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Mirrors > Home > ILE Home > Th. List > ltexprlempr | Unicode version |
Description: Our constructed difference is a positive real. Lemma for ltexpri 7548. (Contributed by Jim Kingdon, 17-Dec-2019.) |
Ref | Expression |
---|---|
ltexprlem.1 |
Ref | Expression |
---|---|
ltexprlempr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltexprlem.1 | . . . 4 | |
2 | 1 | ltexprlemm 7535 | . . 3 |
3 | ssrab2 3225 | . . . . . 6 | |
4 | nqex 7298 | . . . . . . 7 | |
5 | 4 | elpw2 4133 | . . . . . 6 |
6 | 3, 5 | mpbir 145 | . . . . 5 |
7 | ssrab2 3225 | . . . . . 6 | |
8 | 4 | elpw2 4133 | . . . . . 6 |
9 | 7, 8 | mpbir 145 | . . . . 5 |
10 | opelxpi 4633 | . . . . 5 | |
11 | 6, 9, 10 | mp2an 423 | . . . 4 |
12 | 1, 11 | eqeltri 2237 | . . 3 |
13 | 2, 12 | jctil 310 | . 2 |
14 | 1 | ltexprlemrnd 7540 | . . 3 |
15 | 1 | ltexprlemdisj 7541 | . . 3 |
16 | 1 | ltexprlemloc 7542 | . . 3 |
17 | 14, 15, 16 | 3jca 1166 | . 2 |
18 | elnp1st2nd 7411 | . 2 | |
19 | 13, 17, 18 | sylanbrc 414 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 w3a 967 wceq 1342 wex 1479 wcel 2135 wral 2442 wrex 2443 crab 2446 wss 3114 cpw 3556 cop 3576 class class class wbr 3979 cxp 4599 cfv 5185 (class class class)co 5839 c1st 6101 c2nd 6102 cnq 7215 cplq 7217 cltq 7220 cnp 7226 cltp 7230 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4094 ax-sep 4097 ax-nul 4105 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-iinf 4562 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2726 df-sbc 2950 df-csb 3044 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-int 3822 df-iun 3865 df-br 3980 df-opab 4041 df-mpt 4042 df-tr 4078 df-eprel 4264 df-id 4268 df-po 4271 df-iso 4272 df-iord 4341 df-on 4343 df-suc 4346 df-iom 4565 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-f1 5190 df-fo 5191 df-f1o 5192 df-fv 5193 df-ov 5842 df-oprab 5843 df-mpo 5844 df-1st 6103 df-2nd 6104 df-recs 6267 df-irdg 6332 df-1o 6378 df-2o 6379 df-oadd 6382 df-omul 6383 df-er 6495 df-ec 6497 df-qs 6501 df-ni 7239 df-pli 7240 df-mi 7241 df-lti 7242 df-plpq 7279 df-mpq 7280 df-enq 7282 df-nqqs 7283 df-plqqs 7284 df-mqqs 7285 df-1nqqs 7286 df-rq 7287 df-ltnqqs 7288 df-enq0 7359 df-nq0 7360 df-0nq0 7361 df-plq0 7362 df-mq0 7363 df-inp 7401 df-iltp 7405 |
This theorem is referenced by: ltexprlemfl 7544 ltexprlemrl 7545 ltexprlemfu 7546 ltexprlemru 7547 ltexpri 7548 |
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