expaddzap - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > expaddzap		Unicode version

Theorem expaddzap 10728

Description: Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)

**Assertion**
Ref	Expression
expaddzap	$/\$ # $/\$ $/\$

**Proof of Theorem expaddzap**
Step	Hyp	Ref	Expression
1		elznn0nn 9386	. . 3 $\/$ $/\$
2		elznn0nn 9386	. . . 4 $\/$ $/\$
3		expadd 10726	. . . . . . . 8 $/\$ $/\$
4	3	3expia 1208	. . . . . . 7 $/\$
5	4	adantlr 477	. . . . . 6 $/\$ # $/\$
6		expaddzaplem 10727	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
7	6	3expia 1208	. . . . . 6 $/\$ # $/\$ $/\$
8	5, 7	jaodan 799	. . . . 5 $/\$ # $/\$ $\/$ $/\$
9		expaddzaplem 10727	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
10		simp3 1002	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
11	10	nn0cnd 9350	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
12		simp2l 1026	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
13	12	recnd 8101	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
14	11, 13	addcomd 8223	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
15	14	oveq2d 5960	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
16		simp1l 1024	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17		expcl 10702	. . . . . . . . . . 11 $/\$
18	16, 10, 17	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
19		simp1r 1025	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ #
20	13	negnegd 8374	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
21		simp2r 1027	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$
22	21	nnnn0d 9348	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$
23		nn0negz 9406	. . . . . . . . . . . . 13
24	22, 23	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
25	20, 24	eqeltrrd 2283	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
26		expclzap 10709	. . . . . . . . . . 11 $/\$ # $/\$
27	16, 19, 25, 26	syl3anc 1250	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
28	18, 27	mulcomd 8094	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$
29	9, 15, 28	3eqtr4d 2248	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
30	29	3expia 1208	. . . . . . 7 $/\$ # $/\$ $/\$
31	30	impancom 260	. . . . . 6 $/\$ # $/\$ $/\$
32		simp2l 1026	. . . . . . . . . . . . . . 15 $/\$ # $/\$ $/\$ $/\$ $/\$
33	32	recnd 8101	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
34		simp3l 1028	. . . . . . . . . . . . . . 15 $/\$ # $/\$ $/\$ $/\$ $/\$
35	34	recnd 8101	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
36	33, 35	negdid 8396	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
37	36	oveq2d 5960	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$ $/\$
38		simp1l 1024	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
39		simp2r 1027	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
40	39	nnnn0d 9348	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
41		simp3r 1029	. . . . . . . . . . . . . 14 $/\$ # $/\$ $/\$ $/\$ $/\$
42	41	nnnn0d 9348	. . . . . . . . . . . . 13 $/\$ # $/\$ $/\$ $/\$ $/\$
43		expadd 10726	. . . . . . . . . . . . 13 $/\$ $/\$
44	38, 40, 42, 43	syl3anc 1250	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$ $/\$
45	37, 44	eqtrd 2238	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
46	45	oveq2d 5960	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
47		1t1e1 9189	. . . . . . . . . . 11
48	47	oveq1i 5954	. . . . . . . . . 10
49	46, 48	eqtr4di 2256	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
50		expcl 10702	. . . . . . . . . . 11 $/\$
51	38, 40, 50	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
52		simp1r 1025	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$ #
53	40	nn0zd 9493	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
54		expap0i 10716	. . . . . . . . . . 11 $/\$ # $/\$ #
55	38, 52, 53, 54	syl3anc 1250	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
56		expcl 10702	. . . . . . . . . . 11 $/\$
57	38, 42, 56	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
58	42	nn0zd 9493	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
59		expap0i 10716	. . . . . . . . . . 11 $/\$ # $/\$ #
60	38, 52, 58, 59	syl3anc 1250	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$ #
61		ax-1cn 8018	. . . . . . . . . . 11
62		divmuldivap 8785	. . . . . . . . . . 11 $/\$ $/\$ $/\$ # $/\$ $/\$ #
63	61, 61, 62	mpanl12 436	. . . . . . . . . 10 $/\$ # $/\$ $/\$ #
64	51, 55, 57, 60, 63	syl22anc 1251	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
65	49, 64	eqtr4d 2241	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
66	33, 35	addcld 8092	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
67	40, 42	nn0addcld 9352	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
68	36, 67	eqeltrd 2282	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
69		expineg2 10693	. . . . . . . . 9 $/\$ # $/\$ $/\$
70	38, 52, 66, 68, 69	syl22anc 1251	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
71		expineg2 10693	. . . . . . . . . 10 $/\$ # $/\$ $/\$
72	38, 52, 33, 40, 71	syl22anc 1251	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
73		expineg2 10693	. . . . . . . . . 10 $/\$ # $/\$ $/\$
74	38, 52, 35, 42, 73	syl22anc 1251	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
75	72, 74	oveq12d 5962	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
76	65, 70, 75	3eqtr4d 2248	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
77	76	3expia 1208	. . . . . 6 $/\$ # $/\$ $/\$ $/\$
78	31, 77	jaodan 799	. . . . 5 $/\$ # $/\$ $\/$ $/\$ $/\$
79	8, 78	jaod 719	. . . 4 $/\$ # $/\$ $\/$ $/\$ $\/$ $/\$
80	2, 79	sylan2b 287	. . 3 $/\$ # $/\$ $\/$ $/\$
81	1, 80	biimtrid 152	. 2 $/\$ # $/\$
82	81	impr 379	1 $/\$ # $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $\/$ wo 710 $/\$ w3a 981

wceq 1373

wcel 2176 class class class wbr 4044 (class class class)co 5944

cc 7923

cr 7924

cc0 7925

c1 7926

caddc 7928

cmul 7930

cneg 8244 # cap 8654

cdiv 8745

cn 9036

cn0 9295

cz 9372

cexp 10683

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-mulext 8043

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-po 4343 df-iso 4344 df-iord 4413 df-on 4415 df-ilim 4416 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-recs 6391 df-frec 6477 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 df-div 8746 df-inn 9037 df-n0 9296 df-z 9373 df-uz 9649 df-seqfrec 10593 df-exp 10684

This theorem is referenced by: m1expeven 10731 expsubap 10732 expp1zap 10733 pcaddlem 12662 expghmap 14369 lgseisenlem4 15550 lgsquadlem1 15554 lgsquad2lem1 15558

W3C validator