Theorem relexpmulg 38002

Description: With ordered exponents, the composition of powers of a relation is the relation raised to the product of exponents. (Contributed by RP, 13-Jun-2020.)

Assertion

Ref	Expression
relexpmulg	|- ( ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) /\ ( J e. NN0 /\ K e. NN0 ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) )

Proof of Theorem relexpmulg

Step	Hyp	Ref	Expression
1		elnn0 11294	. . . 4 |- ( J e. NN0 <-> ( J e. NN \/ J = 0 ) )
2		elnn0 11294	. . . . . 6 |- ( K e. NN0 <-> ( K e. NN \/ K = 0 ) )
3		relexpmulnn 38001	. . . . . . . . . 10 |- ( ( ( R e. V /\ I = ( J x. K ) ) /\ ( J e. NN /\ K e. NN ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) )
4	3	3adantl3 1219	. . . . . . . . 9 |- ( ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) /\ ( J e. NN /\ K e. NN ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) )
5	4	expcom 451	. . . . . . . 8 |- ( ( J e. NN /\ K e. NN ) -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) )
6	5	expcom 451	. . . . . . 7 |- ( K e. NN -> ( J e. NN -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) )
7		simprr 796	. . . . . . . . . . . . 13 |- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> I = ( J x. K ) )
8		simpll 790	. . . . . . . . . . . . . 14 |- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> K = 0 )
9	8	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> ( J x. K ) = ( J x. 0 ) )
10		simplr 792	. . . . . . . . . . . . . . 15 |- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> J e. NN )
11	10	nncnd 11036	. . . . . . . . . . . . . 14 |- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> J e. CC )
12	11	mul01d 10235	. . . . . . . . . . . . 13 |- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> ( J x. 0 ) = 0 )
13	7, 9, 12	3eqtrd 2660	. . . . . . . . . . . 12 |- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> I = 0 )
14		simpl 473	. . . . . . . . . . . . 13 |- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> ( K = 0 /\ J e. NN ) )
15		nnnle0 11051	. . . . . . . . . . . . . . 15 |- ( J e. NN -> -. J <_ 0 )
16	15	adantl 482	. . . . . . . . . . . . . 14 |- ( ( K = 0 /\ J e. NN ) -> -. J <_ 0 )
17		simpl 473	. . . . . . . . . . . . . . 15 |- ( ( K = 0 /\ J e. NN ) -> K = 0 )
18	17	breq2d 4665	. . . . . . . . . . . . . 14 |- ( ( K = 0 /\ J e. NN ) -> ( J <_ K <-> J <_ 0 ) )
19	16, 18	mtbird 315	. . . . . . . . . . . . 13 |- ( ( K = 0 /\ J e. NN ) -> -. J <_ K )
20	14, 19	syl 17	. . . . . . . . . . . 12 |- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> -. J <_ K )
21		mth8 158	. . . . . . . . . . . 12 |- ( I = 0 -> ( -. J <_ K -> -. ( I = 0 -> J <_ K ) ) )
22	13, 20, 21	sylc 65	. . . . . . . . . . 11 |- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> -. ( I = 0 -> J <_ K ) )
23	22	pm2.21d 118	. . . . . . . . . 10 |- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> ( ( I = 0 -> J <_ K ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) )
24	23	exp32 631	. . . . . . . . 9 |- ( ( K = 0 /\ J e. NN ) -> ( R e. V -> ( I = ( J x. K ) -> ( ( I = 0 -> J <_ K ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) ) )
25	24	3impd 1281	. . . . . . . 8 |- ( ( K = 0 /\ J e. NN ) -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) )
26	25	ex 450	. . . . . . 7 |- ( K = 0 -> ( J e. NN -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) )
27	6, 26	jaoi 394	. . . . . 6 |- ( ( K e. NN \/ K = 0 ) -> ( J e. NN -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) )
28	2, 27	sylbi 207	. . . . 5 |- ( K e. NN0 -> ( J e. NN -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) )
29		simplr 792	. . . . . . . . . . 11 |- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> J = 0 )
30	29	oveq2d 6666	. . . . . . . . . 10 |- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( R ^r J ) = ( R ^r 0 ) )
31		simpr1 1067	. . . . . . . . . . 11 |- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> R e. V )
32		relexp0g 13762	. . . . . . . . . . 11 |- ( R e. V -> ( R ^r 0 ) = ( _I |` ( dom R u. ran R ) ) )
33	31, 32	syl 17	. . . . . . . . . 10 |- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( R ^r 0 ) = ( _I |` ( dom R u. ran R ) ) )
34	30, 33	eqtrd 2656	. . . . . . . . 9 |- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( R ^r J ) = ( _I |` ( dom R u. ran R ) ) )
35	34	oveq1d 6665	. . . . . . . 8 |- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( ( R ^r J ) ^r K ) = ( ( _I |` ( dom R u. ran R ) ) ^r K ) )
36		dmexg 7097	. . . . . . . . . . 11 |- ( R e. V -> dom R e. _V )
37		rnexg 7098	. . . . . . . . . . 11 |- ( R e. V -> ran R e. _V )
38		unexg 6959	. . . . . . . . . . 11 |- ( ( dom R e. _V /\ ran R e. _V ) -> ( dom R u. ran R ) e. _V )
39	36, 37, 38	syl2anc 693	. . . . . . . . . 10 |- ( R e. V -> ( dom R u. ran R ) e. _V )
40	31, 39	syl 17	. . . . . . . . 9 |- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( dom R u. ran R ) e. _V )
41		simpll 790	. . . . . . . . 9 |- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> K e. NN0 )
42		relexpiidm 37996	. . . . . . . . 9 |- ( ( ( dom R u. ran R ) e. _V /\ K e. NN0 ) -> ( ( _I |` ( dom R u. ran R ) ) ^r K ) = ( _I |` ( dom R u. ran R ) ) )
43	40, 41, 42	syl2anc 693	. . . . . . . 8 |- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( ( _I |` ( dom R u. ran R ) ) ^r K ) = ( _I |` ( dom R u. ran R ) ) )
44		simpr2 1068	. . . . . . . . . . 11 |- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> I = ( J x. K ) )
45	29	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( J x. K ) = ( 0 x. K ) )
46	41	nn0cnd 11353	. . . . . . . . . . . 12 |- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> K e. CC )
47	46	mul02d 10234	. . . . . . . . . . 11 |- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( 0 x. K ) = 0 )
48	44, 45, 47	3eqtrd 2660	. . . . . . . . . 10 |- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> I = 0 )
49	48	oveq2d 6666	. . . . . . . . 9 |- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( R ^r I ) = ( R ^r 0 ) )
50	49, 33	eqtr2d 2657	. . . . . . . 8 |- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( _I |` ( dom R u. ran R ) ) = ( R ^r I ) )
51	35, 43, 50	3eqtrd 2660	. . . . . . 7 |- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) )
52	51	ex 450	. . . . . 6 |- ( ( K e. NN0 /\ J = 0 ) -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) )
53	52	ex 450	. . . . 5 |- ( K e. NN0 -> ( J = 0 -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) )
54	28, 53	jaod 395	. . . 4 |- ( K e. NN0 -> ( ( J e. NN \/ J = 0 ) -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) )
55	1, 54	syl5bi 232	. . 3 |- ( K e. NN0 -> ( J e. NN0 -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) )
56	55	impcom 446	. 2 |- ( ( J e. NN0 /\ K e. NN0 ) -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) )
57	56	impcom 446	1 |- ( ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) /\ ( J e. NN0 /\ K e. NN0 ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   class class class wbr 4653   _I cid 5023   dom cdm 5114   ran crn 5115   |` cres 5116  (class class class)co 6650   0cc0 9936   x. cmul 9941   <_ cle 10075   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: (None)