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Mirrors > Home > ILE Home > Th. List > addsrpr | Unicode version |
Description: Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
addsrpr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 4394 | . . . 4 | |
2 | enrex 6914 | . . . . 5 | |
3 | 2 | ecelqsi 6183 | . . . 4 |
4 | 1, 3 | syl 14 | . . 3 |
5 | opelxpi 4394 | . . . 4 | |
6 | 2 | ecelqsi 6183 | . . . 4 |
7 | 5, 6 | syl 14 | . . 3 |
8 | 4, 7 | anim12i 331 | . 2 |
9 | eqid 2081 | . . . 4 | |
10 | eqid 2081 | . . . 4 | |
11 | 9, 10 | pm3.2i 266 | . . 3 |
12 | eqid 2081 | . . 3 | |
13 | opeq12 3572 | . . . . . . . . 9 | |
14 | 13 | eceq1d 6165 | . . . . . . . 8 |
15 | 14 | eqeq2d 2092 | . . . . . . 7 |
16 | 15 | anbi1d 452 | . . . . . 6 |
17 | simpl 107 | . . . . . . . . . 10 | |
18 | 17 | oveq1d 5547 | . . . . . . . . 9 |
19 | simpr 108 | . . . . . . . . . 10 | |
20 | 19 | oveq1d 5547 | . . . . . . . . 9 |
21 | 18, 20 | opeq12d 3578 | . . . . . . . 8 |
22 | 21 | eceq1d 6165 | . . . . . . 7 |
23 | 22 | eqeq2d 2092 | . . . . . 6 |
24 | 16, 23 | anbi12d 456 | . . . . 5 |
25 | 24 | spc2egv 2687 | . . . 4 |
26 | opeq12 3572 | . . . . . . . . . 10 | |
27 | 26 | eceq1d 6165 | . . . . . . . . 9 |
28 | 27 | eqeq2d 2092 | . . . . . . . 8 |
29 | 28 | anbi2d 451 | . . . . . . 7 |
30 | simpl 107 | . . . . . . . . . . 11 | |
31 | 30 | oveq2d 5548 | . . . . . . . . . 10 |
32 | simpr 108 | . . . . . . . . . . 11 | |
33 | 32 | oveq2d 5548 | . . . . . . . . . 10 |
34 | 31, 33 | opeq12d 3578 | . . . . . . . . 9 |
35 | 34 | eceq1d 6165 | . . . . . . . 8 |
36 | 35 | eqeq2d 2092 | . . . . . . 7 |
37 | 29, 36 | anbi12d 456 | . . . . . 6 |
38 | 37 | spc2egv 2687 | . . . . 5 |
39 | 38 | 2eximdv 1803 | . . . 4 |
40 | 25, 39 | sylan9 401 | . . 3 |
41 | 11, 12, 40 | mp2ani 422 | . 2 |
42 | ecexg 6133 | . . . 4 | |
43 | 2, 42 | ax-mp 7 | . . 3 |
44 | simp1 938 | . . . . . . . 8 | |
45 | 44 | eqeq1d 2089 | . . . . . . 7 |
46 | simp2 939 | . . . . . . . 8 | |
47 | 46 | eqeq1d 2089 | . . . . . . 7 |
48 | 45, 47 | anbi12d 456 | . . . . . 6 |
49 | simp3 940 | . . . . . . 7 | |
50 | 49 | eqeq1d 2089 | . . . . . 6 |
51 | 48, 50 | anbi12d 456 | . . . . 5 |
52 | 51 | 4exbidv 1791 | . . . 4 |
53 | addsrmo 6920 | . . . 4 | |
54 | df-plr 6905 | . . . . 5 | |
55 | df-nr 6904 | . . . . . . . . 9 | |
56 | 55 | eleq2i 2145 | . . . . . . . 8 |
57 | 55 | eleq2i 2145 | . . . . . . . 8 |
58 | 56, 57 | anbi12i 447 | . . . . . . 7 |
59 | 58 | anbi1i 445 | . . . . . 6 |
60 | 59 | oprabbii 5580 | . . . . 5 |
61 | 54, 60 | eqtri 2101 | . . . 4 |
62 | 52, 53, 61 | ovig 5642 | . . 3 |
63 | 43, 62 | mp3an3 1257 | . 2 |
64 | 8, 41, 63 | sylc 61 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wceq 1284 wex 1421 wcel 1433 cvv 2601 cop 3401 cxp 4361 (class class class)co 5532 coprab 5533 cec 6127 cqs 6128 cnp 6481 cpp 6483 cer 6486 cnr 6487 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-iplp 6658 df-enr 6903 df-nr 6904 df-plr 6905 |
This theorem is referenced by: addclsr 6930 addcomsrg 6932 addasssrg 6933 distrsrg 6936 m1p1sr 6937 0idsr 6944 ltasrg 6947 prsradd 6962 pitonnlem2 7015 |
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