Theorem expaddzaplem 11019

Description: Lemma for expaddzap 11020. (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 1052	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> A e. CC )
2		simp3 1030	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. NN0 )
3		expcl 10994	. . . 4 |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
4	1, 2, 3	syl2anc 415	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ N ) e. CC )
5		simp2r 1055	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. NN )
6	5	nnnn0d 9620	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. NN0 )
7		expcl 10994	. . . 4 |- ( ( A e. CC /\ -u M e. NN0 ) -> ( A ^ -u M ) e. CC )
8	1, 6, 7	syl2anc 415	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ -u M ) e. CC )
9		simp1r 1053	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> A # 0 )
10	5	nnzd 9767	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. ZZ )
11		expap0i 11008	. . . 4 |- ( ( A e. CC /\ A # 0 /\ -u M e. ZZ ) -> ( A ^ -u M ) # 0 )
12	1, 9, 10, 11	syl3anc 1278	. . 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 9124	. 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 1054	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. RR )
15	14	recnd 8354	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. CC )
16	15	negnegd 8628	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u -u M = M )
17		nnnegz 9647	. . . . . . . . . 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 2316	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. ZZ )
20	2	nn0zd 9766	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. ZZ )
21	19, 20	zaddcld 9772	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( M + N ) e. ZZ )
22		expclzap 11001	. . . . . . 7 |- ( ( A e. CC /\ A # 0 /\ ( M + N ) e. ZZ ) -> ( A ^ ( M + N ) ) e. CC )
23	1, 9, 21, 22	syl3anc 1278	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( M + N ) ) e. CC )
24	23	adantr 276	. . . . 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 276	. . . . 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 276	. . . . 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 9126	. . . 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 276	. . . . . . 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 110	. . . . . . 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 276	. . . . . . 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 11018	. . . . . . 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 1278	. . . . . 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 9769	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( M + N ) e. CC )
34	33, 15	negsubd 8643	. . . . . . . . 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 9622	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. CC )
36	15, 35	pncan2d 8639	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( M + N ) - M ) = N )
37	34, 36	eqtrd 2271	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( M + N ) + -u M ) = N )
38	37	adantr 276	. . . . . . 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 6101	. . . . . 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 2273	. . . . 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 6100	. . . 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 2273	. . 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 276	. . . . 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 276	. . . . 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 276	. . . . 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 110	. . . . 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 10985	. . . . 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 1279	. . . 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 9770	. . . . . . . . . 10 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u ( M + N ) e. ZZ )
50		expclzap 11001	. . . . . . . . . 10 |- ( ( A e. CC /\ A # 0 /\ -u ( M + N ) e. ZZ ) -> ( A ^ -u ( M + N ) ) e. CC )
51	1, 9, 49, 50	syl3anc 1278	. . . . . . . . 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 276	. . . . . . . 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 276	. . . . . . . 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 11008	. . . . . . . . . 10 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) # 0 )
55	1, 9, 20, 54	syl3anc 1278	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ N ) # 0 )
56	55	adantr 276	. . . . . . . 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 9126	. . . . . . 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 276	. . . . . . . . . 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 11018	. . . . . . . . . 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 1278	. . . . . . . . 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 8651	. . . . . . . . . . . . 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 6100	. . . . . . . . . . . 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 8624	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. CC )
64	63, 35	npcand 8641	. . . . . . . . . . . 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 2271	. . . . . . . . . . 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 276	. . . . . . . . . 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 6101	. . . . . . . . 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 2273	. . . . . . . 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 6100	. . . . . . 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 2273	. . . . . 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 6101	. . . . 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 9137	. . . . . 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 276	. . . . 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 2271	. . . 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 2271	. . 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 9659	. . . . 5 |- ( ( M + N ) e. ZZ <-> ( ( M + N ) e. RR /\ ( ( M + N ) e. NN0 \/ -u ( M + N ) e. NN0 ) ) )
77	76	simprbi 275	. . . 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 810	. 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 10985	. . . 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 1279	. . 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 6100	. 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 2281	1 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   - cmin 8497   -ucneg 8498   # cap 8909   / cdiv 9002   NNcn 9304   NN0cn0 9563   ZZcz 9644   ^cexp 10975

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-seqfrec 10885  df-exp 10976

This theorem is used by:  expaddzap  11020