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Mirrors > Home > ILE Home > Th. List > mulsrpr | Unicode version |
Description: Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
mulsrpr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 4652 | . . . 4 | |
2 | enrex 7711 | . . . . 5 | |
3 | 2 | ecelqsi 6579 | . . . 4 |
4 | 1, 3 | syl 14 | . . 3 |
5 | opelxpi 4652 | . . . 4 | |
6 | 2 | ecelqsi 6579 | . . . 4 |
7 | 5, 6 | syl 14 | . . 3 |
8 | 4, 7 | anim12i 338 | . 2 |
9 | eqid 2175 | . . . 4 | |
10 | eqid 2175 | . . . 4 | |
11 | 9, 10 | pm3.2i 272 | . . 3 |
12 | eqid 2175 | . . 3 | |
13 | opeq12 3776 | . . . . . . . . 9 | |
14 | 13 | eceq1d 6561 | . . . . . . . 8 |
15 | 14 | eqeq2d 2187 | . . . . . . 7 |
16 | 15 | anbi1d 465 | . . . . . 6 |
17 | simpl 109 | . . . . . . . . . . 11 | |
18 | 17 | oveq1d 5880 | . . . . . . . . . 10 |
19 | simpr 110 | . . . . . . . . . . 11 | |
20 | 19 | oveq1d 5880 | . . . . . . . . . 10 |
21 | 18, 20 | oveq12d 5883 | . . . . . . . . 9 |
22 | 17 | oveq1d 5880 | . . . . . . . . . 10 |
23 | 19 | oveq1d 5880 | . . . . . . . . . 10 |
24 | 22, 23 | oveq12d 5883 | . . . . . . . . 9 |
25 | 21, 24 | opeq12d 3782 | . . . . . . . 8 |
26 | 25 | eceq1d 6561 | . . . . . . 7 |
27 | 26 | eqeq2d 2187 | . . . . . 6 |
28 | 16, 27 | anbi12d 473 | . . . . 5 |
29 | 28 | spc2egv 2825 | . . . 4 |
30 | opeq12 3776 | . . . . . . . . . 10 | |
31 | 30 | eceq1d 6561 | . . . . . . . . 9 |
32 | 31 | eqeq2d 2187 | . . . . . . . 8 |
33 | 32 | anbi2d 464 | . . . . . . 7 |
34 | simpl 109 | . . . . . . . . . . . 12 | |
35 | 34 | oveq2d 5881 | . . . . . . . . . . 11 |
36 | simpr 110 | . . . . . . . . . . . 12 | |
37 | 36 | oveq2d 5881 | . . . . . . . . . . 11 |
38 | 35, 37 | oveq12d 5883 | . . . . . . . . . 10 |
39 | 36 | oveq2d 5881 | . . . . . . . . . . 11 |
40 | 34 | oveq2d 5881 | . . . . . . . . . . 11 |
41 | 39, 40 | oveq12d 5883 | . . . . . . . . . 10 |
42 | 38, 41 | opeq12d 3782 | . . . . . . . . 9 |
43 | 42 | eceq1d 6561 | . . . . . . . 8 |
44 | 43 | eqeq2d 2187 | . . . . . . 7 |
45 | 33, 44 | anbi12d 473 | . . . . . 6 |
46 | 45 | spc2egv 2825 | . . . . 5 |
47 | 46 | 2eximdv 1880 | . . . 4 |
48 | 29, 47 | sylan9 409 | . . 3 |
49 | 11, 12, 48 | mp2ani 432 | . 2 |
50 | ecexg 6529 | . . . 4 | |
51 | 2, 50 | ax-mp 5 | . . 3 |
52 | simp1 997 | . . . . . . . 8 | |
53 | 52 | eqeq1d 2184 | . . . . . . 7 |
54 | simp2 998 | . . . . . . . 8 | |
55 | 54 | eqeq1d 2184 | . . . . . . 7 |
56 | 53, 55 | anbi12d 473 | . . . . . 6 |
57 | simp3 999 | . . . . . . 7 | |
58 | 57 | eqeq1d 2184 | . . . . . 6 |
59 | 56, 58 | anbi12d 473 | . . . . 5 |
60 | 59 | 4exbidv 1868 | . . . 4 |
61 | mulsrmo 7718 | . . . 4 | |
62 | df-mr 7703 | . . . . 5 | |
63 | df-nr 7701 | . . . . . . . . 9 | |
64 | 63 | eleq2i 2242 | . . . . . . . 8 |
65 | 63 | eleq2i 2242 | . . . . . . . 8 |
66 | 64, 65 | anbi12i 460 | . . . . . . 7 |
67 | 66 | anbi1i 458 | . . . . . 6 |
68 | 67 | oprabbii 5920 | . . . . 5 |
69 | 62, 68 | eqtri 2196 | . . . 4 |
70 | 60, 61, 69 | ovig 5986 | . . 3 |
71 | 51, 70 | mp3an3 1326 | . 2 |
72 | 8, 49, 71 | sylc 62 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 w3a 978 wceq 1353 wex 1490 wcel 2146 cvv 2735 cop 3592 cxp 4618 (class class class)co 5865 coprab 5866 cec 6523 cqs 6524 cnp 7265 cpp 7267 cmp 7268 cer 7270 cnr 7271 cmr 7276 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-eprel 4283 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-irdg 6361 df-1o 6407 df-2o 6408 df-oadd 6411 df-omul 6412 df-er 6525 df-ec 6527 df-qs 6531 df-ni 7278 df-pli 7279 df-mi 7280 df-lti 7281 df-plpq 7318 df-mpq 7319 df-enq 7321 df-nqqs 7322 df-plqqs 7323 df-mqqs 7324 df-1nqqs 7325 df-rq 7326 df-ltnqqs 7327 df-enq0 7398 df-nq0 7399 df-0nq0 7400 df-plq0 7401 df-mq0 7402 df-inp 7440 df-iplp 7442 df-imp 7443 df-enr 7700 df-nr 7701 df-mr 7703 |
This theorem is referenced by: mulclsr 7728 mulcomsrg 7731 mulasssrg 7732 distrsrg 7733 m1m1sr 7735 1idsr 7742 00sr 7743 recexgt0sr 7747 mulgt0sr 7752 mulextsr1 7755 recidpirq 7832 |
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