Theorem expaddzaplem 10462

Description: Lemma for expaddzap 10463. (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 1006	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> A e. CC )
2		simp3 984	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. NN0 )
3		expcl 10437	. . . 4 |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
4	1, 2, 3	syl2anc 409	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ N ) e. CC )
5		simp2r 1009	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. NN )
6	5	nnnn0d 9143	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. NN0 )
7		expcl 10437	. . . 4 |- ( ( A e. CC /\ -u M e. NN0 ) -> ( A ^ -u M ) e. CC )
8	1, 6, 7	syl2anc 409	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ -u M ) e. CC )
9		simp1r 1007	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> A # 0 )
10	5	nnzd 9285	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. ZZ )
11		expap0i 10451	. . . 4 |- ( ( A e. CC /\ A # 0 /\ -u M e. ZZ ) -> ( A ^ -u M ) # 0 )
12	1, 9, 10, 11	syl3anc 1220	. . 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 8667	. 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 1008	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. RR )
15	14	recnd 7906	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. CC )
16	15	negnegd 8177	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u -u M = M )
17		nnnegz 9170	. . . . . . . . . 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 2235	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. ZZ )
20	2	nn0zd 9284	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. ZZ )
21	19, 20	zaddcld 9290	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( M + N ) e. ZZ )
22		expclzap 10444	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ ( M + N ) e. ZZ ) -> ( A ^ ( M + N ) ) e. CC )
23	1, 9, 21, 22	syl3anc 1220	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( M + N ) ) e. CC )
24	23	adantr 274	. . . . 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 274	. . . . 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 274	. . . . 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 8669	. . . 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 274	. . . . . . 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 109	. . . . . . 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 274	. . . . . . 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 10461	. . . . . . 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 1220	. . . . . 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 9287	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( M + N ) e. CC )
34	33, 15	negsubd 8192	. . . . . . . . 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 9145	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. CC )
36	15, 35	pncan2d 8188	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( M + N ) - M ) = N )
37	34, 36	eqtrd 2190	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( M + N ) + -u M ) = N )
38	37	adantr 274	. . . . . . 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 5840	. . . . . 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 2192	. . . . 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 5839	. . . 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 2192	. . 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 274	. . . . 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 274	. . . . 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 274	. . . . 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 109	. . . . 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 10428	. . . . 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 1221	. . . 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 9288	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u ( M + N ) e. ZZ )
50		expclzap 10444	. . . . . . . . . 10 |- ( ( A e. CC /\ A # 0 /\ -u ( M + N ) e. ZZ ) -> ( A ^ -u ( M + N ) ) e. CC )
51	1, 9, 49, 50	syl3anc 1220	. . . . . . . . 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 274	. . . . . . . 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 274	. . . . . . . 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 10451	. . . . . . . . . 10 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) # 0 )
55	1, 9, 20, 54	syl3anc 1220	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ N ) # 0 )
56	55	adantr 274	. . . . . . . 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 8669	. . . . . . 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 274	. . . . . . . . . 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 10461	. . . . . . . . . 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 1220	. . . . . . . . 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 8200	. . . . . . . . . . . . 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 5839	. . . . . . . . . . . 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 8173	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. CC )
64	63, 35	npcand 8190	. . . . . . . . . . . 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 2190	. . . . . . . . . . 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 274	. . . . . . . . . 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 5840	. . . . . . . . 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 2192	. . . . . . . 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 5839	. . . . . . 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 2192	. . . . . 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 5840	. . . . 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 8680	. . . . . 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 274	. . . . 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 2190	. . . 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 2190	. . 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 9182	. . . . 5 |- ( ( M + N ) e. ZZ <-> ( ( M + N ) e. RR /\ ( ( M + N ) e. NN0 \/ -u ( M + N ) e. NN0 ) ) )
77	76	simprbi 273	. . . 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 788	. 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 10428	. . . 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 1221	. . 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 5839	. 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 2200	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 103   \/ wo 698   /\ w3a 963   = wceq 1335   e. wcel 2128   class class class wbr 3965  (class class class)co 5824   CCcc 7730   RRcr 7731   0cc0 7732   1c1 7733   + caddc 7735   x. cmul 7737   - cmin 8046   -ucneg 8047   # cap 8456   / cdiv 8545   NNcn 8833   NN0cn0 9090   ZZcz 9167   ^cexp 10418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-n0 9091  df-z 9168  df-uz 9440  df-seqfrec 10345  df-exp 10419

This theorem is referenced by:  expaddzap  10463