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 4618 | . . . 4 | |
2 | enrex 7657 | . . . . 5 | |
3 | 2 | ecelqsi 6534 | . . . 4 |
4 | 1, 3 | syl 14 | . . 3 |
5 | opelxpi 4618 | . . . 4 | |
6 | 2 | ecelqsi 6534 | . . . 4 |
7 | 5, 6 | syl 14 | . . 3 |
8 | 4, 7 | anim12i 336 | . 2 |
9 | eqid 2157 | . . . 4 | |
10 | eqid 2157 | . . . 4 | |
11 | 9, 10 | pm3.2i 270 | . . 3 |
12 | eqid 2157 | . . 3 | |
13 | opeq12 3743 | . . . . . . . . 9 | |
14 | 13 | eceq1d 6516 | . . . . . . . 8 |
15 | 14 | eqeq2d 2169 | . . . . . . 7 |
16 | 15 | anbi1d 461 | . . . . . 6 |
17 | simpl 108 | . . . . . . . . . 10 | |
18 | 17 | oveq1d 5839 | . . . . . . . . 9 |
19 | simpr 109 | . . . . . . . . . 10 | |
20 | 19 | oveq1d 5839 | . . . . . . . . 9 |
21 | 18, 20 | opeq12d 3749 | . . . . . . . 8 |
22 | 21 | eceq1d 6516 | . . . . . . 7 |
23 | 22 | eqeq2d 2169 | . . . . . 6 |
24 | 16, 23 | anbi12d 465 | . . . . 5 |
25 | 24 | spc2egv 2802 | . . . 4 |
26 | opeq12 3743 | . . . . . . . . . 10 | |
27 | 26 | eceq1d 6516 | . . . . . . . . 9 |
28 | 27 | eqeq2d 2169 | . . . . . . . 8 |
29 | 28 | anbi2d 460 | . . . . . . 7 |
30 | simpl 108 | . . . . . . . . . . 11 | |
31 | 30 | oveq2d 5840 | . . . . . . . . . 10 |
32 | simpr 109 | . . . . . . . . . . 11 | |
33 | 32 | oveq2d 5840 | . . . . . . . . . 10 |
34 | 31, 33 | opeq12d 3749 | . . . . . . . . 9 |
35 | 34 | eceq1d 6516 | . . . . . . . 8 |
36 | 35 | eqeq2d 2169 | . . . . . . 7 |
37 | 29, 36 | anbi12d 465 | . . . . . 6 |
38 | 37 | spc2egv 2802 | . . . . 5 |
39 | 38 | 2eximdv 1862 | . . . 4 |
40 | 25, 39 | sylan9 407 | . . 3 |
41 | 11, 12, 40 | mp2ani 429 | . 2 |
42 | ecexg 6484 | . . . 4 | |
43 | 2, 42 | ax-mp 5 | . . 3 |
44 | simp1 982 | . . . . . . . 8 | |
45 | 44 | eqeq1d 2166 | . . . . . . 7 |
46 | simp2 983 | . . . . . . . 8 | |
47 | 46 | eqeq1d 2166 | . . . . . . 7 |
48 | 45, 47 | anbi12d 465 | . . . . . 6 |
49 | simp3 984 | . . . . . . 7 | |
50 | 49 | eqeq1d 2166 | . . . . . 6 |
51 | 48, 50 | anbi12d 465 | . . . . 5 |
52 | 51 | 4exbidv 1850 | . . . 4 |
53 | addsrmo 7663 | . . . 4 | |
54 | df-plr 7648 | . . . . 5 | |
55 | df-nr 7647 | . . . . . . . . 9 | |
56 | 55 | eleq2i 2224 | . . . . . . . 8 |
57 | 55 | eleq2i 2224 | . . . . . . . 8 |
58 | 56, 57 | anbi12i 456 | . . . . . . 7 |
59 | 58 | anbi1i 454 | . . . . . 6 |
60 | 59 | oprabbii 5876 | . . . . 5 |
61 | 54, 60 | eqtri 2178 | . . . 4 |
62 | 52, 53, 61 | ovig 5942 | . . 3 |
63 | 43, 62 | mp3an3 1308 | . 2 |
64 | 8, 41, 63 | sylc 62 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 963 wceq 1335 wex 1472 wcel 2128 cvv 2712 cop 3563 cxp 4584 (class class class)co 5824 coprab 5825 cec 6478 cqs 6479 cnp 7211 cpp 7213 cer 7216 cnr 7217 cplr 7221 |
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 4496 ax-iinf 4547 |
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 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-1o 6363 df-2o 6364 df-oadd 6367 df-omul 6368 df-er 6480 df-ec 6482 df-qs 6486 df-ni 7224 df-pli 7225 df-mi 7226 df-lti 7227 df-plpq 7264 df-mpq 7265 df-enq 7267 df-nqqs 7268 df-plqqs 7269 df-mqqs 7270 df-1nqqs 7271 df-rq 7272 df-ltnqqs 7273 df-enq0 7344 df-nq0 7345 df-0nq0 7346 df-plq0 7347 df-mq0 7348 df-inp 7386 df-iplp 7388 df-enr 7646 df-nr 7647 df-plr 7648 |
This theorem is referenced by: addclsr 7673 addcomsrg 7675 addasssrg 7676 distrsrg 7679 m1p1sr 7680 0idsr 7687 ltasrg 7690 prsradd 7706 pitonnlem2 7767 |
Copyright terms: Public domain | W3C validator |