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Mirrors > Home > MPE Home > Th. List > ltaddpr | Structured version Visualization version Unicode version |
Description: The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ltaddpr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prn0 9811 | . . . . 5 | |
2 | n0 3931 | . . . . 5 | |
3 | 1, 2 | sylib 208 | . . . 4 |
4 | 3 | adantl 482 | . . 3 |
5 | addclpr 9840 | . . . . . . . . . . . 12 | |
6 | 5 | adantr 481 | . . . . . . . . . . 11 |
7 | df-plp 9805 | . . . . . . . . . . . . 13 | |
8 | addclnq 9767 | . . . . . . . . . . . . 13 | |
9 | 7, 8 | genpprecl 9823 | . . . . . . . . . . . 12 |
10 | 9 | imp 445 | . . . . . . . . . . 11 |
11 | elprnq 9813 | . . . . . . . . . . . . 13 | |
12 | addnqf 9770 | . . . . . . . . . . . . . . 15 | |
13 | 12 | fdmi 6052 | . . . . . . . . . . . . . 14 |
14 | 0nnq 9746 | . . . . . . . . . . . . . 14 | |
15 | 13, 14 | ndmovrcl 6820 | . . . . . . . . . . . . 13 |
16 | ltaddnq 9796 | . . . . . . . . . . . . 13 | |
17 | 11, 15, 16 | 3syl 18 | . . . . . . . . . . . 12 |
18 | prcdnq 9815 | . . . . . . . . . . . 12 | |
19 | 17, 18 | mpd 15 | . . . . . . . . . . 11 |
20 | 6, 10, 19 | syl2anc 693 | . . . . . . . . . 10 |
21 | 20 | exp32 631 | . . . . . . . . 9 |
22 | 21 | com23 86 | . . . . . . . 8 |
23 | 22 | alrimdv 1857 | . . . . . . 7 |
24 | dfss2 3591 | . . . . . . 7 | |
25 | 23, 24 | syl6ibr 242 | . . . . . 6 |
26 | vex 3203 | . . . . . . . . 9 | |
27 | 26 | prlem934 9855 | . . . . . . . 8 |
28 | 27 | adantr 481 | . . . . . . 7 |
29 | eleq2 2690 | . . . . . . . . . . . . 13 | |
30 | 29 | biimprcd 240 | . . . . . . . . . . . 12 |
31 | 30 | con3d 148 | . . . . . . . . . . 11 |
32 | 9, 31 | syl6 35 | . . . . . . . . . 10 |
33 | 32 | expd 452 | . . . . . . . . 9 |
34 | 33 | com34 91 | . . . . . . . 8 |
35 | 34 | rexlimdv 3030 | . . . . . . 7 |
36 | 28, 35 | mpd 15 | . . . . . 6 |
37 | 25, 36 | jcad 555 | . . . . 5 |
38 | dfpss2 3692 | . . . . 5 | |
39 | 37, 38 | syl6ibr 242 | . . . 4 |
40 | 39 | exlimdv 1861 | . . 3 |
41 | 4, 40 | mpd 15 | . 2 |
42 | ltprord 9852 | . . 3 | |
43 | 5, 42 | syldan 487 | . 2 |
44 | 41, 43 | mpbird 247 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wal 1481 wceq 1483 wex 1704 wcel 1990 wne 2794 wrex 2913 wss 3574 wpss 3575 c0 3915 class class class wbr 4653 cxp 5112 (class class class)co 6650 cnq 9674 cplq 9677 cltq 9680 cnp 9681 cpp 9683 cltp 9685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-ni 9694 df-pli 9695 df-mi 9696 df-lti 9697 df-plpq 9730 df-mpq 9731 df-ltpq 9732 df-enq 9733 df-nq 9734 df-erq 9735 df-plq 9736 df-mq 9737 df-1nq 9738 df-rq 9739 df-ltnq 9740 df-np 9803 df-plp 9805 df-ltp 9807 |
This theorem is referenced by: ltaddpr2 9857 ltexprlem7 9864 ltaprlem 9866 0lt1sr 9916 mappsrpr 9929 |
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