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Mirrors > Home > ILE Home > Th. List > ltexprlemlol | Unicode version |
Description: The lower cut of our constructed difference is lower. Lemma for ltexpri 7533. (Contributed by Jim Kingdon, 21-Dec-2019.) |
Ref | Expression |
---|---|
ltexprlem.1 |
Ref | Expression |
---|---|
ltexprlemlol |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 520 | . . . . . 6 | |
2 | simprrr 530 | . . . . . . 7 | |
3 | 2 | simpld 111 | . . . . . 6 |
4 | simprl 521 | . . . . . . . 8 | |
5 | simpll 519 | . . . . . . . . 9 | |
6 | ltrelpr 7425 | . . . . . . . . . . . 12 | |
7 | 6 | brel 4638 | . . . . . . . . . . 11 |
8 | 7 | simpld 111 | . . . . . . . . . 10 |
9 | prop 7395 | . . . . . . . . . . 11 | |
10 | elprnqu 7402 | . . . . . . . . . . 11 | |
11 | 9, 10 | sylan 281 | . . . . . . . . . 10 |
12 | 8, 11 | sylan 281 | . . . . . . . . 9 |
13 | 5, 3, 12 | syl2anc 409 | . . . . . . . 8 |
14 | ltanqi 7322 | . . . . . . . 8 | |
15 | 4, 13, 14 | syl2anc 409 | . . . . . . 7 |
16 | 7 | simprd 113 | . . . . . . . . 9 |
17 | 5, 16 | syl 14 | . . . . . . . 8 |
18 | 2 | simprd 113 | . . . . . . . 8 |
19 | prop 7395 | . . . . . . . . 9 | |
20 | prcdnql 7404 | . . . . . . . . 9 | |
21 | 19, 20 | sylan 281 | . . . . . . . 8 |
22 | 17, 18, 21 | syl2anc 409 | . . . . . . 7 |
23 | 15, 22 | mpd 13 | . . . . . 6 |
24 | 1, 3, 23 | jca32 308 | . . . . 5 |
25 | 24 | eximi 1580 | . . . 4 |
26 | ltexprlem.1 | . . . . . . . . . 10 | |
27 | 26 | ltexprlemell 7518 | . . . . . . . . 9 |
28 | 19.42v 1886 | . . . . . . . . 9 | |
29 | 27, 28 | bitr4i 186 | . . . . . . . 8 |
30 | 29 | anbi2i 453 | . . . . . . 7 |
31 | 19.42v 1886 | . . . . . . 7 | |
32 | 30, 31 | bitr4i 186 | . . . . . 6 |
33 | 32 | anbi2i 453 | . . . . 5 |
34 | 19.42v 1886 | . . . . 5 | |
35 | 33, 34 | bitr4i 186 | . . . 4 |
36 | 26 | ltexprlemell 7518 | . . . . 5 |
37 | 19.42v 1886 | . . . . 5 | |
38 | 36, 37 | bitr4i 186 | . . . 4 |
39 | 25, 35, 38 | 3imtr4i 200 | . . 3 |
40 | 39 | ex 114 | . 2 |
41 | 40 | rexlimdvw 2578 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wex 1472 wcel 2128 wrex 2436 crab 2439 cop 3563 class class class wbr 3965 cfv 5170 (class class class)co 5824 c1st 6086 c2nd 6087 cnq 7200 cplq 7202 cltq 7205 cnp 7211 cltp 7215 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-eprel 4249 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-oadd 6367 df-omul 6368 df-er 6480 df-ec 6482 df-qs 6486 df-ni 7224 df-pli 7225 df-mi 7226 df-lti 7227 df-plpq 7264 df-enq 7267 df-nqqs 7268 df-plqqs 7269 df-ltnqqs 7273 df-inp 7386 df-iltp 7390 |
This theorem is referenced by: ltexprlemrnd 7525 |
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