Theorem relexpxpmin 36321

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 10878	. . . . 5 |- ( K e. NN0 <-> ( K e. NN \/ K = 0 ) )
2		elnn0 10878	. . . . . . 7 |- ( J e. NN0 <-> ( J e. NN \/ J = 0 ) )
3		ifeqor 3927	. . . . . . . . . . 11 |- ( if ( J < K , J , K ) = J \/ if ( J < K , J , K ) = K )
4		andi 879	. . . . . . . . . . . 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 198	. . . . . . . . . . 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 678	. . . . . . . . . 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 2472	. . . . . . . . . . 11 |- ( ( I = if ( J < K , J , K ) /\ if ( J < K , J , K ) = J ) -> I = J )
8		eqtr 2472	. . . . . . . . . . 11 |- ( ( I = if ( J < K , J , K ) /\ if ( J < K , J , K ) = K ) -> I = K )
9	7, 8	orim12i 519	. . . . . . . . . 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 36307	. . . . . . . . . . . . . . 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 431	. . . . . . . . . . . . . 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 1173	. . . . . . . . . . . . 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 36307	. . . . . . . . . . . . . . . 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 431	. . . . . . . . . . . . . . 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 1172	. . . . . . . . . . . . . 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 6310	. . . . . . . . . . . . 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 1012	. . . . . . . . . . . . . . 15 |- ( ( ( I = J /\ J e. NN /\ K e. NN ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> I = J )
18	17	oveq2d 6311	. . . . . . . . . . . . . 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 2487	. . . . . . . . . . . . 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 2497	. . . . . . . . . . . 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 1226	. . . . . . . . . . 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 1172	. . . . . . . . . . . . 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 1012	. . . . . . . . . . . . . 14 |- ( ( ( I = K /\ J e. NN /\ K e. NN ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> I = K )
24	23	eqcomd 2459	. . . . . . . . . . . . 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 6313	. . . . . . . . . . . 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 1226	. . . . . . . . . . 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 381	. . . . . . . . . 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 83	. . . . . . . 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 1011	. . . . . . . . . . 11 |- ( ( K e. NN /\ J = 0 /\ I = if ( J < K , J , K ) ) -> I = if ( J < K , J , K ) )
31		simp2 1010	. . . . . . . . . . . . 13 |- ( ( K e. NN /\ J = 0 /\ I = if ( J < K , J , K ) ) -> J = 0 )
32		simp1 1009	. . . . . . . . . . . . . 14 |- ( ( K e. NN /\ J = 0 /\ I = if ( J < K , J , K ) ) -> K e. NN )
33	32	nngt0d 10660	. . . . . . . . . . . . 13 |- ( ( K e. NN /\ J = 0 /\ I = if ( J < K , J , K ) ) -> 0 < K )
34	31, 33	eqbrtrd 4426	. . . . . . . . . . . 12 |- ( ( K e. NN /\ J = 0 /\ I = if ( J < K , J , K ) ) -> J < K )
35	34	iftrued 3891	. . . . . . . . . . 11 |- ( ( K e. NN /\ J = 0 /\ I = if ( J < K , J , K ) ) -> if ( J < K , J , K ) = J )
36	30, 35, 31	3eqtrd 2491	. . . . . . . . . 10 |- ( ( K e. NN /\ J = 0 /\ I = if ( J < K , J , K ) ) -> I = 0 )
37		simpr1 1015	. . . . . . . . . . . . . . 15 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> A e. U )
38		simpr2 1016	. . . . . . . . . . . . . . 15 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> B e. V )
39		xpexg 6598	. . . . . . . . . . . . . . 15 |- ( ( A e. U /\ B e. V ) -> ( A X. B ) e. _V )
40	37, 38, 39	syl2anc 667	. . . . . . . . . . . . . 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 6729	. . . . . . . . . . . . . . 15 |- ( ( A X. B ) e. _V -> dom ( A X. B ) e. _V )
42		rnexg 6730	. . . . . . . . . . . . . . 15 |- ( ( A X. B ) e. _V -> ran ( A X. B ) e. _V )
43	41, 42	jca 535	. . . . . . . . . . . . . 14 |- ( ( A X. B ) e. _V -> ( dom ( A X. B ) e. _V /\ ran ( A X. B ) e. _V ) )
44		unexg 6597	. . . . . . . . . . . . . 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 1012	. . . . . . . . . . . . . 14 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> K e. NN )
47	46	nnnn0d 10932	. . . . . . . . . . . . 13 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> K e. NN0 )
48		relexpiidm 36308	. . . . . . . . . . . . 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 667	. . . . . . . . . . . 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 1013	. . . . . . . . . . . . . . 15 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> J = 0 )
51	50	oveq2d 6311	. . . . . . . . . . . . . 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 13097	. . . . . . . . . . . . . . 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 2487	. . . . . . . . . . . . 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 6310	. . . . . . . . . . . 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 1014	. . . . . . . . . . . . . 14 |- ( ( ( K e. NN /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> I = 0 )
57	56	oveq2d 6311	. . . . . . . . . . . . 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 2487	. . . . . . . . . . . 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 2497	. . . . . . . . . . 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 436	. . . . . . . . . 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 1314	. . . . . . . . 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 1208	. . . . . . . 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 382	. . . . . . 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 221	. . . . . 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 1009	. . . . . . . 8 |- ( ( K = 0 /\ J e. NN0 /\ I = if ( J < K , J , K ) ) -> K = 0 )
66	2	biimpi 198	. . . . . . . . 9 |- ( J e. NN0 -> ( J e. NN \/ J = 0 ) )
67	66	3ad2ant2 1031	. . . . . . . 8 |- ( ( K = 0 /\ J e. NN0 /\ I = if ( J < K , J , K ) ) -> ( J e. NN \/ J = 0 ) )
68		simp3 1011	. . . . . . . . 9 |- ( ( K = 0 /\ J e. NN0 /\ I = if ( J < K , J , K ) ) -> I = if ( J < K , J , K ) )
69		nn0nlt0 10903	. . . . . . . . . . . 12 |- ( J e. NN0 -> -. J < 0 )
70	69	3ad2ant2 1031	. . . . . . . . . . 11 |- ( ( K = 0 /\ J e. NN0 /\ I = if ( J < K , J , K ) ) -> -. J < 0 )
71	65	breq2d 4417	. . . . . . . . . . 11 |- ( ( K = 0 /\ J e. NN0 /\ I = if ( J < K , J , K ) ) -> ( J < K <-> J < 0 ) )
72	70, 71	mtbird 303	. . . . . . . . . 10 |- ( ( K = 0 /\ J e. NN0 /\ I = if ( J < K , J , K ) ) -> -. J < K )
73	72	iffalsed 3894	. . . . . . . . 9 |- ( ( K = 0 /\ J e. NN0 /\ I = if ( J < K , J , K ) ) -> if ( J < K , J , K ) = K )
74	68, 73, 65	3eqtrd 2491	. . . . . . . 8 |- ( ( K = 0 /\ J e. NN0 /\ I = if ( J < K , J , K ) ) -> I = 0 )
75	13	3ad2ant2 1031	. . . . . . . . . . . . 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 431	. . . . . . . . . . . 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 6310	. . . . . . . . . . 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 1012	. . . . . . . . . . . 12 |- ( ( ( K = 0 /\ J e. NN /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> K = 0 )
79	78	oveq2d 6311	. . . . . . . . . . 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 1014	. . . . . . . . . . . 12 |- ( ( ( K = 0 /\ J e. NN /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> I = 0 )
81	80	oveq2d 6311	. . . . . . . . . . 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 2497	. . . . . . . . . 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 1226	. . . . . . . . 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 1015	. . . . . . . . . . . . 13 |- ( ( ( K = 0 /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> A e. U )
85		simpr2 1016	. . . . . . . . . . . . 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 667	. . . . . . . . . . . 12 |- ( ( ( K = 0 /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> ( A X. B ) e. _V )
87		relexp0idm 36319	. . . . . . . . . . . 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 1013	. . . . . . . . . . . . 13 |- ( ( ( K = 0 /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> J = 0 )
90	89	oveq2d 6311	. . . . . . . . . . . 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 1012	. . . . . . . . . . . 12 |- ( ( ( K = 0 /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> K = 0 )
92	90, 91	oveq12d 6313	. . . . . . . . . . 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 1014	. . . . . . . . . . . 12 |- ( ( ( K = 0 /\ J = 0 /\ I = 0 ) /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) ) -> I = 0 )
94	93	oveq2d 6311	. . . . . . . . . . 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 2497	. . . . . . . . . 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 1226	. . . . . . . . 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 382	. . . . . . . 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 63	. . . . . . 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 1208	. . . . . 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 381	. . . . 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 199	. . . 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 83	. . 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 1203	. 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 432	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 370   /\ wa 371   /\ w3a 986   = wceq 1446   e. wcel 1889   =/= wne 2624   _Vcvv 3047   u. cun 3404   i^i cin 3405   (/)c0 3733   ifcif 3883   class class class wbr 4405   _I cid 4747   X. cxp 4835   dom cdm 4837   ran crn 4838   |` cres 4839  (class class class)co 6295   0cc0 9544   < clt 9680   NNcn 10616   NN0cn0 10876   ^r crelexp 13095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-seq 12221  df-relexp 13096

This theorem is referenced by: (None)