Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4571 | . . . 4 | |
2 | enrex 7545 | . . . . 5 | |
3 | 2 | ecelqsi 6483 | . . . 4 |
4 | 1, 3 | syl 14 | . . 3 |
5 | opelxpi 4571 | . . . 4 | |
6 | 2 | ecelqsi 6483 | . . . 4 |
7 | 5, 6 | syl 14 | . . 3 |
8 | 4, 7 | anim12i 336 | . 2 |
9 | eqid 2139 | . . . 4 | |
10 | eqid 2139 | . . . 4 | |
11 | 9, 10 | pm3.2i 270 | . . 3 |
12 | eqid 2139 | . . 3 | |
13 | opeq12 3707 | . . . . . . . . 9 | |
14 | 13 | eceq1d 6465 | . . . . . . . 8 |
15 | 14 | eqeq2d 2151 | . . . . . . 7 |
16 | 15 | anbi1d 460 | . . . . . 6 |
17 | simpl 108 | . . . . . . . . . 10 | |
18 | 17 | oveq1d 5789 | . . . . . . . . 9 |
19 | simpr 109 | . . . . . . . . . 10 | |
20 | 19 | oveq1d 5789 | . . . . . . . . 9 |
21 | 18, 20 | opeq12d 3713 | . . . . . . . 8 |
22 | 21 | eceq1d 6465 | . . . . . . 7 |
23 | 22 | eqeq2d 2151 | . . . . . 6 |
24 | 16, 23 | anbi12d 464 | . . . . 5 |
25 | 24 | spc2egv 2775 | . . . 4 |
26 | opeq12 3707 | . . . . . . . . . 10 | |
27 | 26 | eceq1d 6465 | . . . . . . . . 9 |
28 | 27 | eqeq2d 2151 | . . . . . . . 8 |
29 | 28 | anbi2d 459 | . . . . . . 7 |
30 | simpl 108 | . . . . . . . . . . 11 | |
31 | 30 | oveq2d 5790 | . . . . . . . . . 10 |
32 | simpr 109 | . . . . . . . . . . 11 | |
33 | 32 | oveq2d 5790 | . . . . . . . . . 10 |
34 | 31, 33 | opeq12d 3713 | . . . . . . . . 9 |
35 | 34 | eceq1d 6465 | . . . . . . . 8 |
36 | 35 | eqeq2d 2151 | . . . . . . 7 |
37 | 29, 36 | anbi12d 464 | . . . . . 6 |
38 | 37 | spc2egv 2775 | . . . . 5 |
39 | 38 | 2eximdv 1854 | . . . 4 |
40 | 25, 39 | sylan9 406 | . . 3 |
41 | 11, 12, 40 | mp2ani 428 | . 2 |
42 | ecexg 6433 | . . . 4 | |
43 | 2, 42 | ax-mp 5 | . . 3 |
44 | simp1 981 | . . . . . . . 8 | |
45 | 44 | eqeq1d 2148 | . . . . . . 7 |
46 | simp2 982 | . . . . . . . 8 | |
47 | 46 | eqeq1d 2148 | . . . . . . 7 |
48 | 45, 47 | anbi12d 464 | . . . . . 6 |
49 | simp3 983 | . . . . . . 7 | |
50 | 49 | eqeq1d 2148 | . . . . . 6 |
51 | 48, 50 | anbi12d 464 | . . . . 5 |
52 | 51 | 4exbidv 1842 | . . . 4 |
53 | addsrmo 7551 | . . . 4 | |
54 | df-plr 7536 | . . . . 5 | |
55 | df-nr 7535 | . . . . . . . . 9 | |
56 | 55 | eleq2i 2206 | . . . . . . . 8 |
57 | 55 | eleq2i 2206 | . . . . . . . 8 |
58 | 56, 57 | anbi12i 455 | . . . . . . 7 |
59 | 58 | anbi1i 453 | . . . . . 6 |
60 | 59 | oprabbii 5826 | . . . . 5 |
61 | 54, 60 | eqtri 2160 | . . . 4 |
62 | 52, 53, 61 | ovig 5892 | . . 3 |
63 | 43, 62 | mp3an3 1304 | . 2 |
64 | 8, 41, 63 | sylc 62 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wex 1468 wcel 1480 cvv 2686 cop 3530 cxp 4537 (class class class)co 5774 coprab 5775 cec 6427 cqs 6428 cnp 7099 cpp 7101 cer 7104 cnr 7105 cplr 7109 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-2o 6314 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-enq0 7232 df-nq0 7233 df-0nq0 7234 df-plq0 7235 df-mq0 7236 df-inp 7274 df-iplp 7276 df-enr 7534 df-nr 7535 df-plr 7536 |
This theorem is referenced by: addclsr 7561 addcomsrg 7563 addasssrg 7564 distrsrg 7567 m1p1sr 7568 0idsr 7575 ltasrg 7578 prsradd 7594 pitonnlem2 7655 |
Copyright terms: Public domain | W3C validator |