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Mirrors > Home > MPE Home > Th. List > ledivp1 | Unicode version |
Description: Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 28-Sep-2005.) |
Ref | Expression |
---|---|
ledivp1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 756 | . . 3 | |
2 | peano2re 9774 | . . . 4 | |
3 | 2 | ad2antrl 727 | . . 3 |
4 | simpll 753 | . . . . 5 | |
5 | ltp1 10405 | . . . . . . . . 9 | |
6 | 0re 9617 | . . . . . . . . . . 11 | |
7 | lelttr 9696 | . . . . . . . . . . 11 | |
8 | 6, 7 | mp3an1 1311 | . . . . . . . . . 10 |
9 | 2, 8 | mpdan 668 | . . . . . . . . 9 |
10 | 5, 9 | mpan2d 674 | . . . . . . . 8 |
11 | 10 | imp 429 | . . . . . . 7 |
12 | 11 | gt0ne0d 10142 | . . . . . 6 |
13 | 12 | adantl 466 | . . . . 5 |
14 | 4, 3, 13 | redivcld 10397 | . . . 4 |
15 | 2 | adantr 465 | . . . . . 6 |
16 | 15, 11 | jca 532 | . . . . 5 |
17 | divge0 10436 | . . . . 5 | |
18 | 16, 17 | sylan2 474 | . . . 4 |
19 | 14, 18 | jca 532 | . . 3 |
20 | lep1 10406 | . . . 4 | |
21 | 20 | ad2antrl 727 | . . 3 |
22 | lemul2a 10422 | . . 3 | |
23 | 1, 3, 19, 21, 22 | syl31anc 1231 | . 2 |
24 | recn 9603 | . . . 4 | |
25 | 24 | ad2antrr 725 | . . 3 |
26 | 2 | recnd 9643 | . . . 4 |
27 | 26 | ad2antrl 727 | . . 3 |
28 | 25, 27, 13 | divcan1d 10346 | . 2 |
29 | 23, 28 | breqtrd 4476 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 /\ wa 369
e. wcel 1818 =/= wne 2652 class class class wbr 4452
(class class class)co 6296 cc 9511 cr 9512 0 cc0 9513 1 c1 9514
caddc 9516 cmul 9518 clt 9649 cle 9650 cdiv 10231 |
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-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 |
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-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-op 4036 df-uni 4250 df-br 4453 df-opab 4511 df-mpt 4512 df-id 4800 df-po 4805 df-so 4806 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-er 7330 df-en 7537 df-dom 7538 df-sdom 7539 df-pnf 9651 df-mnf 9652 df-xr 9653 df-ltxr 9654 df-le 9655 df-sub 9830 df-neg 9831 df-div 10232 |
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