Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE 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.) |
Ref | Expression |
---|---|
ltaddpr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prop 7430 | . . . 4 | |
2 | prml 7432 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | 3 | adantl 275 | . 2 |
5 | prop 7430 | . . . . 5 | |
6 | prarloc 7458 | . . . . 5 | |
7 | 5, 6 | sylan 281 | . . . 4 |
8 | 7 | ad2ant2r 506 | . . 3 |
9 | elprnqu 7437 | . . . . . . . . . . 11 | |
10 | 5, 9 | sylan 281 | . . . . . . . . . 10 |
11 | 10 | adantlr 474 | . . . . . . . . 9 |
12 | 11 | ad2ant2rl 508 | . . . . . . . 8 |
13 | 12 | adantr 274 | . . . . . . 7 |
14 | simplrr 531 | . . . . . . 7 | |
15 | simprl 526 | . . . . . . . . . . . . 13 | |
16 | simplr 525 | . . . . . . . . . . . . 13 | |
17 | 15, 16 | jca 304 | . . . . . . . . . . . 12 |
18 | df-iplp 7423 | . . . . . . . . . . . . 13 | |
19 | addclnq 7330 | . . . . . . . . . . . . 13 | |
20 | 18, 19 | genpprecll 7469 | . . . . . . . . . . . 12 |
21 | 17, 20 | syl5 32 | . . . . . . . . . . 11 |
22 | 21 | imdistani 443 | . . . . . . . . . 10 |
23 | addclpr 7492 | . . . . . . . . . . 11 | |
24 | prop 7430 | . . . . . . . . . . . 12 | |
25 | prcdnql 7439 | . . . . . . . . . . . 12 | |
26 | 24, 25 | sylan 281 | . . . . . . . . . . 11 |
27 | 23, 26 | sylan 281 | . . . . . . . . . 10 |
28 | 22, 27 | syl 14 | . . . . . . . . 9 |
29 | 28 | anassrs 398 | . . . . . . . 8 |
30 | 29 | imp 123 | . . . . . . 7 |
31 | rspe 2519 | . . . . . . 7 | |
32 | 13, 14, 30, 31 | syl12anc 1231 | . . . . . 6 |
33 | ltdfpr 7461 | . . . . . . . 8 | |
34 | 23, 33 | syldan 280 | . . . . . . 7 |
35 | 34 | ad3antrrr 489 | . . . . . 6 |
36 | 32, 35 | mpbird 166 | . . . . 5 |
37 | 36 | ex 114 | . . . 4 |
38 | 37 | rexlimdvva 2595 | . . 3 |
39 | 8, 38 | mpd 13 | . 2 |
40 | 4, 39 | rexlimddv 2592 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2141 wrex 2449 cop 3584 class class class wbr 3987 cfv 5196 (class class class)co 5851 c1st 6115 c2nd 6116 cnq 7235 cplq 7237 cltq 7240 cnp 7246 cpp 7248 cltp 7250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-eprel 4272 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-irdg 6347 df-1o 6393 df-2o 6394 df-oadd 6397 df-omul 6398 df-er 6511 df-ec 6513 df-qs 6517 df-ni 7259 df-pli 7260 df-mi 7261 df-lti 7262 df-plpq 7299 df-mpq 7300 df-enq 7302 df-nqqs 7303 df-plqqs 7304 df-mqqs 7305 df-1nqqs 7306 df-rq 7307 df-ltnqqs 7308 df-enq0 7379 df-nq0 7380 df-0nq0 7381 df-plq0 7382 df-mq0 7383 df-inp 7421 df-iplp 7423 df-iltp 7425 |
This theorem is referenced by: ltexprlemrl 7565 ltaprlem 7573 ltaprg 7574 prplnqu 7575 ltmprr 7597 caucvgprprlemnkltj 7644 caucvgprprlemnkeqj 7645 caucvgprprlemnbj 7648 0lt1sr 7720 recexgt0sr 7728 mulgt0sr 7733 archsr 7737 prsrpos 7740 mappsrprg 7759 pitoregt0 7804 |
Copyright terms: Public domain | W3C validator |