Theorem relexpcnv 13775

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

Assertion

Ref	Expression
relexpcnv	|- ( ( N e. NN0 /\ R e. V ) -> `' ( R ^r N ) = ( `' R ^r N ) )

Proof of Theorem relexpcnv

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

Step	Hyp	Ref	Expression
1		elnn0 11294	. . 3 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		oveq2 6658	. . . . . . . 8 |- ( n = 1 -> ( R ^r n ) = ( R ^r 1 ) )
3	2	cnveqd 5298	. . . . . . 7 |- ( n = 1 -> `' ( R ^r n ) = `' ( R ^r 1 ) )
4		oveq2 6658	. . . . . . 7 |- ( n = 1 -> ( `' R ^r n ) = ( `' R ^r 1 ) )
5	3, 4	eqeq12d 2637	. . . . . 6 |- ( n = 1 -> ( `' ( R ^r n ) = ( `' R ^r n ) <-> `' ( R ^r 1 ) = ( `' R ^r 1 ) ) )
6	5	imbi2d 330	. . . . 5 |- ( n = 1 -> ( ( R e. V -> `' ( R ^r n ) = ( `' R ^r n ) ) <-> ( R e. V -> `' ( R ^r 1 ) = ( `' R ^r 1 ) ) ) )
7		oveq2 6658	. . . . . . . 8 |- ( n = m -> ( R ^r n ) = ( R ^r m ) )
8	7	cnveqd 5298	. . . . . . 7 |- ( n = m -> `' ( R ^r n ) = `' ( R ^r m ) )
9		oveq2 6658	. . . . . . 7 |- ( n = m -> ( `' R ^r n ) = ( `' R ^r m ) )
10	8, 9	eqeq12d 2637	. . . . . 6 |- ( n = m -> ( `' ( R ^r n ) = ( `' R ^r n ) <-> `' ( R ^r m ) = ( `' R ^r m ) ) )
11	10	imbi2d 330	. . . . 5 |- ( n = m -> ( ( R e. V -> `' ( R ^r n ) = ( `' R ^r n ) ) <-> ( R e. V -> `' ( R ^r m ) = ( `' R ^r m ) ) ) )
12		oveq2 6658	. . . . . . . 8 |- ( n = ( m + 1 ) -> ( R ^r n ) = ( R ^r ( m + 1 ) ) )
13	12	cnveqd 5298	. . . . . . 7 |- ( n = ( m + 1 ) -> `' ( R ^r n ) = `' ( R ^r ( m + 1 ) ) )
14		oveq2 6658	. . . . . . 7 |- ( n = ( m + 1 ) -> ( `' R ^r n ) = ( `' R ^r ( m + 1 ) ) )
15	13, 14	eqeq12d 2637	. . . . . 6 |- ( n = ( m + 1 ) -> ( `' ( R ^r n ) = ( `' R ^r n ) <-> `' ( R ^r ( m + 1 ) ) = ( `' R ^r ( m + 1 ) ) ) )
16	15	imbi2d 330	. . . . 5 |- ( n = ( m + 1 ) -> ( ( R e. V -> `' ( R ^r n ) = ( `' R ^r n ) ) <-> ( R e. V -> `' ( R ^r ( m + 1 ) ) = ( `' R ^r ( m + 1 ) ) ) ) )
17		oveq2 6658	. . . . . . . 8 |- ( n = N -> ( R ^r n ) = ( R ^r N ) )
18	17	cnveqd 5298	. . . . . . 7 |- ( n = N -> `' ( R ^r n ) = `' ( R ^r N ) )
19		oveq2 6658	. . . . . . 7 |- ( n = N -> ( `' R ^r n ) = ( `' R ^r N ) )
20	18, 19	eqeq12d 2637	. . . . . 6 |- ( n = N -> ( `' ( R ^r n ) = ( `' R ^r n ) <-> `' ( R ^r N ) = ( `' R ^r N ) ) )
21	20	imbi2d 330	. . . . 5 |- ( n = N -> ( ( R e. V -> `' ( R ^r n ) = ( `' R ^r n ) ) <-> ( R e. V -> `' ( R ^r N ) = ( `' R ^r N ) ) ) )
22		relexp1g 13766	. . . . . . 7 |- ( R e. V -> ( R ^r 1 ) = R )
23	22	cnveqd 5298	. . . . . 6 |- ( R e. V -> `' ( R ^r 1 ) = `' R )
24		cnvexg 7112	. . . . . . 7 |- ( R e. V -> `' R e. _V )
25		relexp1g 13766	. . . . . . 7 |- ( `' R e. _V -> ( `' R ^r 1 ) = `' R )
26	24, 25	syl 17	. . . . . 6 |- ( R e. V -> ( `' R ^r 1 ) = `' R )
27	23, 26	eqtr4d 2659	. . . . 5 |- ( R e. V -> `' ( R ^r 1 ) = ( `' R ^r 1 ) )
28		cnvco 5308	. . . . . . . . 9 |- `' ( ( R ^r m ) o. R ) = ( `' R o. `' ( R ^r m ) )
29		simp3 1063	. . . . . . . . . 10 |- ( ( m e. NN /\ R e. V /\ `' ( R ^r m ) = ( `' R ^r m ) ) -> `' ( R ^r m ) = ( `' R ^r m ) )
30	29	coeq2d 5284	. . . . . . . . 9 |- ( ( m e. NN /\ R e. V /\ `' ( R ^r m ) = ( `' R ^r m ) ) -> ( `' R o. `' ( R ^r m ) ) = ( `' R o. ( `' R ^r m ) ) )
31	28, 30	syl5eq 2668	. . . . . . . 8 |- ( ( m e. NN /\ R e. V /\ `' ( R ^r m ) = ( `' R ^r m ) ) -> `' ( ( R ^r m ) o. R ) = ( `' R o. ( `' R ^r m ) ) )
32		simp2 1062	. . . . . . . . . 10 |- ( ( m e. NN /\ R e. V /\ `' ( R ^r m ) = ( `' R ^r m ) ) -> R e. V )
33		simp1 1061	. . . . . . . . . 10 |- ( ( m e. NN /\ R e. V /\ `' ( R ^r m ) = ( `' R ^r m ) ) -> m e. NN )
34		relexpsucnnr 13765	. . . . . . . . . 10 |- ( ( R e. V /\ m e. NN ) -> ( R ^r ( m + 1 ) ) = ( ( R ^r m ) o. R ) )
35	32, 33, 34	syl2anc 693	. . . . . . . . 9 |- ( ( m e. NN /\ R e. V /\ `' ( R ^r m ) = ( `' R ^r m ) ) -> ( R ^r ( m + 1 ) ) = ( ( R ^r m ) o. R ) )
36	35	cnveqd 5298	. . . . . . . 8 |- ( ( m e. NN /\ R e. V /\ `' ( R ^r m ) = ( `' R ^r m ) ) -> `' ( R ^r ( m + 1 ) ) = `' ( ( R ^r m ) o. R ) )
37	32, 24	syl 17	. . . . . . . . 9 |- ( ( m e. NN /\ R e. V /\ `' ( R ^r m ) = ( `' R ^r m ) ) -> `' R e. _V )
38		relexpsucnnl 13772	. . . . . . . . 9 |- ( ( `' R e. _V /\ m e. NN ) -> ( `' R ^r ( m + 1 ) ) = ( `' R o. ( `' R ^r m ) ) )
39	37, 33, 38	syl2anc 693	. . . . . . . 8 |- ( ( m e. NN /\ R e. V /\ `' ( R ^r m ) = ( `' R ^r m ) ) -> ( `' R ^r ( m + 1 ) ) = ( `' R o. ( `' R ^r m ) ) )
40	31, 36, 39	3eqtr4d 2666	. . . . . . 7 |- ( ( m e. NN /\ R e. V /\ `' ( R ^r m ) = ( `' R ^r m ) ) -> `' ( R ^r ( m + 1 ) ) = ( `' R ^r ( m + 1 ) ) )
41	40	3exp 1264	. . . . . 6 |- ( m e. NN -> ( R e. V -> ( `' ( R ^r m ) = ( `' R ^r m ) -> `' ( R ^r ( m + 1 ) ) = ( `' R ^r ( m + 1 ) ) ) ) )
42	41	a2d 29	. . . . 5 |- ( m e. NN -> ( ( R e. V -> `' ( R ^r m ) = ( `' R ^r m ) ) -> ( R e. V -> `' ( R ^r ( m + 1 ) ) = ( `' R ^r ( m + 1 ) ) ) ) )
43	6, 11, 16, 21, 27, 42	nnind 11038	. . . 4 |- ( N e. NN -> ( R e. V -> `' ( R ^r N ) = ( `' R ^r N ) ) )
44		cnvresid 5968	. . . . . . 7 |- `' ( _I |` ( dom R u. ran R ) ) = ( _I |` ( dom R u. ran R ) )
45		uncom 3757	. . . . . . . . 9 |- ( dom R u. ran R ) = ( ran R u. dom R )
46		df-rn 5125	. . . . . . . . . 10 |- ran R = dom `' R
47		dfdm4 5316	. . . . . . . . . 10 |- dom R = ran `' R
48	46, 47	uneq12i 3765	. . . . . . . . 9 |- ( ran R u. dom R ) = ( dom `' R u. ran `' R )
49	45, 48	eqtri 2644	. . . . . . . 8 |- ( dom R u. ran R ) = ( dom `' R u. ran `' R )
50	49	reseq2i 5393	. . . . . . 7 |- ( _I |` ( dom R u. ran R ) ) = ( _I |` ( dom `' R u. ran `' R ) )
51	44, 50	eqtri 2644	. . . . . 6 |- `' ( _I |` ( dom R u. ran R ) ) = ( _I |` ( dom `' R u. ran `' R ) )
52		oveq2 6658	. . . . . . . 8 |- ( N = 0 -> ( R ^r N ) = ( R ^r 0 ) )
53		relexp0g 13762	. . . . . . . 8 |- ( R e. V -> ( R ^r 0 ) = ( _I |` ( dom R u. ran R ) ) )
54	52, 53	sylan9eq 2676	. . . . . . 7 |- ( ( N = 0 /\ R e. V ) -> ( R ^r N ) = ( _I |` ( dom R u. ran R ) ) )
55	54	cnveqd 5298	. . . . . 6 |- ( ( N = 0 /\ R e. V ) -> `' ( R ^r N ) = `' ( _I |` ( dom R u. ran R ) ) )
56		oveq2 6658	. . . . . . . 8 |- ( N = 0 -> ( `' R ^r N ) = ( `' R ^r 0 ) )
57	56	adantr 481	. . . . . . 7 |- ( ( N = 0 /\ R e. V ) -> ( `' R ^r N ) = ( `' R ^r 0 ) )
58		simpr 477	. . . . . . . 8 |- ( ( N = 0 /\ R e. V ) -> R e. V )
59		relexp0g 13762	. . . . . . . 8 |- ( `' R e. _V -> ( `' R ^r 0 ) = ( _I |` ( dom `' R u. ran `' R ) ) )
60	58, 24, 59	3syl 18	. . . . . . 7 |- ( ( N = 0 /\ R e. V ) -> ( `' R ^r 0 ) = ( _I |` ( dom `' R u. ran `' R ) ) )
61	57, 60	eqtrd 2656	. . . . . 6 |- ( ( N = 0 /\ R e. V ) -> ( `' R ^r N ) = ( _I |` ( dom `' R u. ran `' R ) ) )
62	51, 55, 61	3eqtr4a 2682	. . . . 5 |- ( ( N = 0 /\ R e. V ) -> `' ( R ^r N ) = ( `' R ^r N ) )
63	62	ex 450	. . . 4 |- ( N = 0 -> ( R e. V -> `' ( R ^r N ) = ( `' R ^r N ) ) )
64	43, 63	jaoi 394	. . 3 |- ( ( N e. NN \/ N = 0 ) -> ( R e. V -> `' ( R ^r N ) = ( `' R ^r N ) ) )
65	1, 64	sylbi 207	. 2 |- ( N e. NN0 -> ( R e. V -> `' ( R ^r N ) = ( `' R ^r N ) ) )
66	65	imp 445	1 |- ( ( N e. NN0 /\ R e. V ) -> `' ( R ^r N ) = ( `' R ^r N ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   _I cid 5023   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   NNcn 11020   NN0cn0 11292   ^r crelexp 13760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-relexp 13761

This theorem is referenced by:  relexpcnvd  13776  relexpnnrn  13785  relexpaddg  13793  relexpaddss  38010  cnvtrclfv  38016