Theorem relexpcnv 15068

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 12501	. . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		oveq2 7418	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
3	2	cnveqd 5861	. . . . . . 7 ⊢ (𝑛 = 1 → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟1))
4		oveq2 7418	. . . . . . 7 ⊢ (𝑛 = 1 → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟1))
5	3, 4	eqeq12d 2779	. . . . . 6 ⊢ (𝑛 = 1 → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟1) = (◡𝑅↑𝑟1)))
6	5	imbi2d 343	. . . . 5 ⊢ (𝑛 = 1 → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟1) = (◡𝑅↑𝑟1))))
7		oveq2 7418	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑚))
8	7	cnveqd 5861	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟𝑚))
9		oveq2 7418	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑚))
10	8, 9	eqeq12d 2779	. . . . . 6 ⊢ (𝑛 = 𝑚 → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)))
11	10	imbi2d 343	. . . . 5 ⊢ (𝑛 = 𝑚 → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚))))
12		oveq2 7418	. . . . . . . 8 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
13	12	cnveqd 5861	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟(𝑚 + 1)))
14		oveq2 7418	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟(𝑚 + 1)))
15	13, 14	eqeq12d 2779	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟(𝑚 + 1)) = (◡𝑅↑𝑟(𝑚 + 1))))
16	15	imbi2d 343	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟(𝑚 + 1)) = (◡𝑅↑𝑟(𝑚 + 1)))))
17		oveq2 7418	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
18	17	cnveqd 5861	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟𝑁))
19		oveq2 7418	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑁))
20	18, 19	eqeq12d 2779	. . . . . 6 ⊢ (𝑛 = 𝑁 → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
21	20	imbi2d 343	. . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))))
22		relexp1g 15059	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
23	22	cnveqd 5861	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟1) = ◡𝑅)
24		cnvexg 7917	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ◡𝑅 ∈ V)
25		relexp1g 15059	. . . . . . 7 ⊢ (◡𝑅 ∈ V → (◡𝑅↑𝑟1) = ◡𝑅)
26	24, 25	syl 18	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (◡𝑅↑𝑟1) = ◡𝑅)
27	23, 26	eqtr4d 2801	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟1) = (◡𝑅↑𝑟1))
28		cnvco 5875	. . . . . . . . 9 ⊢ ◡((𝑅↑𝑟𝑚) ∘ 𝑅) = (◡𝑅 ∘ ◡(𝑅↑𝑟𝑚))
29		simp3 1156	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚))
30	29	coeq2d 5848	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → (◡𝑅 ∘ ◡(𝑅↑𝑟𝑚)) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
31	28, 30	eqtrid 2810	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → ◡((𝑅↑𝑟𝑚) ∘ 𝑅) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
32		simp2 1155	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → 𝑅 ∈ 𝑉)
33		simp1 1154	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → 𝑚 ∈ ℕ)
34		relexpsucnnr 15058	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
35	32, 33, 34	syl2anc 595	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
36	35	cnveqd 5861	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → ◡(𝑅↑𝑟(𝑚 + 1)) = ◡((𝑅↑𝑟𝑚) ∘ 𝑅))
37	32, 24	syl 18	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → ◡𝑅 ∈ V)
38		relexpsucnnl 15063	. . . . . . . . 9 ⊢ ((◡𝑅 ∈ V ∧ 𝑚 ∈ ℕ) → (◡𝑅↑𝑟(𝑚 + 1)) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
39	37, 33, 38	syl2anc 595	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → (◡𝑅↑𝑟(𝑚 + 1)) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
40	31, 36, 39	3eqtr4d 2808	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → ◡(𝑅↑𝑟(𝑚 + 1)) = (◡𝑅↑𝑟(𝑚 + 1)))
41	40	3exp 1137	. . . . . 6 ⊢ (𝑚 ∈ ℕ → (𝑅 ∈ 𝑉 → (◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚) → ◡(𝑅↑𝑟(𝑚 + 1)) = (◡𝑅↑𝑟(𝑚 + 1)))))
42	41	a2d 30	. . . . 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 12246	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
44		cnvresid 6615	. . . . . . 7 ⊢ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
45		uncom 4112	. . . . . . . . 9 ⊢ (dom 𝑅 ∪ ran 𝑅) = (ran 𝑅 ∪ dom 𝑅)
46		df-rn 5672	. . . . . . . . . 10 ⊢ ran 𝑅 = dom ◡𝑅
47		dfdm4 5885	. . . . . . . . . 10 ⊢ dom 𝑅 = ran ◡𝑅
48	46, 47	uneq12i 4120	. . . . . . . . 9 ⊢ (ran 𝑅 ∪ dom 𝑅) = (dom ◡𝑅 ∪ ran ◡𝑅)
49	45, 48	eqtri 2786	. . . . . . . 8 ⊢ (dom 𝑅 ∪ ran 𝑅) = (dom ◡𝑅 ∪ ran ◡𝑅)
50	49	reseq2i 5975	. . . . . . 7 ⊢ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))
51	44, 50	eqtri 2786	. . . . . 6 ⊢ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))
52		oveq2 7418	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
53		relexp0g 15055	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
54	52, 53	sylan9eq 2818	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
55	54	cnveqd 5861	. . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)))
56		oveq2 7418	. . . . . . . 8 ⊢ (𝑁 = 0 → (◡𝑅↑𝑟𝑁) = (◡𝑅↑𝑟0))
57	56	adantr 485	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟𝑁) = (◡𝑅↑𝑟0))
58		simpr 489	. . . . . . . 8 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
59		relexp0g 15055	. . . . . . . 8 ⊢ (◡𝑅 ∈ V → (◡𝑅↑𝑟0) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅)))
60	58, 24, 59	3syl 19	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟0) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅)))
61	57, 60	eqtrd 2798	. . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟𝑁) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅)))
62	51, 55, 61	3eqtr4a 2824	. . . . 5 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))
63	62	ex 417	. . . 4 ⊢ (𝑁 = 0 → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
64	43, 63	jaoi 870	. . 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 411	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∪ cun 3903   I cid 5555  ^◡ccnv 5660  dom cdm 5661  ran crn 5662   ↾ cres 5663   ∘ ccom 5665  (class class class)co 7410  0cc0 11095  1c1 11096   + caddc 11098  ℕcn 12228  ℕ₀cn0 12499  ↑_𝑟crelexp 15052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  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 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-n0 12500  df-z 12587  df-uz 12858  df-seq 14034  df-relexp 15053

This theorem is referenced by:  relexpcnvd  15069  relexpnnrn  15078  relexpaddg  15086  cnvtrclfv  44470