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Mirrors > Home > ILE Home > Th. List > modqexp | Unicode version |
Description: Exponentiation property of the modulo operation, see theorem 5.2(c) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.) |
Ref | Expression |
---|---|
modqexp.a | |
modqexp.b | |
modqexp.c | |
modqexp.dq | |
modqexp.dgt0 | |
modqexp.mod |
Ref | Expression |
---|---|
modqexp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modqexp.c | . 2 | |
2 | oveq2 5834 | . . . . . 6 | |
3 | 2 | oveq1d 5841 | . . . . 5 |
4 | oveq2 5834 | . . . . . 6 | |
5 | 4 | oveq1d 5841 | . . . . 5 |
6 | 3, 5 | eqeq12d 2172 | . . . 4 |
7 | 6 | imbi2d 229 | . . 3 |
8 | oveq2 5834 | . . . . . 6 | |
9 | 8 | oveq1d 5841 | . . . . 5 |
10 | oveq2 5834 | . . . . . 6 | |
11 | 10 | oveq1d 5841 | . . . . 5 |
12 | 9, 11 | eqeq12d 2172 | . . . 4 |
13 | 12 | imbi2d 229 | . . 3 |
14 | oveq2 5834 | . . . . . 6 | |
15 | 14 | oveq1d 5841 | . . . . 5 |
16 | oveq2 5834 | . . . . . 6 | |
17 | 16 | oveq1d 5841 | . . . . 5 |
18 | 15, 17 | eqeq12d 2172 | . . . 4 |
19 | 18 | imbi2d 229 | . . 3 |
20 | oveq2 5834 | . . . . . 6 | |
21 | 20 | oveq1d 5841 | . . . . 5 |
22 | oveq2 5834 | . . . . . 6 | |
23 | 22 | oveq1d 5841 | . . . . 5 |
24 | 21, 23 | eqeq12d 2172 | . . . 4 |
25 | 24 | imbi2d 229 | . . 3 |
26 | modqexp.a | . . . . . . 7 | |
27 | 26 | zcnd 9292 | . . . . . 6 |
28 | exp0 10432 | . . . . . 6 | |
29 | 27, 28 | syl 14 | . . . . 5 |
30 | modqexp.b | . . . . . . 7 | |
31 | 30 | zcnd 9292 | . . . . . 6 |
32 | exp0 10432 | . . . . . 6 | |
33 | 31, 32 | syl 14 | . . . . 5 |
34 | 29, 33 | eqtr4d 2193 | . . . 4 |
35 | 34 | oveq1d 5841 | . . 3 |
36 | zexpcl 10443 | . . . . . . . . . 10 | |
37 | 26, 36 | sylan 281 | . . . . . . . . 9 |
38 | 37 | adantr 274 | . . . . . . . 8 |
39 | zexpcl 10443 | . . . . . . . . . 10 | |
40 | 30, 39 | sylan 281 | . . . . . . . . 9 |
41 | 40 | adantr 274 | . . . . . . . 8 |
42 | 26 | ad2antrr 480 | . . . . . . . 8 |
43 | 30 | ad2antrr 480 | . . . . . . . 8 |
44 | modqexp.dq | . . . . . . . . 9 | |
45 | 44 | ad2antrr 480 | . . . . . . . 8 |
46 | modqexp.dgt0 | . . . . . . . . 9 | |
47 | 46 | ad2antrr 480 | . . . . . . . 8 |
48 | simpr 109 | . . . . . . . 8 | |
49 | modqexp.mod | . . . . . . . . 9 | |
50 | 49 | ad2antrr 480 | . . . . . . . 8 |
51 | 38, 41, 42, 43, 45, 47, 48, 50 | modqmul12d 10286 | . . . . . . 7 |
52 | 27 | ad2antrr 480 | . . . . . . . . 9 |
53 | simpr 109 | . . . . . . . . . 10 | |
54 | 53 | adantr 274 | . . . . . . . . 9 |
55 | expp1 10435 | . . . . . . . . 9 | |
56 | 52, 54, 55 | syl2anc 409 | . . . . . . . 8 |
57 | 56 | oveq1d 5841 | . . . . . . 7 |
58 | 31 | ad2antrr 480 | . . . . . . . . 9 |
59 | expp1 10435 | . . . . . . . . 9 | |
60 | 58, 54, 59 | syl2anc 409 | . . . . . . . 8 |
61 | 60 | oveq1d 5841 | . . . . . . 7 |
62 | 51, 57, 61 | 3eqtr4d 2200 | . . . . . 6 |
63 | 62 | ex 114 | . . . . 5 |
64 | 63 | expcom 115 | . . . 4 |
65 | 64 | a2d 26 | . . 3 |
66 | 7, 13, 19, 25, 35, 65 | nn0ind 9283 | . 2 |
67 | 1, 66 | mpcom 36 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 class class class wbr 3967 (class class class)co 5826 cc 7732 cc0 7734 c1 7735 caddc 7737 cmul 7739 clt 7914 cn0 9095 cz 9172 cq 9534 cmo 10230 cexp 10427 |
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 4081 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-iinf 4549 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-mulrcl 7833 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-1rid 7841 ax-0id 7842 ax-rnegex 7843 ax-precex 7844 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-apti 7849 ax-pre-ltadd 7850 ax-pre-mulgt0 7851 ax-pre-mulext 7852 ax-arch 7853 |
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 3396 df-if 3507 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-tr 4065 df-id 4255 df-po 4258 df-iso 4259 df-iord 4328 df-on 4330 df-ilim 4331 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-recs 6254 df-frec 6340 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-reap 8454 df-ap 8461 df-div 8550 df-inn 8839 df-n0 9096 df-z 9173 df-uz 9445 df-q 9535 df-rp 9567 df-fl 10178 df-mod 10231 df-seqfrec 10354 df-exp 10428 |
This theorem is referenced by: dvdsmodexp 11702 odzdvds 12135 |
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