . . . . . . 8 ⊢ (dom 𝑅 ∪ ran 𝑅) = (dom ◡𝑅 ∪ ran ◡𝑅)
50	49	reseq2i 5815	. . . . . . 7 ⊢ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))
51	44, 50	eqtri 2821	. . . . . 6 ⊢ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))
52		oveq2 7143	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
53		relexp0g 14373	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
54	52, 53	sylan9eq 2853	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
55	54	cnveqd 5710	. . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)))
56		oveq2 7143	. . . . . . . 8 ⊢ (𝑁 = 0 → (◡𝑅↑𝑟𝑁) = (◡𝑅↑𝑟0))
57	56	adantr 484	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟𝑁) = (◡𝑅↑𝑟0))
58		simpr 488	. . . . . . . 8 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
59		relexp0g 14373	. . . . . . . 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 2833	. . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟𝑁) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅)))
62	51, 55, 61	3eqtr4a 2859	. . . . 5 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))
63	62	ex 416	. . . 4 ⊢ (𝑁 = 0 → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
64	43, 63	jaoi 854	. . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
65	1, 64	sylbi 220	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
66	65	imp 410	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∪ cun 3879   I cid 5424  ^◡ccnv 5518  dom cdm 5519  ran crn 5520   ↾ cres 5521   ∘ ccom 5523  (class class class)co 7135  0cc0 10526  1c1 10527   + caddc 10529  ℕcn 11625  ℕ₀cn0 11885  ↑_𝑟crelexp 14370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-seq 13365  df-relexp 14371

This theorem is referenced by:  relexpcnvd  14387  relexpnnrn  14396  relexpaddg  14404  relexpaddss  40419  cnvtrclfv  40425