Theorem relexpcnv 14958

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 12403	. . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		oveq2 7366	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
3	2	cnveqd 5824	. . . . . . 7 ⊢ (𝑛 = 1 → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟1))
4		oveq2 7366	. . . . . . 7 ⊢ (𝑛 = 1 → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟1))
5	3, 4	eqeq12d 2752	. . . . . 6 ⊢ (𝑛 = 1 → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟1) = (◡𝑅↑𝑟1)))
6	5	imbi2d 340	. . . . 5 ⊢ (𝑛 = 1 → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟1) = (◡𝑅↑𝑟1))))
7		oveq2 7366	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑚))
8	7	cnveqd 5824	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟𝑚))
9		oveq2 7366	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑚))
10	8, 9	eqeq12d 2752	. . . . . 6 ⊢ (𝑛 = 𝑚 → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)))
11	10	imbi2d 340	. . . . 5 ⊢ (𝑛 = 𝑚 → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚))))
12		oveq2 7366	. . . . . . . 8 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
13	12	cnveqd 5824	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟(𝑚 + 1)))
14		oveq2 7366	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟(𝑚 + 1)))
15	13, 14	eqeq12d 2752	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟(𝑚 + 1)) = (◡𝑅↑𝑟(𝑚 + 1))))
16	15	imbi2d 340	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟(𝑚 + 1)) = (◡𝑅↑𝑟(𝑚 + 1)))))
17		oveq2 7366	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
18	17	cnveqd 5824	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟𝑁))
19		oveq2 7366	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑁))
20	18, 19	eqeq12d 2752	. . . . . 6 ⊢ (𝑛 = 𝑁 → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
21	20	imbi2d 340	. . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))))
22		relexp1g 14949	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
23	22	cnveqd 5824	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟1) = ◡𝑅)
24		cnvexg 7866	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ◡𝑅 ∈ V)
25		relexp1g 14949	. . . . . . 7 ⊢ (◡𝑅 ∈ V → (◡𝑅↑𝑟1) = ◡𝑅)
26	24, 25	syl 17	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (◡𝑅↑𝑟1) = ◡𝑅)
27	23, 26	eqtr4d 2774	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟1) = (◡𝑅↑𝑟1))
28		cnvco 5834	. . . . . . . . 9 ⊢ ◡((𝑅↑𝑟𝑚) ∘ 𝑅) = (◡𝑅 ∘ ◡(𝑅↑𝑟𝑚))
29		simp3 1138	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚))
30	29	coeq2d 5811	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → (◡𝑅 ∘ ◡(𝑅↑𝑟𝑚)) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
31	28, 30	eqtrid 2783	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → ◡((𝑅↑𝑟𝑚) ∘ 𝑅) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
32		simp2 1137	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → 𝑅 ∈ 𝑉)
33		simp1 1136	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → 𝑚 ∈ ℕ)
34		relexpsucnnr 14948	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
35	32, 33, 34	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
36	35	cnveqd 5824	. . . . . . . 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 14953	. . . . . . . . 9 ⊢ ((◡𝑅 ∈ V ∧ 𝑚 ∈ ℕ) → (◡𝑅↑𝑟(𝑚 + 1)) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
39	37, 33, 38	syl2anc 584	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → (◡𝑅↑𝑟(𝑚 + 1)) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
40	31, 36, 39	3eqtr4d 2781	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → ◡(𝑅↑𝑟(𝑚 + 1)) = (◡𝑅↑𝑟(𝑚 + 1)))
41	40	3exp 1119	. . . . . 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 12163	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
44		cnvresid 6571	. . . . . . 7 ⊢ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
45		uncom 4110	. . . . . . . . 9 ⊢ (dom 𝑅 ∪ ran 𝑅) = (ran 𝑅 ∪ dom 𝑅)
46		df-rn 5635	. . . . . . . . . 10 ⊢ ran 𝑅 = dom ◡𝑅
47		dfdm4 5844	. . . . . . . . . 10 ⊢ dom 𝑅 = ran ◡𝑅
48	46, 47	uneq12i 4118	. . . . . . . . 9 ⊢ (ran 𝑅 ∪ dom 𝑅) = (dom ◡𝑅 ∪ ran ◡𝑅)
49	45, 48	eqtri 2759	. . . . . . . 8 ⊢ (dom 𝑅 ∪ ran 𝑅) = (dom ◡𝑅 ∪ ran ◡𝑅)
50	49	reseq2i 5935	. . . . . . 7 ⊢ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))
51	44, 50	eqtri 2759	. . . . . 6 ⊢ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))
52		oveq2 7366	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
53		relexp0g 14945	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
54	52, 53	sylan9eq 2791	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
55	54	cnveqd 5824	. . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)))
56		oveq2 7366	. . . . . . . 8 ⊢ (𝑁 = 0 → (◡𝑅↑𝑟𝑁) = (◡𝑅↑𝑟0))
57	56	adantr 480	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟𝑁) = (◡𝑅↑𝑟0))
58		simpr 484	. . . . . . . 8 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
59		relexp0g 14945	. . . . . . . 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 2771	. . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟𝑁) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅)))
62	51, 55, 61	3eqtr4a 2797	. . . . 5 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))
63	62	ex 412	. . . 4 ⊢ (𝑁 = 0 → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
64	43, 63	jaoi 857	. . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
65	1, 64	sylbi 217	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
66	65	imp 406	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ∪ cun 3899   I cid 5518  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   ↾ cres 5626   ∘ ccom 5628  (class class class)co 7358  0cc0 11026  1c1 11027   + caddc 11029  ℕcn 12145  ℕ₀cn0 12401  ↑_𝑟crelexp 14942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-n0 12402  df-z 12489  df-uz 12752  df-seq 13925  df-relexp 14943

This theorem is referenced by:  relexpcnvd  14959  relexpnnrn  14968  relexpaddg  14976  cnvtrclfv  43975