Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > srpospr | Unicode version |
Description: Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Ref | Expression |
---|---|
srpospr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 7649 | . . 3 | |
2 | breq2 3971 | . . . 4 | |
3 | eqeq2 2167 | . . . . 5 | |
4 | 3 | reubidv 2640 | . . . 4 |
5 | 2, 4 | imbi12d 233 | . . 3 |
6 | gt0srpr 7670 | . . . . . . . 8 | |
7 | 6 | biimpi 119 | . . . . . . 7 |
8 | 7 | adantl 275 | . . . . . 6 |
9 | lteupri 7539 | . . . . . 6 | |
10 | 8, 9 | syl 14 | . . . . 5 |
11 | simpr 109 | . . . . . . . . 9 | |
12 | 1pr 7476 | . . . . . . . . . 10 | |
13 | 12 | a1i 9 | . . . . . . . . 9 |
14 | addclpr 7459 | . . . . . . . . 9 | |
15 | 11, 13, 14 | syl2anc 409 | . . . . . . . 8 |
16 | simplll 523 | . . . . . . . 8 | |
17 | simpllr 524 | . . . . . . . 8 | |
18 | enreceq 7658 | . . . . . . . 8 | |
19 | 15, 13, 16, 17, 18 | syl22anc 1221 | . . . . . . 7 |
20 | addcomprg 7500 | . . . . . . . . . . . 12 | |
21 | 11, 13, 20 | syl2anc 409 | . . . . . . . . . . 11 |
22 | 21 | oveq1d 5841 | . . . . . . . . . 10 |
23 | addassprg 7501 | . . . . . . . . . . 11 | |
24 | 13, 11, 17, 23 | syl3anc 1220 | . . . . . . . . . 10 |
25 | 22, 24 | eqtrd 2190 | . . . . . . . . 9 |
26 | 25 | eqeq1d 2166 | . . . . . . . 8 |
27 | addclpr 7459 | . . . . . . . . . . 11 | |
28 | 11, 17, 27 | syl2anc 409 | . . . . . . . . . 10 |
29 | addcanprg 7538 | . . . . . . . . . 10 | |
30 | 13, 28, 16, 29 | syl3anc 1220 | . . . . . . . . 9 |
31 | oveq2 5834 | . . . . . . . . 9 | |
32 | 30, 31 | impbid1 141 | . . . . . . . 8 |
33 | 26, 32 | bitrd 187 | . . . . . . 7 |
34 | addcomprg 7500 | . . . . . . . . 9 | |
35 | 11, 17, 34 | syl2anc 409 | . . . . . . . 8 |
36 | 35 | eqeq1d 2166 | . . . . . . 7 |
37 | 19, 33, 36 | 3bitrrd 214 | . . . . . 6 |
38 | 37 | reubidva 2639 | . . . . 5 |
39 | 10, 38 | mpbid 146 | . . . 4 |
40 | 39 | ex 114 | . . 3 |
41 | 1, 5, 40 | ecoptocl 6569 | . 2 |
42 | 41 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 wreu 2437 cop 3564 class class class wbr 3967 (class class class)co 5826 cec 6480 cnp 7213 c1p 7214 cpp 7215 cltp 7217 cer 7218 cnr 7219 c0r 7220 cltr 7225 |
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 4081 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-iinf 4549 |
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-rmo 2443 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 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-tr 4065 df-eprel 4251 df-id 4255 df-po 4258 df-iso 4259 df-iord 4328 df-on 4330 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-ov 5829 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-recs 6254 df-irdg 6319 df-1o 6365 df-2o 6366 df-oadd 6369 df-omul 6370 df-er 6482 df-ec 6484 df-qs 6488 df-ni 7226 df-pli 7227 df-mi 7228 df-lti 7229 df-plpq 7266 df-mpq 7267 df-enq 7269 df-nqqs 7270 df-plqqs 7271 df-mqqs 7272 df-1nqqs 7273 df-rq 7274 df-ltnqqs 7275 df-enq0 7346 df-nq0 7347 df-0nq0 7348 df-plq0 7349 df-mq0 7350 df-inp 7388 df-i1p 7389 df-iplp 7390 df-iltp 7392 df-enr 7648 df-nr 7649 df-ltr 7652 df-0r 7653 |
This theorem is referenced by: prsrriota 7710 caucvgsrlemcl 7711 caucvgsrlemgt1 7717 |
Copyright terms: Public domain | W3C validator |