Theorem relexpxpmin 38009

Description: The composition of powers of a cross-product of non-disjoint sets is the cross product raised to the minimum exponent. (Contributed by RP, 13-Jun-2020.)

Assertion

Ref	Expression
relexpxpmin	|- ( ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) /\ ( I = if ( J < K , J , K ) /\ J e. NN0 /\ K e. NN0 ) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) )

Proof of Theorem relexpxpmin

Step	Hyp	Ref	Expression
1		elnn0 11294	. . . . 5 |- ( K e. NN0 <-> ( K e. NN \/ K = 0 ) )
2		elnn0 11294	. . . . . . 7 |- ( J e. NN0 <-> ( J e. NN \/ J = 0 ) )
3		ifeqor 4132	. . . . . . . . . . 11 |- ( if ( J < K , J , K ) = J \/ if ( J < K , J , K ) = K )
4		andi 911	. . . . . . . . . . . 12 |- ( ( I = if ( J < K , J , K ) /\ ( if ( J < K , J , K ) = J \/ if ( J < K , J , K ) = K ) ) <-> ( ( I = if ( J < K , J , K ) /\ if ( J < K , J , K ) = J ) \/ ( I = if ( J < K , J , K ) /\ if ( J < K , J , K ) = K ) ) )
5	4	biimpi 206	. . . . . . . . . . 11 |- ( ( I = if ( J < K , J , K ) /\ ( if ( J < K , J , K ) = J \/ if ( J < K , J , K ) = K ) ) -> ( ( I = if ( J < K , J , K ) /\ if ( J < K , J , K ) = J ) \/ ( I = if ( J < K , J , K ) /\ if ( J < K , J , K ) = K ) ) )
6	3, 5	mpan2 707	. . . . . . . . . 10 |- ( I = if ( J < K , J , K ) -> ( ( I = if ( J < K , J , K ) /\ if ( J < K , J , K ) = J ) \/ ( I = if ( J < K , J , K ) /\ if ( J < K , J , K ) = K ) ) )
7		eqtr 2641	. . . . . . . . . . 11 |- ( ( I = if ( J < K , J , K ) /\ if ( J < K , J , K ) = J ) -> I = J )
8		eqtr 2641	. . . . . . . . . . 11 |- ( ( I = if ( J < K , J , K ) /\ if ( J < K , J , K ) = K ) -> I = K )
9	7, 8	orim12i 538	. . . . . . . . . 10 |- ( ( ( I = if ( J < K , J , K ) /\ if ( J < K , J , K ) = J ) \/ ( I = if ( J < K , J , K ) /\ if ( J < K , J , K ) = K ) ) -> ( I = J \/ I = K ) )
10		relexpxpnnidm 37995	. . . . . . . . . . . . . . 15 |- ( K e. NN -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( A X. B ) ^r K ) = ( A X. B ) ) )
11	10	imp 445	. . . . . . . . . . . . . 14 |- ( ( K e. NN /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( A X. B ) ^r K ) = ( A X. B ) )
12	11	3ad2antl3 1225	. . . . . . . . . . . . 13 |- ( ( ( I = J /\ J e. NN /\ K e. NN ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( A X. B ) ^r K ) = ( A X. B ) )
13		relexpxpnnidm 37995	. . . . . . . . . . . . . . . 16 |- ( J e. NN -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( A X. B ) ^r J ) = ( A X. B ) ) )
14	13	imp 445	. . . . . . . . . . . . . . 15 |- ( ( J e. NN /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( A X. B ) ^r J ) = ( A X. B ) )
15	14	3ad2antl2 1224	. . . . . . . . . . . . . 14 |- ( ( ( I = J /\ J e. NN /\ K e. NN ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( A X. B ) ^r J ) = ( A X. B ) )
16	15	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ( ( I = J /\ J e. NN /\ K e. NN ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r K ) )
17		simpl1 1064	. . . . . . . . . . . . . . 15 |- ( ( ( I = J /\ J e. NN /\ K e. NN ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> I = J )
18	17	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( ( ( I = J /\ J e. NN /\ K e. NN ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( A X. B ) ^r I ) = ( ( A X. B ) ^r J ) )
19	18, 15	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ( I = J /\ J e. NN /\ K e. NN ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( A X. B ) ^r I ) = ( A X. B ) )
20	12, 16, 19	3eqtr4d 2666	. . . . . . . . . . . 12 |- ( ( ( I = J /\ J e. NN /\ K e. NN ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) )
21	20	3exp1 1283	. . . . . . . . . . 11 |- ( I = J -> ( J e. NN -> ( K e. NN -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) ) ) )
22	14	3ad2antl2 1224	. . . . . . . . . . . . 13 |- ( ( ( I = K /\ J e. NN /\ K e. NN ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( A X. B ) ^r J ) = ( A X. B ) )
23		simpl1 1064	. . . . . . . . . . . . . 14 |- ( ( ( I = K /\ J e. NN /\ K e. NN ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> I = K )
24	23	eqcomd 2628	. . . . . . . . . . . . 13 |- ( ( ( I = K /\ J e. NN /\ K e. NN ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> K = I )
25	22, 24	oveq12d 6668	. . . . . . . . . . . 12 |- ( ( ( I = K /\ J e. NN /\ K e. NN ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) )
26	25	3exp1 1283	. . . . . . . . . . 11 |- ( I = K -> ( J e. NN -> ( K e. NN -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) ) ) )
27	21, 26	jaoi 394	. . . . . . . . . 10 |- ( ( I = J \/ I = K ) -> ( J e. NN -> ( K e. NN -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) ) ) )
28	6, 9, 27	3syl 18	. . . . . . . . 9 |- ( I = if ( J < K , J , K ) -> ( J e. NN -> ( K e. NN -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) ) ) )
29	28	com13 88	. . . . . . . 8 |- ( K e. NN -> ( J e. NN -> ( I = if ( J < K , J , K ) -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) ) ) )
30		simp3 1063	. . . . . . . . . . 11 |- ( ( K e. NN /\ J = 0 /\ I = if ( J < K , J , K ) ) -> I = if ( J < K , J , K ) )
31		simp2 1062	. . . . . . . . . . . . 13 |- ( ( K e. NN /\ J = 0 /\ I = if ( J < K , J , K ) ) -> J = 0 )
32		simp1 1061	. . . . . . . . . . . . . 14 |- ( ( K e. NN /\ J = 0 /\ I = if ( J < K , J , K ) ) -> K e. NN )
33	32	nngt0d 11064	. . . . . . . . . . . . 13 |- ( ( K e. NN /\ J = 0 /\ I = if ( J < K , J , K ) ) -> 0 < K )
34	31, 33	eqbrtrd 4675	. . . . . . . . . . . 12 |- ( ( K e. NN /\ J = 0 /\ I = if ( J < K , J , K ) ) -> J < K )
35	34	iftrued 4094	. . . . . . . . . . 11 |- ( ( K e. NN /\ J = 0 /\ I = if ( J < K , J , K ) ) -> if ( J < K , J , K ) = J )
36	30, 35, 31	3eqtrd 2660	. . . . . . . . . 10 |- ( ( K e. NN /\ J = 0 /\ I = if ( J < K , J , K ) ) -> I = 0 )
37		simpr1 1067	. . . . . . . . . . . . . . 15 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> A e. U )
38		simpr2 1068	. . . . . . . . . . . . . . 15 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> B e. V )
39		xpexg 6960	. . . . . . . . . . . . . . 15 |- ( ( A e. U /\ B e. V ) -> ( A X. B ) e. _V )
40	37, 38, 39	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( A X. B ) e. _V )
41		dmexg 7097	. . . . . . . . . . . . . . 15 |- ( ( A X. B ) e. _V -> dom ( A X. B ) e. _V )
42		rnexg 7098	. . . . . . . . . . . . . . 15 |- ( ( A X. B ) e. _V -> ran ( A X. B ) e. _V )
43	41, 42	jca 554	. . . . . . . . . . . . . 14 |- ( ( A X. B ) e. _V -> ( dom ( A X. B ) e. _V /\ ran ( A X. B ) e. _V ) )
44		unexg 6959	. . . . . . . . . . . . . 14 |- ( ( dom ( A X. B ) e. _V /\ ran ( A X. B ) e. _V ) -> ( dom ( A X. B ) u. ran ( A X. B ) ) e. _V )
45	40, 43, 44	3syl 18	. . . . . . . . . . . . 13 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( dom ( A X. B ) u. ran ( A X. B ) ) e. _V )
46		simpl1 1064	. . . . . . . . . . . . . 14 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> K e. NN )
47	46	nnnn0d 11351	. . . . . . . . . . . . 13 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> K e. NN0 )
48		relexpiidm 37996	. . . . . . . . . . . . 13 |- ( ( ( dom ( A X. B ) u. ran ( A X. B ) ) e. _V /\ K e. NN0 ) -> ( ( _I |` ( dom ( A X. B ) u. ran ( A X. B ) ) ) ^r K ) = ( _I |` ( dom ( A X. B ) u. ran ( A X. B ) ) ) )
49	45, 47, 48	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( _I |` ( dom ( A X. B ) u. ran ( A X. B ) ) ) ^r K ) = ( _I |` ( dom ( A X. B ) u. ran ( A X. B ) ) ) )
50		simpl2 1065	. . . . . . . . . . . . . . 15 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> J = 0 )
51	50	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( A X. B ) ^r J ) = ( ( A X. B ) ^r 0 ) )
52		relexp0g 13762	. . . . . . . . . . . . . . 15 |- ( ( A X. B ) e. _V -> ( ( A X. B ) ^r 0 ) = ( _I |` ( dom ( A X. B ) u. ran ( A X. B ) ) ) )
53	40, 52	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( A X. B ) ^r 0 ) = ( _I |` ( dom ( A X. B ) u. ran ( A X. B ) ) ) )
54	51, 53	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( A X. B ) ^r J ) = ( _I |` ( dom ( A X. B ) u. ran ( A X. B ) ) ) )
55	54	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( _I |` ( dom ( A X. B ) u. ran ( A X. B ) ) ) ^r K ) )
56		simpl3 1066	. . . . . . . . . . . . . 14 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> I = 0 )
57	56	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( A X. B ) ^r I ) = ( ( A X. B ) ^r 0 ) )
58	57, 53	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( A X. B ) ^r I ) = ( _I |` ( dom ( A X. B ) u. ran ( A X. B ) ) ) )
59	49, 55, 58	3eqtr4d 2666	. . . . . . . . . . 11 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) )
60	59	ex 450	. . . . . . . . . 10 |- ( ( K e. NN /\ J = 0 /\ I = 0 ) -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) )
61	36, 60	syld3an3 1371	. . . . . . . . 9 |- ( ( K e. NN /\ J = 0 /\ I = if ( J < K , J , K ) ) -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) )
62	61	3exp 1264	. . . . . . . 8 |- ( K e. NN -> ( J = 0 -> ( I = if ( J < K , J , K ) -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) ) ) )
63	29, 62	jaod 395	. . . . . . 7 |- ( K e. NN -> ( ( J e. NN \/ J = 0 ) -> ( I = if ( J < K , J , K ) -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) ) ) )
64	2, 63	syl5bi 232	. . . . . 6 |- ( K e. NN -> ( J e. NN0 -> ( I = if ( J < K , J , K ) -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) ) ) )
65		simp1 1061	. . . . . . . 8 |- ( ( K = 0 /\ J e. NN0 /\ I = if ( J < K , J , K ) ) -> K = 0 )
66	2	biimpi 206	. . . . . . . . 9 |- ( J e. NN0 -> ( J e. NN \/ J = 0 ) )
67	66	3ad2ant2 1083	. . . . . . . 8 |- ( ( K = 0 /\ J e. NN0 /\ I = if ( J < K , J , K ) ) -> ( J e. NN \/ J = 0 ) )
68		simp3 1063	. . . . . . . . 9 |- ( ( K = 0 /\ J e. NN0 /\ I = if ( J < K , J , K ) ) -> I = if ( J < K , J , K ) )
69		nn0nlt0 11319	. . . . . . . . . . . 12 |- ( J e. NN0 -> -. J < 0 )
70	69	3ad2ant2 1083	. . . . . . . . . . 11 |- ( ( K = 0 /\ J e. NN0 /\ I = if ( J < K , J , K ) ) -> -. J < 0 )
71	65	breq2d 4665	. . . . . . . . . . 11 |- ( ( K = 0 /\ J e. NN0 /\ I = if ( J < K , J , K ) ) -> ( J < K <-> J < 0 ) )
72	70, 71	mtbird 315	. . . . . . . . . 10 |- ( ( K = 0 /\ J e. NN0 /\ I = if ( J < K , J , K ) ) -> -. J < K )
73	72	iffalsed 4097	. . . . . . . . 9 |- ( ( K = 0 /\ J e. NN0 /\ I = if ( J < K , J , K ) ) -> if ( J < K , J , K ) = K )
74	68, 73, 65	3eqtrd 2660	. . . . . . . 8 |- ( ( K = 0 /\ J e. NN0 /\ I = if ( J < K , J , K ) ) -> I = 0 )
75	13	3ad2ant2 1083	. . . . . . . . . . . . 13 |- ( ( K = 0 /\ J e. NN /\ I = 0 ) -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( A X. B ) ^r J ) = ( A X. B ) ) )
76	75	imp 445	. . . . . . . . . . . 12 |- ( ( ( K = 0 /\ J e. NN /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( A X. B ) ^r J ) = ( A X. B ) )
77	76	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ( K = 0 /\ J e. NN /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( ( A X. B ) ^r J ) ^r 0 ) = ( ( A X. B ) ^r 0 ) )
78		simpl1 1064	. . . . . . . . . . . 12 |- ( ( ( K = 0 /\ J e. NN /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> K = 0 )
79	78	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( K = 0 /\ J e. NN /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( ( A X. B ) ^r J ) ^r 0 ) )
80		simpl3 1066	. . . . . . . . . . . 12 |- ( ( ( K = 0 /\ J e. NN /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> I = 0 )
81	80	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( K = 0 /\ J e. NN /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( A X. B ) ^r I ) = ( ( A X. B ) ^r 0 ) )
82	77, 79, 81	3eqtr4d 2666	. . . . . . . . . 10 |- ( ( ( K = 0 /\ J e. NN /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) )
83	82	3exp1 1283	. . . . . . . . 9 |- ( K = 0 -> ( J e. NN -> ( I = 0 -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) ) ) )
84		simpr1 1067	. . . . . . . . . . . . 13 |- ( ( ( K = 0 /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> A e. U )
85		simpr2 1068	. . . . . . . . . . . . 13 |- ( ( ( K = 0 /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> B e. V )
86	84, 85, 39	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( K = 0 /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( A X. B ) e. _V )
87		relexp0idm 38007	. . . . . . . . . . . 12 |- ( ( A X. B ) e. _V -> ( ( ( A X. B ) ^r 0 ) ^r 0 ) = ( ( A X. B ) ^r 0 ) )
88	86, 87	syl 17	. . . . . . . . . . 11 |- ( ( ( K = 0 /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( ( A X. B ) ^r 0 ) ^r 0 ) = ( ( A X. B ) ^r 0 ) )
89		simpl2 1065	. . . . . . . . . . . . 13 |- ( ( ( K = 0 /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> J = 0 )
90	89	oveq2d 6666	. . . . . . . . . . . 12 |- ( ( ( K = 0 /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( A X. B ) ^r J ) = ( ( A X. B ) ^r 0 ) )
91		simpl1 1064	. . . . . . . . . . . 12 |- ( ( ( K = 0 /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> K = 0 )
92	90, 91	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ( K = 0 /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( ( A X. B ) ^r 0 ) ^r 0 ) )
93		simpl3 1066	. . . . . . . . . . . 12 |- ( ( ( K = 0 /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> I = 0 )
94	93	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( K = 0 /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( A X. B ) ^r I ) = ( ( A X. B ) ^r 0 ) )
95	88, 92, 94	3eqtr4d 2666	. . . . . . . . . 10 |- ( ( ( K = 0 /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) )
96	95	3exp1 1283	. . . . . . . . 9 |- ( K = 0 -> ( J = 0 -> ( I = 0 -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) ) ) )
97	83, 96	jaod 395	. . . . . . . 8 |- ( K = 0 -> ( ( J e. NN \/ J = 0 ) -> ( I = 0 -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) ) ) )
98	65, 67, 74, 97	syl3c 66	. . . . . . 7 |- ( ( K = 0 /\ J e. NN0 /\ I = if ( J < K , J , K ) ) -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) )
99	98	3exp 1264	. . . . . 6 |- ( K = 0 -> ( J e. NN0 -> ( I = if ( J < K , J , K ) -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) ) ) )
100	64, 99	jaoi 394	. . . . 5 |- ( ( K e. NN \/ K = 0 ) -> ( J e. NN0 -> ( I = if ( J < K , J , K ) -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) ) ) )
101	1, 100	sylbi 207	. . . 4 |- ( K e. NN0 -> ( J e. NN0 -> ( I = if ( J < K , J , K ) -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) ) ) )
102	101	com13 88	. . 3 |- ( I = if ( J < K , J , K ) -> ( J e. NN0 -> ( K e. NN0 -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) ) ) )
103	102	3imp 1256	. 2 |- ( ( I = if ( J < K , J , K ) /\ J e. NN0 /\ K e. NN0 ) -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) ) )
104	103	impcom 446	1 |- ( ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) /\ ( I = if ( J < K , J , K ) /\ J e. NN0 /\ K e. NN0 ) ) -> ( ( ( A X. B ) ^r J ) ^r K ) = ( ( A X. B ) ^r I ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   u. cun 3572   i^i cin 3573   (/)c0 3915   ifcif 4086   class class class wbr 4653   _I cid 5023   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116  (class class class)co 6650   0cc0 9936   < clt 10074   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-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: (None)