Theorem relexpcnv 15020

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 12510	. . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		oveq2 7432	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
3	2	cnveqd 5880	. . . . . . 7 ⊢ (𝑛 = 1 → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟1))
4		oveq2 7432	. . . . . . 7 ⊢ (𝑛 = 1 → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟1))
5	3, 4	eqeq12d 2743	. . . . . 6 ⊢ (𝑛 = 1 → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟1) = (◡𝑅↑𝑟1)))
6	5	imbi2d 339	. . . . 5 ⊢ (𝑛 = 1 → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟1) = (◡𝑅↑𝑟1))))
7		oveq2 7432	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑚))
8	7	cnveqd 5880	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟𝑚))
9		oveq2 7432	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑚))
10	8, 9	eqeq12d 2743	. . . . . 6 ⊢ (𝑛 = 𝑚 → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)))
11	10	imbi2d 339	. . . . 5 ⊢ (𝑛 = 𝑚 → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚))))
12		oveq2 7432	. . . . . . . 8 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
13	12	cnveqd 5880	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟(𝑚 + 1)))
14		oveq2 7432	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟(𝑚 + 1)))
15	13, 14	eqeq12d 2743	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟(𝑚 + 1)) = (◡𝑅↑𝑟(𝑚 + 1))))
16	15	imbi2d 339	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟(𝑚 + 1)) = (◡𝑅↑𝑟(𝑚 + 1)))))
17		oveq2 7432	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
18	17	cnveqd 5880	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ◡(𝑅↑𝑟𝑛) = ◡(𝑅↑𝑟𝑁))
19		oveq2 7432	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (◡𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑁))
20	18, 19	eqeq12d 2743	. . . . . 6 ⊢ (𝑛 = 𝑁 → (◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) ↔ ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
21	20	imbi2d 339	. . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) ↔ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))))
22		relexp1g 15011	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
23	22	cnveqd 5880	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟1) = ◡𝑅)
24		cnvexg 7936	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ◡𝑅 ∈ V)
25		relexp1g 15011	. . . . . . 7 ⊢ (◡𝑅 ∈ V → (◡𝑅↑𝑟1) = ◡𝑅)
26	24, 25	syl 17	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (◡𝑅↑𝑟1) = ◡𝑅)
27	23, 26	eqtr4d 2770	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟1) = (◡𝑅↑𝑟1))
28		cnvco 5890	. . . . . . . . 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 5867	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → (◡𝑅 ∘ ◡(𝑅↑𝑟𝑚)) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
31	28, 30	eqtrid 2779	. . . . . . . 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 15010	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
35	32, 33, 34	syl2anc 582	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
36	35	cnveqd 5880	. . . . . . . 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 15015	. . . . . . . . 9 ⊢ ((◡𝑅 ∈ V ∧ 𝑚 ∈ ℕ) → (◡𝑅↑𝑟(𝑚 + 1)) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
39	37, 33, 38	syl2anc 582	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ ◡(𝑅↑𝑟𝑚) = (◡𝑅↑𝑟𝑚)) → (◡𝑅↑𝑟(𝑚 + 1)) = (◡𝑅 ∘ (◡𝑅↑𝑟𝑚)))
40	31, 36, 39	3eqtr4d 2777	. . . . . . 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 12266	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
44		cnvresid 6635	. . . . . . 7 ⊢ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
45		uncom 4152	. . . . . . . . 9 ⊢ (dom 𝑅 ∪ ran 𝑅) = (ran 𝑅 ∪ dom 𝑅)
46		df-rn 5691	. . . . . . . . . 10 ⊢ ran 𝑅 = dom ◡𝑅
47		dfdm4 5900	. . . . . . . . . 10 ⊢ dom 𝑅 = ran ◡𝑅
48	46, 47	uneq12i 4160	. . . . . . . . 9 ⊢ (ran 𝑅 ∪ dom 𝑅) = (dom ◡𝑅 ∪ ran ◡𝑅)
49	45, 48	eqtri 2755	. . . . . . . 8 ⊢ (dom 𝑅 ∪ ran 𝑅) = (dom ◡𝑅 ∪ ran ◡𝑅)
50	49	reseq2i 5984	. . . . . . 7 ⊢ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))
51	44, 50	eqtri 2755	. . . . . 6 ⊢ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))
52		oveq2 7432	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
53		relexp0g 15007	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
54	52, 53	sylan9eq 2787	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
55	54	cnveqd 5880	. . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)))
56		oveq2 7432	. . . . . . . 8 ⊢ (𝑁 = 0 → (◡𝑅↑𝑟𝑁) = (◡𝑅↑𝑟0))
57	56	adantr 479	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟𝑁) = (◡𝑅↑𝑟0))
58		simpr 483	. . . . . . . 8 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
59		relexp0g 15007	. . . . . . . 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 2767	. . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟𝑁) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅)))
62	51, 55, 61	3eqtr4a 2793	. . . . 5 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))
63	62	ex 411	. . . 4 ⊢ (𝑁 = 0 → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
64	43, 63	jaoi 855	. . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
65	1, 64	sylbi 216	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑅 ∈ 𝑉 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)))
66	65	imp 405	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))

Syntax hints:   → wi 4   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3471   ∪ cun 3945   I cid 5577  ^◡ccnv 5679  dom cdm 5680  ran crn 5681   ↾ cres 5682   ∘ ccom 5684  (class class class)co 7424  0cc0 11144  1c1 11145   + caddc 11147  ℕcn 12248  ℕ₀cn0 12508  ↑_𝑟crelexp 15004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-resscn 11201  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-mulcom 11208  ax-addass 11209  ax-mulass 11210  ax-distr 11211  ax-i2m1 11212  ax-1ne0 11213  ax-1rid 11214  ax-rnegex 11215  ax-rrecex 11216  ax-cnre 11217  ax-pre-lttri 11218  ax-pre-lttrn 11219  ax-pre-ltadd 11220  ax-pre-mulgt0 11221

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-er 8729  df-en 8969  df-dom 8970  df-sdom 8971  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-sub 11482  df-neg 11483  df-nn 12249  df-n0 12509  df-z 12595  df-uz 12859  df-seq 14005  df-relexp 15005

This theorem is referenced by:  relexpcnvd  15021  relexpnnrn  15030  relexpaddg  15038  cnvtrclfv  43157

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