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Mirrors > Home > MPE Home > Th. List > gcdcllem1 | Unicode version |
Description: Lemma for gcdn0cl 14152, gcddvds 14153 and dvdslegcd 14154. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
gcdcllem1.1 |
Ref | Expression |
---|---|
gcdcllem1 |
S
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 10919 | . . . . 5 | |
2 | ssel 3497 | . . . . . . 7 | |
3 | 1dvds 13998 | . . . . . . 7 | |
4 | 2, 3 | syl6 33 | . . . . . 6 |
5 | 4 | ralrimiv 2869 | . . . . 5 |
6 | breq1 4455 | . . . . . . . 8 | |
7 | 6 | ralbidv 2896 | . . . . . . 7 |
8 | gcdcllem1.1 | . . . . . . 7 | |
9 | 7, 8 | elrab2 3259 | . . . . . 6 |
10 | 9 | biimpri 206 | . . . . 5 |
11 | 1, 5, 10 | sylancr 663 | . . . 4 |
12 | ne0i 3790 | . . . 4 | |
13 | 11, 12 | syl 16 | . . 3 |
14 | 13 | adantr 465 | . 2 |
15 | neeq1 2738 | . . . 4 | |
16 | 15 | cbvrexv 3085 | . . 3 |
17 | breq1 4455 | . . . . . . . . . . . . 13 | |
18 | 17 | ralbidv 2896 | . . . . . . . . . . . 12 |
19 | 18, 8 | elrab2 3259 | . . . . . . . . . . 11 |
20 | 19 | simprbi 464 | . . . . . . . . . 10 |
21 | 20 | adantl 466 | . . . . . . . . 9 |
22 | 19 | simplbi 460 | . . . . . . . . . 10 |
23 | ssel2 3498 | . . . . . . . . . . . . . . 15 | |
24 | dvdsleabs 14032 | . . . . . . . . . . . . . . . 16 | |
25 | 24 | 3expia 1198 | . . . . . . . . . . . . . . 15 |
26 | 23, 25 | sylan2 474 | . . . . . . . . . . . . . 14 |
27 | 26 | anassrs 648 | . . . . . . . . . . . . 13 |
28 | 27 | com23 78 | . . . . . . . . . . . 12 |
29 | 28 | ralrimiva 2871 | . . . . . . . . . . 11 |
30 | 29 | ancoms 453 | . . . . . . . . . 10 |
31 | 22, 30 | sylan2 474 | . . . . . . . . 9 |
32 | r19.26 2984 | . . . . . . . . . 10 | |
33 | pm3.35 587 | . . . . . . . . . . 11 | |
34 | 33 | ralimi 2850 | . . . . . . . . . 10 |
35 | 32, 34 | sylbir 213 | . . . . . . . . 9 |
36 | 21, 31, 35 | syl2anc 661 | . . . . . . . 8 |
37 | 36 | ralrimiva 2871 | . . . . . . 7 |
38 | fveq2 5871 | . . . . . . . . . . . 12 | |
39 | 38 | breq2d 4464 | . . . . . . . . . . 11 |
40 | 15, 39 | imbi12d 320 | . . . . . . . . . 10 |
41 | 40 | cbvralv 3084 | . . . . . . . . 9 |
42 | 41 | ralbii 2888 | . . . . . . . 8 |
43 | ralcom 3018 | . . . . . . . 8 | |
44 | r19.21v 2862 | . . . . . . . . 9 | |
45 | 44 | ralbii 2888 | . . . . . . . 8 |
46 | 42, 43, 45 | 3bitri 271 | . . . . . . 7 |
47 | 37, 46 | sylib 196 | . . . . . 6 |
48 | ssel2 3498 | . . . . . . . . . . 11 | |
49 | nn0abscl 13145 | . . . . . . . . . . 11 | |
50 | 48, 49 | syl 16 | . . . . . . . . . 10 |
51 | 50 | nn0zd 10992 | . . . . . . . . 9 |
52 | breq2 4456 | . . . . . . . . . . 11 | |
53 | 52 | ralbidv 2896 | . . . . . . . . . 10 |
54 | 53 | adantl 466 | . . . . . . . . 9 |
55 | 51, 54 | rspcedv 3214 | . . . . . . . 8 |
56 | 55 | imim2d 52 | . . . . . . 7 |
57 | 56 | ralimdva 2865 | . . . . . 6 |
58 | 47, 57 | mpd 15 | . . . . 5 |
59 | r19.23v 2937 | . . . . 5 | |
60 | 58, 59 | sylib 196 | . . . 4 |
61 | 60 | imp 429 | . . 3 |
62 | 16, 61 | sylan2b 475 | . 2 |
63 | 14, 62 | jca 532 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 <-> wb 184
/\ wa 369 = wceq 1395 e. wcel 1818
=/= wne 2652 A. wral 2807 E. wrex 2808
{ crab 2811 C_ wss 3475 c0 3784 class class class wbr 4452
` cfv 5593 0 cc0 9513 1 c1 9514
cle 9650 cn0 10820
cz 10889 cabs 13067 cdvds 13986 |
This theorem is referenced by: gcdcllem3 14151 |
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-2nd 6801 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-2 10619 df-3 10620 df-n0 10821 df-z 10890 df-uz 11111 df-rp 11250 df-seq 12108 df-exp 12167 df-cj 12932 df-re 12933 df-im 12934 df-sqrt 13068 df-abs 13069 df-dvds 13987 |
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