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Mirrors > Home > MPE Home > Th. List > modmulnn | Unicode version |
Description: Move a positive integer in and out of a floor in the first argument of a modulo operation. (Contributed by NM, 2-Jan-2009.) |
Ref | Expression |
---|---|
modmulnn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnre 10568 | . . . . 5 | |
2 | reflcl 11933 | . . . . 5 | |
3 | remulcl 9598 | . . . . 5 | |
4 | 1, 2, 3 | syl2an 477 | . . . 4 |
5 | 4 | 3adant3 1016 | . . 3 |
6 | remulcl 9598 | . . . . . 6 | |
7 | 1, 6 | sylan 471 | . . . . 5 |
8 | reflcl 11933 | . . . . 5 | |
9 | 7, 8 | syl 16 | . . . 4 |
10 | 9 | 3adant3 1016 | . . 3 |
11 | nnmulcl 10584 | . . . . . 6 | |
12 | 11 | nnred 10576 | . . . . 5 |
13 | 12 | 3adant2 1015 | . . . 4 |
14 | nncn 10569 | . . . . . . . . 9 | |
15 | nnne0 10593 | . . . . . . . . 9 | |
16 | 14, 15 | jca 532 | . . . . . . . 8 |
17 | nncn 10569 | . . . . . . . . 9 | |
18 | nnne0 10593 | . . . . . . . . 9 | |
19 | 17, 18 | jca 532 | . . . . . . . 8 |
20 | mulne0 10216 | . . . . . . . 8 | |
21 | 16, 19, 20 | syl2an 477 | . . . . . . 7 |
22 | 21 | 3adant2 1015 | . . . . . 6 |
23 | 5, 13, 22 | redivcld 10397 | . . . . 5 |
24 | reflcl 11933 | . . . . 5 | |
25 | 23, 24 | syl 16 | . . . 4 |
26 | 13, 25 | remulcld 9645 | . . 3 |
27 | nnnn0 10827 | . . . . 5 | |
28 | flmulnn0 11960 | . . . . 5 | |
29 | 27, 28 | sylan 471 | . . . 4 |
30 | 29 | 3adant3 1016 | . . 3 |
31 | 5, 10, 26, 30 | lesub1dd 10193 | . 2 |
32 | 11 | nnrpd 11284 | . . . 4 |
33 | 32 | 3adant2 1015 | . . 3 |
34 | modval 11998 | . . 3 | |
35 | 5, 33, 34 | syl2anc 661 | . 2 |
36 | modval 11998 | . . . 4 | |
37 | 10, 33, 36 | syl2anc 661 | . . 3 |
38 | 7 | 3adant3 1016 | . . . . . . 7 |
39 | 11 | 3adant2 1015 | . . . . . . 7 |
40 | fldiv 11987 | . . . . . . 7 | |
41 | 38, 39, 40 | syl2anc 661 | . . . . . 6 |
42 | fldiv 11987 | . . . . . . . . 9 | |
43 | 42 | 3adant3 1016 | . . . . . . . 8 |
44 | 2 | recnd 9643 | . . . . . . . . . 10 |
45 | divcan5 10271 | . . . . . . . . . 10 | |
46 | 44, 19, 16, 45 | syl3an 1270 | . . . . . . . . 9 |
47 | 46 | fveq2d 5875 | . . . . . . . 8 |
48 | recn 9603 | . . . . . . . . . 10 | |
49 | divcan5 10271 | . . . . . . . . . 10 | |
50 | 48, 19, 16, 49 | syl3an 1270 | . . . . . . . . 9 |
51 | 50 | fveq2d 5875 | . . . . . . . 8 |
52 | 43, 47, 51 | 3eqtr4rd 2509 | . . . . . . 7 |
53 | 52 | 3comr 1204 | . . . . . 6 |
54 | 41, 53 | eqtrd 2498 | . . . . 5 |
55 | 54 | oveq2d 6312 | . . . 4 |
56 | 55 | oveq2d 6312 | . . 3 |
57 | 37, 56 | eqtrd 2498 | . 2 |
58 | 31, 35, 57 | 3brtr4d 4482 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 /\ wa 369
/\ w3a 973 = wceq 1395 e. wcel 1818
=/= wne 2652 class class class wbr 4452
` cfv 5593 (class class class)co 6296
cc 9511 cr 9512 0 cc0 9513 cmul 9518 cle 9650 cmin 9828 cdiv 10231 cn 10561 cn0 10820
crp 11249
cfl 11927
cmo 11996 |
This theorem is referenced by: digit1 12300 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 ax-cnex 9569 ax-resscn 9570 ax-1cn 9571 ax-icn 9572 ax-addcl 9573 ax-addrcl 9574 ax-mulcl 9575 ax-mulrcl 9576 ax-mulcom 9577 ax-addass 9578 ax-mulass 9579 ax-distr 9580 ax-i2m1 9581 ax-1ne0 9582 ax-1rid 9583 ax-rnegex 9584 ax-rrecex 9585 ax-cnre 9586 ax-pre-lttri 9587 ax-pre-lttrn 9588 ax-pre-ltadd 9589 ax-pre-mulgt0 9590 ax-pre-sup 9591 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-nel 2655 df-ral 2812 df-rex 2813 df-reu 2814 df-rmo 2815 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-pss 3491 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-tp 4034 df-op 4036 df-uni 4250 df-iun 4332 df-br 4453 df-opab 4511 df-mpt 4512 df-tr 4546 df-eprel 4796 df-id 4800 df-po 4805 df-so 4806 df-fr 4843 df-we 4845 df-ord 4886 df-on 4887 df-lim 4888 df-suc 4889 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-riota 6257 df-ov 6299 df-oprab 6300 df-mpt2 6301 df-om 6701 df-recs 7061 df-rdg 7095 df-er 7330 df-en 7537 df-dom 7538 df-sdom 7539 df-sup 7921 df-pnf 9651 df-mnf 9652 df-xr 9653 df-ltxr 9654 df-le 9655 df-sub 9830 df-neg 9831 df-div 10232 df-nn 10562 df-n0 10821 df-z 10890 df-uz 11111 df-rp 11250 df-fl 11929 df-mod 11997 |
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