Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > recexprlempr | Unicode version |
Description: is a positive real. Lemma for recexpr 7589. (Contributed by Jim Kingdon, 27-Dec-2019.) |
Ref | Expression |
---|---|
recexpr.1 |
Ref | Expression |
---|---|
recexprlempr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recexpr.1 | . . . 4 | |
2 | 1 | recexprlemm 7575 | . . 3 |
3 | ltrelnq 7316 | . . . . . . . . . . 11 | |
4 | 3 | brel 4661 | . . . . . . . . . 10 |
5 | 4 | simpld 111 | . . . . . . . . 9 |
6 | 5 | adantr 274 | . . . . . . . 8 |
7 | 6 | exlimiv 1591 | . . . . . . 7 |
8 | 7 | abssi 3222 | . . . . . 6 |
9 | nqex 7314 | . . . . . . 7 | |
10 | 9 | elpw2 4141 | . . . . . 6 |
11 | 8, 10 | mpbir 145 | . . . . 5 |
12 | 3 | brel 4661 | . . . . . . . . . 10 |
13 | 12 | simprd 113 | . . . . . . . . 9 |
14 | 13 | adantr 274 | . . . . . . . 8 |
15 | 14 | exlimiv 1591 | . . . . . . 7 |
16 | 15 | abssi 3222 | . . . . . 6 |
17 | 9 | elpw2 4141 | . . . . . 6 |
18 | 16, 17 | mpbir 145 | . . . . 5 |
19 | opelxpi 4641 | . . . . 5 | |
20 | 11, 18, 19 | mp2an 424 | . . . 4 |
21 | 1, 20 | eqeltri 2243 | . . 3 |
22 | 2, 21 | jctil 310 | . 2 |
23 | 1 | recexprlemrnd 7580 | . . 3 |
24 | 1 | recexprlemdisj 7581 | . . 3 |
25 | 1 | recexprlemloc 7582 | . . 3 |
26 | 23, 24, 25 | 3jca 1172 | . 2 |
27 | elnp1st2nd 7427 | . 2 | |
28 | 22, 26, 27 | sylanbrc 415 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 703 w3a 973 wceq 1348 wex 1485 wcel 2141 cab 2156 wral 2448 wrex 2449 wss 3121 cpw 3564 cop 3584 class class class wbr 3987 cxp 4607 cfv 5196 c1st 6115 c2nd 6116 cnq 7231 crq 7235 cltq 7236 cnp 7242 |
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-oadd 6397 df-omul 6398 df-er 6510 df-ec 6512 df-qs 6516 df-ni 7255 df-pli 7256 df-mi 7257 df-lti 7258 df-plpq 7295 df-mpq 7296 df-enq 7298 df-nqqs 7299 df-plqqs 7300 df-mqqs 7301 df-1nqqs 7302 df-rq 7303 df-ltnqqs 7304 df-inp 7417 |
This theorem is referenced by: recexprlem1ssl 7584 recexprlem1ssu 7585 recexprlemss1l 7586 recexprlemss1u 7587 recexprlemex 7588 recexpr 7589 |
Copyright terms: Public domain | W3C validator |