Theorem relexp01min 38005

Description: With exponents limited to 0 and 1, the composition of powers of a relation is the relation raised to the minimum of exponents. (Contributed by RP, 12-Jun-2020.)

Assertion

Ref	Expression
relexp01min	|- ( ( ( R e. V /\ I = if ( J < K , J , K ) ) /\ ( J e. { 0 , 1 } /\ K e. { 0 , 1 } ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) )

Proof of Theorem relexp01min

Step	Hyp	Ref	Expression
1		elpri 4197	. . 3 |- ( J e. { 0 , 1 } -> ( J = 0 \/ J = 1 ) )
2		elpri 4197	. . 3 |- ( K e. { 0 , 1 } -> ( K = 0 \/ K = 1 ) )
3		dmresi 5457	. . . . . . . . . . 11 |- dom ( _I |` ( dom R u. ran R ) ) = ( dom R u. ran R )
4		rnresi 5479	. . . . . . . . . . 11 |- ran ( _I |` ( dom R u. ran R ) ) = ( dom R u. ran R )
5	3, 4	uneq12i 3765	. . . . . . . . . 10 |- ( dom ( _I |` ( dom R u. ran R ) ) u. ran ( _I |` ( dom R u. ran R ) ) ) = ( ( dom R u. ran R ) u. ( dom R u. ran R ) )
6		unidm 3756	. . . . . . . . . 10 |- ( ( dom R u. ran R ) u. ( dom R u. ran R ) ) = ( dom R u. ran R )
7	5, 6	eqtri 2644	. . . . . . . . 9 |- ( dom ( _I |` ( dom R u. ran R ) ) u. ran ( _I |` ( dom R u. ran R ) ) ) = ( dom R u. ran R )
8	7	reseq2i 5393	. . . . . . . 8 |- ( _I |` ( dom ( _I |` ( dom R u. ran R ) ) u. ran ( _I |` ( dom R u. ran R ) ) ) ) = ( _I |` ( dom R u. ran R ) )
9		simp1 1061	. . . . . . . . . . . 12 |- ( ( J = 0 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> J = 0 )
10	9	oveq2d 6666	. . . . . . . . . . 11 |- ( ( J = 0 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( R ^r J ) = ( R ^r 0 ) )
11		simp3l 1089	. . . . . . . . . . . 12 |- ( ( J = 0 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> R e. V )
12		relexp0g 13762	. . . . . . . . . . . 12 |- ( R e. V -> ( R ^r 0 ) = ( _I |` ( dom R u. ran R ) ) )
13	11, 12	syl 17	. . . . . . . . . . 11 |- ( ( J = 0 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( R ^r 0 ) = ( _I |` ( dom R u. ran R ) ) )
14	10, 13	eqtrd 2656	. . . . . . . . . 10 |- ( ( J = 0 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( R ^r J ) = ( _I |` ( dom R u. ran R ) ) )
15		simp2 1062	. . . . . . . . . 10 |- ( ( J = 0 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> K = 0 )
16	14, 15	oveq12d 6668	. . . . . . . . 9 |- ( ( J = 0 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( ( R ^r J ) ^r K ) = ( ( _I |` ( dom R u. ran R ) ) ^r 0 ) )
17		dmexg 7097	. . . . . . . . . . . 12 |- ( R e. V -> dom R e. _V )
18		rnexg 7098	. . . . . . . . . . . 12 |- ( R e. V -> ran R e. _V )
19		unexg 6959	. . . . . . . . . . . 12 |- ( ( dom R e. _V /\ ran R e. _V ) -> ( dom R u. ran R ) e. _V )
20	17, 18, 19	syl2anc 693	. . . . . . . . . . 11 |- ( R e. V -> ( dom R u. ran R ) e. _V )
21	20	resiexd 6480	. . . . . . . . . 10 |- ( R e. V -> ( _I |` ( dom R u. ran R ) ) e. _V )
22		relexp0g 13762	. . . . . . . . . 10 |- ( ( _I |` ( dom R u. ran R ) ) e. _V -> ( ( _I |` ( dom R u. ran R ) ) ^r 0 ) = ( _I |` ( dom ( _I |` ( dom R u. ran R ) ) u. ran ( _I |` ( dom R u. ran R ) ) ) ) )
23	11, 21, 22	3syl 18	. . . . . . . . 9 |- ( ( J = 0 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( ( _I |` ( dom R u. ran R ) ) ^r 0 ) = ( _I |` ( dom ( _I |` ( dom R u. ran R ) ) u. ran ( _I |` ( dom R u. ran R ) ) ) ) )
24	16, 23	eqtrd 2656	. . . . . . . 8 |- ( ( J = 0 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( ( R ^r J ) ^r K ) = ( _I |` ( dom ( _I |` ( dom R u. ran R ) ) u. ran ( _I |` ( dom R u. ran R ) ) ) ) )
25		simp3r 1090	. . . . . . . . . . 11 |- ( ( J = 0 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> I = if ( J < K , J , K ) )
26		0re 10040	. . . . . . . . . . . . . 14 |- 0 e. RR
27	26	ltnri 10146	. . . . . . . . . . . . 13 |- -. 0 < 0
28	9, 15	breq12d 4666	. . . . . . . . . . . . 13 |- ( ( J = 0 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( J < K <-> 0 < 0 ) )
29	27, 28	mtbiri 317	. . . . . . . . . . . 12 |- ( ( J = 0 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> -. J < K )
30	29	iffalsed 4097	. . . . . . . . . . 11 |- ( ( J = 0 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> if ( J < K , J , K ) = K )
31	25, 30, 15	3eqtrd 2660	. . . . . . . . . 10 |- ( ( J = 0 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> I = 0 )
32	31	oveq2d 6666	. . . . . . . . 9 |- ( ( J = 0 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( R ^r I ) = ( R ^r 0 ) )
33	32, 13	eqtrd 2656	. . . . . . . 8 |- ( ( J = 0 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( R ^r I ) = ( _I |` ( dom R u. ran R ) ) )
34	8, 24, 33	3eqtr4a 2682	. . . . . . 7 |- ( ( J = 0 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) )
35	34	3exp 1264	. . . . . 6 |- ( J = 0 -> ( K = 0 -> ( ( R e. V /\ I = if ( J < K , J , K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) )
36		simp1 1061	. . . . . . . . . . 11 |- ( ( J = 1 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> J = 1 )
37	36	oveq2d 6666	. . . . . . . . . 10 |- ( ( J = 1 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( R ^r J ) = ( R ^r 1 ) )
38		simp3l 1089	. . . . . . . . . . 11 |- ( ( J = 1 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> R e. V )
39		relexp1g 13766	. . . . . . . . . . 11 |- ( R e. V -> ( R ^r 1 ) = R )
40	38, 39	syl 17	. . . . . . . . . 10 |- ( ( J = 1 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( R ^r 1 ) = R )
41	37, 40	eqtrd 2656	. . . . . . . . 9 |- ( ( J = 1 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( R ^r J ) = R )
42		simp2 1062	. . . . . . . . 9 |- ( ( J = 1 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> K = 0 )
43	41, 42	oveq12d 6668	. . . . . . . 8 |- ( ( J = 1 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r 0 ) )
44		simp3r 1090	. . . . . . . . . 10 |- ( ( J = 1 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> I = if ( J < K , J , K ) )
45		0lt1 10550	. . . . . . . . . . . . 13 |- 0 < 1
46		1re 10039	. . . . . . . . . . . . . 14 |- 1 e. RR
47	26, 46	ltnsymi 10156	. . . . . . . . . . . . 13 |- ( 0 < 1 -> -. 1 < 0 )
48	45, 47	mp1i 13	. . . . . . . . . . . 12 |- ( ( J = 1 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> -. 1 < 0 )
49	36, 42	breq12d 4666	. . . . . . . . . . . 12 |- ( ( J = 1 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( J < K <-> 1 < 0 ) )
50	48, 49	mtbird 315	. . . . . . . . . . 11 |- ( ( J = 1 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> -. J < K )
51	50	iffalsed 4097	. . . . . . . . . 10 |- ( ( J = 1 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> if ( J < K , J , K ) = K )
52	44, 51, 42	3eqtrd 2660	. . . . . . . . 9 |- ( ( J = 1 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> I = 0 )
53	52	oveq2d 6666	. . . . . . . 8 |- ( ( J = 1 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( R ^r I ) = ( R ^r 0 ) )
54	43, 53	eqtr4d 2659	. . . . . . 7 |- ( ( J = 1 /\ K = 0 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) )
55	54	3exp 1264	. . . . . 6 |- ( J = 1 -> ( K = 0 -> ( ( R e. V /\ I = if ( J < K , J , K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) )
56	35, 55	jaoi 394	. . . . 5 |- ( ( J = 0 \/ J = 1 ) -> ( K = 0 -> ( ( R e. V /\ I = if ( J < K , J , K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) )
57		ovex 6678	. . . . . . . . 9 |- ( R ^r 0 ) e. _V
58		relexp1g 13766	. . . . . . . . 9 |- ( ( R ^r 0 ) e. _V -> ( ( R ^r 0 ) ^r 1 ) = ( R ^r 0 ) )
59	57, 58	mp1i 13	. . . . . . . 8 |- ( ( J = 0 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( ( R ^r 0 ) ^r 1 ) = ( R ^r 0 ) )
60		simp1 1061	. . . . . . . . . 10 |- ( ( J = 0 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> J = 0 )
61	60	oveq2d 6666	. . . . . . . . 9 |- ( ( J = 0 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( R ^r J ) = ( R ^r 0 ) )
62		simp2 1062	. . . . . . . . 9 |- ( ( J = 0 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> K = 1 )
63	61, 62	oveq12d 6668	. . . . . . . 8 |- ( ( J = 0 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( ( R ^r J ) ^r K ) = ( ( R ^r 0 ) ^r 1 ) )
64		simp3r 1090	. . . . . . . . . 10 |- ( ( J = 0 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> I = if ( J < K , J , K ) )
65	60, 62	breq12d 4666	. . . . . . . . . . . 12 |- ( ( J = 0 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( J < K <-> 0 < 1 ) )
66	45, 65	mpbiri 248	. . . . . . . . . . 11 |- ( ( J = 0 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> J < K )
67	66	iftrued 4094	. . . . . . . . . 10 |- ( ( J = 0 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> if ( J < K , J , K ) = J )
68	64, 67, 60	3eqtrd 2660	. . . . . . . . 9 |- ( ( J = 0 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> I = 0 )
69	68	oveq2d 6666	. . . . . . . 8 |- ( ( J = 0 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( R ^r I ) = ( R ^r 0 ) )
70	59, 63, 69	3eqtr4d 2666	. . . . . . 7 |- ( ( J = 0 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) )
71	70	3exp 1264	. . . . . 6 |- ( J = 0 -> ( K = 1 -> ( ( R e. V /\ I = if ( J < K , J , K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) )
72		simp1 1061	. . . . . . . . . . 11 |- ( ( J = 1 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> J = 1 )
73	72	oveq2d 6666	. . . . . . . . . 10 |- ( ( J = 1 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( R ^r J ) = ( R ^r 1 ) )
74		simp3l 1089	. . . . . . . . . . 11 |- ( ( J = 1 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> R e. V )
75	74, 39	syl 17	. . . . . . . . . 10 |- ( ( J = 1 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( R ^r 1 ) = R )
76	73, 75	eqtrd 2656	. . . . . . . . 9 |- ( ( J = 1 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( R ^r J ) = R )
77		simp2 1062	. . . . . . . . 9 |- ( ( J = 1 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> K = 1 )
78	76, 77	oveq12d 6668	. . . . . . . 8 |- ( ( J = 1 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r 1 ) )
79		simp3r 1090	. . . . . . . . . 10 |- ( ( J = 1 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> I = if ( J < K , J , K ) )
80	46	ltnri 10146	. . . . . . . . . . . 12 |- -. 1 < 1
81	72, 77	breq12d 4666	. . . . . . . . . . . 12 |- ( ( J = 1 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( J < K <-> 1 < 1 ) )
82	80, 81	mtbiri 317	. . . . . . . . . . 11 |- ( ( J = 1 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> -. J < K )
83	82	iffalsed 4097	. . . . . . . . . 10 |- ( ( J = 1 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> if ( J < K , J , K ) = K )
84	79, 83, 77	3eqtrd 2660	. . . . . . . . 9 |- ( ( J = 1 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> I = 1 )
85	84	oveq2d 6666	. . . . . . . 8 |- ( ( J = 1 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( R ^r I ) = ( R ^r 1 ) )
86	78, 85	eqtr4d 2659	. . . . . . 7 |- ( ( J = 1 /\ K = 1 /\ ( R e. V /\ I = if ( J < K , J , K ) ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) )
87	86	3exp 1264	. . . . . 6 |- ( J = 1 -> ( K = 1 -> ( ( R e. V /\ I = if ( J < K , J , K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) )
88	71, 87	jaoi 394	. . . . 5 |- ( ( J = 0 \/ J = 1 ) -> ( K = 1 -> ( ( R e. V /\ I = if ( J < K , J , K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) )
89	56, 88	jaod 395	. . . 4 |- ( ( J = 0 \/ J = 1 ) -> ( ( K = 0 \/ K = 1 ) -> ( ( R e. V /\ I = if ( J < K , J , K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) )
90	89	imp 445	. . 3 |- ( ( ( J = 0 \/ J = 1 ) /\ ( K = 0 \/ K = 1 ) ) -> ( ( R e. V /\ I = if ( J < K , J , K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) )
91	1, 2, 90	syl2an 494	. 2 |- ( ( J e. { 0 , 1 } /\ K e. { 0 , 1 } ) -> ( ( R e. V /\ I = if ( J < K , J , K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) )
92	91	impcom 446	1 |- ( ( ( R e. V /\ I = if ( J < K , J , K ) ) /\ ( J e. { 0 , 1 } /\ K e. { 0 , 1 } ) ) -> ( ( 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   ifcif 4086   {cpr 4179   class class class wbr 4653   _I cid 5023   dom cdm 5114   ran crn 5115   |` cres 5116  (class class class)co 6650   0cc0 9936   1c1 9937   < clt 10074   ^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-rep 4771  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:  relexp1idm  38006  relexp0idm  38007