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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 6744 | . . . 4 | |
2 | addclpr 6727 | . . . 4 | |
3 | 1, 2 | mpan2 415 | . . 3 |
4 | addclpr 6727 | . . . 4 | |
5 | 1, 4 | mpan2 415 | . . 3 |
6 | addsrpr 6922 | . . . . 5 | |
7 | 1, 6 | mpanl2 425 | . . . 4 |
8 | 1, 7 | mpanr2 428 | . . 3 |
9 | 3, 5, 8 | syl2an 283 | . 2 |
10 | simpl 107 | . . . . . . 7 | |
11 | 1 | a1i 9 | . . . . . . 7 |
12 | simpr 108 | . . . . . . 7 | |
13 | addcomprg 6768 | . . . . . . . 8 | |
14 | 13 | adantl 271 | . . . . . . 7 |
15 | addassprg 6769 | . . . . . . . 8 | |
16 | 15 | adantl 271 | . . . . . . 7 |
17 | addclpr 6727 | . . . . . . . 8 | |
18 | 17 | adantl 271 | . . . . . . 7 |
19 | 10, 11, 12, 14, 16, 11, 18 | caov4d 5705 | . . . . . 6 |
20 | addclpr 6727 | . . . . . . 7 | |
21 | addclpr 6727 | . . . . . . . . 9 | |
22 | 1, 1, 21 | mp2an 416 | . . . . . . . 8 |
23 | 22 | a1i 9 | . . . . . . 7 |
24 | addcomprg 6768 | . . . . . . 7 | |
25 | 20, 23, 24 | syl2anc 403 | . . . . . 6 |
26 | 19, 25 | eqtrd 2113 | . . . . 5 |
27 | 26 | oveq1d 5547 | . . . 4 |
28 | addassprg 6769 | . . . . 5 | |
29 | 23, 20, 11, 28 | syl3anc 1169 | . . . 4 |
30 | 27, 29 | eqtrd 2113 | . . 3 |
31 | addclpr 6727 | . . . . 5 | |
32 | 3, 5, 31 | syl2an 283 | . . . 4 |
33 | addclpr 6727 | . . . . 5 | |
34 | 20, 11, 33 | syl2anc 403 | . . . 4 |
35 | enreceq 6913 | . . . . . 6 | |
36 | 1, 35 | mpanr2 428 | . . . . 5 |
37 | 22, 36 | mpanl2 425 | . . . 4 |
38 | 32, 34, 37 | syl2anc 403 | . . 3 |
39 | 30, 38 | mpbird 165 | . 2 |
40 | 9, 39 | eqtr2d 2114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wceq 1284 wcel 1433 cop 3401 (class class class)co 5532 cec 6127 cnp 6481 c1p 6482 cpp 6483 cer 6486 cplr 6491 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-i1p 6657 df-iplp 6658 df-enr 6903 df-nr 6904 df-plr 6905 |
This theorem is referenced by: caucvgsrlemcau 6969 caucvgsrlemgt1 6971 |
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