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Mirrors > Home > ILE Home > Th. List > prsradd | Unicode version |
Description: Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Ref | Expression |
---|---|
prsradd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1pr 7469 | . . . 4 | |
2 | addclpr 7452 | . . . 4 | |
3 | 1, 2 | mpan2 422 | . . 3 |
4 | addclpr 7452 | . . . 4 | |
5 | 1, 4 | mpan2 422 | . . 3 |
6 | addsrpr 7660 | . . . . 5 | |
7 | 1, 6 | mpanl2 432 | . . . 4 |
8 | 1, 7 | mpanr2 435 | . . 3 |
9 | 3, 5, 8 | syl2an 287 | . 2 |
10 | simpl 108 | . . . . . . 7 | |
11 | 1 | a1i 9 | . . . . . . 7 |
12 | simpr 109 | . . . . . . 7 | |
13 | addcomprg 7493 | . . . . . . . 8 | |
14 | 13 | adantl 275 | . . . . . . 7 |
15 | addassprg 7494 | . . . . . . . 8 | |
16 | 15 | adantl 275 | . . . . . . 7 |
17 | addclpr 7452 | . . . . . . . 8 | |
18 | 17 | adantl 275 | . . . . . . 7 |
19 | 10, 11, 12, 14, 16, 11, 18 | caov4d 6002 | . . . . . 6 |
20 | addclpr 7452 | . . . . . . 7 | |
21 | addclpr 7452 | . . . . . . . . 9 | |
22 | 1, 1, 21 | mp2an 423 | . . . . . . . 8 |
23 | 22 | a1i 9 | . . . . . . 7 |
24 | addcomprg 7493 | . . . . . . 7 | |
25 | 20, 23, 24 | syl2anc 409 | . . . . . 6 |
26 | 19, 25 | eqtrd 2190 | . . . . 5 |
27 | 26 | oveq1d 5836 | . . . 4 |
28 | addassprg 7494 | . . . . 5 | |
29 | 23, 20, 11, 28 | syl3anc 1220 | . . . 4 |
30 | 27, 29 | eqtrd 2190 | . . 3 |
31 | addclpr 7452 | . . . . 5 | |
32 | 3, 5, 31 | syl2an 287 | . . . 4 |
33 | addclpr 7452 | . . . . 5 | |
34 | 20, 11, 33 | syl2anc 409 | . . . 4 |
35 | enreceq 7651 | . . . . . 6 | |
36 | 1, 35 | mpanr2 435 | . . . . 5 |
37 | 22, 36 | mpanl2 432 | . . . 4 |
38 | 32, 34, 37 | syl2anc 409 | . . 3 |
39 | 30, 38 | mpbird 166 | . 2 |
40 | 9, 39 | eqtr2d 2191 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wceq 1335 wcel 2128 cop 3563 (class class class)co 5821 cec 6475 cnp 7206 c1p 7207 cpp 7208 cer 7211 cplr 7216 |
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 4495 ax-iinf 4546 |
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-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-irdg 6314 df-1o 6360 df-2o 6361 df-oadd 6364 df-omul 6365 df-er 6477 df-ec 6479 df-qs 6483 df-ni 7219 df-pli 7220 df-mi 7221 df-lti 7222 df-plpq 7259 df-mpq 7260 df-enq 7262 df-nqqs 7263 df-plqqs 7264 df-mqqs 7265 df-1nqqs 7266 df-rq 7267 df-ltnqqs 7268 df-enq0 7339 df-nq0 7340 df-0nq0 7341 df-plq0 7342 df-mq0 7343 df-inp 7381 df-i1p 7382 df-iplp 7383 df-enr 7641 df-nr 7642 df-plr 7643 |
This theorem is referenced by: caucvgsrlemcau 7708 caucvgsrlemgt1 7710 |
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