Theorem expaddzaplem 9835

Description: Lemma for expaddzap 9836. (Contributed by Jim Kingdon, 10-Jun-2020.)

Assertion

Ref	Expression
expaddzaplem	|- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )

Proof of Theorem expaddzaplem

Step	Hyp	Ref	Expression
1		simp1l 963	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> A e. CC )
2		simp3 941	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. NN0 )
3		expcl 9810	. . . 4 |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
4	1, 2, 3	syl2anc 403	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ N ) e. CC )
5		simp2r 966	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. NN )
6	5	nnnn0d 8618	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. NN0 )
7		expcl 9810	. . . 4 |- ( ( A e. CC /\ -u M e. NN0 ) -> ( A ^ -u M ) e. CC )
8	1, 6, 7	syl2anc 403	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ -u M ) e. CC )
9		simp1r 964	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> A # 0 )
10	5	nnzd 8763	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. ZZ )
11		expap0i 9824	. . . 4 |- ( ( A e. CC /\ A # 0 /\ -u M e. ZZ ) -> ( A ^ -u M ) # 0 )
12	1, 9, 10, 11	syl3anc 1170	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ -u M ) # 0 )
13	4, 8, 12	divrecap2d 8158	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( A ^ N ) / ( A ^ -u M ) ) = ( ( 1 / ( A ^ -u M ) ) x. ( A ^ N ) ) )
14		simp2l 965	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. RR )
15	14	recnd 7419	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. CC )
16	15	negnegd 7687	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u -u M = M )
17		nnnegz 8649	. . . . . . . . . 10 |- ( -u M e. NN -> -u -u M e. ZZ )
18	5, 17	syl 14	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u -u M e. ZZ )
19	16, 18	eqeltrrd 2160	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. ZZ )
20	2	nn0zd 8762	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. ZZ )
21	19, 20	zaddcld 8768	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( M + N ) e. ZZ )
22		expclzap 9817	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ ( M + N ) e. ZZ ) -> ( A ^ ( M + N ) ) e. CC )
23	1, 9, 21, 22	syl3anc 1170	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( M + N ) ) e. CC )
24	23	adantr 270	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( A ^ ( M + N ) ) e. CC )
25	8	adantr 270	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( A ^ -u M ) e. CC )
26	12	adantr 270	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( A ^ -u M ) # 0 )
27	24, 25, 26	divcanap4d 8160	. . . 4 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( ( ( A ^ ( M + N ) ) x. ( A ^ -u M ) ) / ( A ^ -u M ) ) = ( A ^ ( M + N ) ) )
28	1	adantr 270	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> A e. CC )
29		simpr 108	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( M + N ) e. NN0 )
30	6	adantr 270	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> -u M e. NN0 )
31		expadd 9834	. . . . . . 7 |- ( ( A e. CC /\ ( M + N ) e. NN0 /\ -u M e. NN0 ) -> ( A ^ ( ( M + N ) + -u M ) ) = ( ( A ^ ( M + N ) ) x. ( A ^ -u M ) ) )
32	28, 29, 30, 31	syl3anc 1170	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( A ^ ( ( M + N ) + -u M ) ) = ( ( A ^ ( M + N ) ) x. ( A ^ -u M ) ) )
33	21	zcnd 8765	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( M + N ) e. CC )
34	33, 15	negsubd 7702	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( M + N ) + -u M ) = ( ( M + N ) - M ) )
35	2	nn0cnd 8620	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. CC )
36	15, 35	pncan2d 7698	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( M + N ) - M ) = N )
37	34, 36	eqtrd 2115	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( M + N ) + -u M ) = N )
38	37	adantr 270	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( ( M + N ) + -u M ) = N )
39	38	oveq2d 5607	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( A ^ ( ( M + N ) + -u M ) ) = ( A ^ N ) )
40	32, 39	eqtr3d 2117	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( ( A ^ ( M + N ) ) x. ( A ^ -u M ) ) = ( A ^ N ) )
41	40	oveq1d 5606	. . . 4 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( ( ( A ^ ( M + N ) ) x. ( A ^ -u M ) ) / ( A ^ -u M ) ) = ( ( A ^ N ) / ( A ^ -u M ) ) )
42	27, 41	eqtr3d 2117	. . 3 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ ( M + N ) e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ N ) / ( A ^ -u M ) ) )
43	1	adantr 270	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> A e. CC )
44	9	adantr 270	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> A # 0 )
45	33	adantr 270	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( M + N ) e. CC )
46		simpr 108	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> -u ( M + N ) e. NN0 )
47		expineg2 9801	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( ( M + N ) e. CC /\ -u ( M + N ) e. NN0 ) ) -> ( A ^ ( M + N ) ) = ( 1 / ( A ^ -u ( M + N ) ) ) )
48	43, 44, 45, 46, 47	syl22anc 1171	. . . 4 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( A ^ ( M + N ) ) = ( 1 / ( A ^ -u ( M + N ) ) ) )
49	21	znegcld 8766	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u ( M + N ) e. ZZ )
50		expclzap 9817	. . . . . . . . . 10 |- ( ( A e. CC /\ A # 0 /\ -u ( M + N ) e. ZZ ) -> ( A ^ -u ( M + N ) ) e. CC )
51	1, 9, 49, 50	syl3anc 1170	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ -u ( M + N ) ) e. CC )
52	51	adantr 270	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( A ^ -u ( M + N ) ) e. CC )
53	4	adantr 270	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( A ^ N ) e. CC )
54		expap0i 9824	. . . . . . . . . 10 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) # 0 )
55	1, 9, 20, 54	syl3anc 1170	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ N ) # 0 )
56	55	adantr 270	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( A ^ N ) # 0 )
57	52, 53, 56	divcanap4d 8160	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( ( ( A ^ -u ( M + N ) ) x. ( A ^ N ) ) / ( A ^ N ) ) = ( A ^ -u ( M + N ) ) )
58	2	adantr 270	. . . . . . . . . 10 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> N e. NN0 )
59		expadd 9834	. . . . . . . . . 10 |- ( ( A e. CC /\ -u ( M + N ) e. NN0 /\ N e. NN0 ) -> ( A ^ ( -u ( M + N ) + N ) ) = ( ( A ^ -u ( M + N ) ) x. ( A ^ N ) ) )
60	43, 46, 58, 59	syl3anc 1170	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( A ^ ( -u ( M + N ) + N ) ) = ( ( A ^ -u ( M + N ) ) x. ( A ^ N ) ) )
61	15, 35	negdi2d 7710	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u ( M + N ) = ( -u M - N ) )
62	61	oveq1d 5606	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( -u ( M + N ) + N ) = ( ( -u M - N ) + N ) )
63	15	negcld 7683	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. CC )
64	63, 35	npcand 7700	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( -u M - N ) + N ) = -u M )
65	62, 64	eqtrd 2115	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( -u ( M + N ) + N ) = -u M )
66	65	adantr 270	. . . . . . . . . 10 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( -u ( M + N ) + N ) = -u M )
67	66	oveq2d 5607	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( A ^ ( -u ( M + N ) + N ) ) = ( A ^ -u M ) )
68	60, 67	eqtr3d 2117	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( ( A ^ -u ( M + N ) ) x. ( A ^ N ) ) = ( A ^ -u M ) )
69	68	oveq1d 5606	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( ( ( A ^ -u ( M + N ) ) x. ( A ^ N ) ) / ( A ^ N ) ) = ( ( A ^ -u M ) / ( A ^ N ) ) )
70	57, 69	eqtr3d 2117	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( A ^ -u ( M + N ) ) = ( ( A ^ -u M ) / ( A ^ N ) ) )
71	70	oveq2d 5607	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( 1 / ( A ^ -u ( M + N ) ) ) = ( 1 / ( ( A ^ -u M ) / ( A ^ N ) ) ) )
72	8, 4, 12, 55	recdivapd 8171	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( 1 / ( ( A ^ -u M ) / ( A ^ N ) ) ) = ( ( A ^ N ) / ( A ^ -u M ) ) )
73	72	adantr 270	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( 1 / ( ( A ^ -u M ) / ( A ^ N ) ) ) = ( ( A ^ N ) / ( A ^ -u M ) ) )
74	71, 73	eqtrd 2115	. . . 4 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( 1 / ( A ^ -u ( M + N ) ) ) = ( ( A ^ N ) / ( A ^ -u M ) ) )
75	48, 74	eqtrd 2115	. . 3 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) /\ -u ( M + N ) e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ N ) / ( A ^ -u M ) ) )
76		elznn0 8661	. . . . 5 |- ( ( M + N ) e. ZZ <-> ( ( M + N ) e. RR /\ ( ( M + N ) e. NN0 \/ -u ( M + N ) e. NN0 ) ) )
77	76	simprbi 269	. . . 4 |- ( ( M + N ) e. ZZ -> ( ( M + N ) e. NN0 \/ -u ( M + N ) e. NN0 ) )
78	21, 77	syl 14	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( M + N ) e. NN0 \/ -u ( M + N ) e. NN0 ) )
79	42, 75, 78	mpjaodan 745	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ N ) / ( A ^ -u M ) ) )
80		expineg2 9801	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. CC /\ -u M e. NN0 ) ) -> ( A ^ M ) = ( 1 / ( A ^ -u M ) ) )
81	1, 9, 15, 6, 80	syl22anc 1171	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ M ) = ( 1 / ( A ^ -u M ) ) )
82	81	oveq1d 5606	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( A ^ M ) x. ( A ^ N ) ) = ( ( 1 / ( A ^ -u M ) ) x. ( A ^ N ) ) )
83	13, 79, 82	3eqtr4d 2125	1 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 662   /\ w3a 920   = wceq 1285   e. wcel 1434   class class class wbr 3811  (class class class)co 5591   CCcc 7251   RRcr 7252   0cc0 7253   1c1 7254   + caddc 7256   x. cmul 7258   - cmin 7556   -ucneg 7557   # cap 7958   / cdiv 8037   NNcn 8316   NN0cn0 8565   ZZcz 8646   ^cexp 9791

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366  ax-cnex 7339  ax-resscn 7340  ax-1cn 7341  ax-1re 7342  ax-icn 7343  ax-addcl 7344  ax-addrcl 7345  ax-mulcl 7346  ax-mulrcl 7347  ax-addcom 7348  ax-mulcom 7349  ax-addass 7350  ax-mulass 7351  ax-distr 7352  ax-i2m1 7353  ax-0lt1 7354  ax-1rid 7355  ax-0id 7356  ax-rnegex 7357  ax-precex 7358  ax-cnre 7359  ax-pre-ltirr 7360  ax-pre-ltwlin 7361  ax-pre-lttrn 7362  ax-pre-apti 7363  ax-pre-ltadd 7364  ax-pre-mulgt0 7365  ax-pre-mulext 7366

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-ilim 4160  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-riota 5547  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-frec 6088  df-pnf 7427  df-mnf 7428  df-xr 7429  df-ltxr 7430  df-le 7431  df-sub 7558  df-neg 7559  df-reap 7952  df-ap 7959  df-div 8038  df-inn 8317  df-n0 8566  df-z 8647  df-uz 8915  df-iseq 9741  df-iexp 9792

This theorem is referenced by:  expaddzap  9836