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Mirrors > Home > MPE Home > Th. List > ltaddpr | 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 9388 | . . . . 5 | |
2 | n0 3794 | . . . . 5 | |
3 | 1, 2 | sylib 196 | . . . 4 |
4 | 3 | adantl 466 | . . 3 |
5 | addclpr 9417 | . . . . . . . . . . . 12 | |
6 | 5 | adantr 465 | . . . . . . . . . . 11 |
7 | df-plp 9382 | . . . . . . . . . . . . 13 | |
8 | addclnq 9344 | . . . . . . . . . . . . 13 | |
9 | 7, 8 | genpprecl 9400 | . . . . . . . . . . . 12 |
10 | 9 | imp 429 | . . . . . . . . . . 11 |
11 | elprnq 9390 | . . . . . . . . . . . . 13 | |
12 | addnqf 9347 | . . . . . . . . . . . . . . 15 | |
13 | 12 | fdmi 5741 | . . . . . . . . . . . . . 14 |
14 | 0nnq 9323 | . . . . . . . . . . . . . 14 | |
15 | 13, 14 | ndmovrcl 6461 | . . . . . . . . . . . . 13 |
16 | ltaddnq 9373 | . . . . . . . . . . . . 13 | |
17 | 11, 15, 16 | 3syl 20 | . . . . . . . . . . . 12 |
18 | prcdnq 9392 | . . . . . . . . . . . 12 | |
19 | 17, 18 | mpd 15 | . . . . . . . . . . 11 |
20 | 6, 10, 19 | syl2anc 661 | . . . . . . . . . 10 |
21 | 20 | exp32 605 | . . . . . . . . 9 |
22 | 21 | com23 78 | . . . . . . . 8 |
23 | 22 | alrimdv 1721 | . . . . . . 7 |
24 | dfss2 3492 | . . . . . . 7 | |
25 | 23, 24 | syl6ibr 227 | . . . . . 6 |
26 | vex 3112 | . . . . . . . . 9 | |
27 | 26 | prlem934 9432 | . . . . . . . 8 |
28 | 27 | adantr 465 | . . . . . . 7 |
29 | eleq2 2530 | . . . . . . . . . . . . 13 | |
30 | 29 | biimprcd 225 | . . . . . . . . . . . 12 |
31 | 30 | con3d 133 | . . . . . . . . . . 11 |
32 | 9, 31 | syl6 33 | . . . . . . . . . 10 |
33 | 32 | expd 436 | . . . . . . . . 9 |
34 | 33 | com34 83 | . . . . . . . 8 |
35 | 34 | rexlimdv 2947 | . . . . . . 7 |
36 | 28, 35 | mpd 15 | . . . . . 6 |
37 | 25, 36 | jcad 533 | . . . . 5 |
38 | dfpss2 3588 | . . . . 5 | |
39 | 37, 38 | syl6ibr 227 | . . . 4 |
40 | 39 | exlimdv 1724 | . . 3 |
41 | 4, 40 | mpd 15 | . 2 |
42 | ltprord 9429 | . . 3 | |
43 | 5, 42 | syldan 470 | . 2 |
44 | 41, 43 | mpbird 232 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -. wn 3 -> wi 4
<-> wb 184 /\ wa 369 A. wal 1393
= wceq 1395 E. wex 1612 e. wcel 1818
=/= wne 2652 E. wrex 2808 C_ wss 3475
C. wpss 3476 c0 3784 class class class wbr 4452
X. cxp 5002 (class class class)co 6296
cnq 9251
cplq 9254
cltq 9257
cnp 9258
cpp 9260
cltp 9262 |
This theorem is referenced by: ltaddpr2 9434 ltexprlem7 9441 ltaprlem 9443 0lt1sr 9493 mappsrpr 9506 |
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-inf2 8079 |
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-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-int 4287 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-ov 6299 df-oprab 6300 df-mpt2 6301 df-om 6701 df-1st 6800 df-2nd 6801 df-recs 7061 df-rdg 7095 df-1o 7149 df-oadd 7153 df-omul 7154 df-er 7330 df-ni 9271 df-pli 9272 df-mi 9273 df-lti 9274 df-plpq 9307 df-mpq 9308 df-ltpq 9309 df-enq 9310 df-nq 9311 df-erq 9312 df-plq 9313 df-mq 9314 df-1nq 9315 df-rq 9316 df-ltnq 9317 df-np 9380 df-plp 9382 df-ltp 9384 |
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