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Mirrors > Home > ILE Home > Th. List > mulresr | Unicode version |
Description: Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) |
Ref | Expression |
---|---|
mulresr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0r 7671 | . . 3 | |
2 | mulcnsr 7756 | . . . 4 | |
3 | 2 | an4s 578 | . . 3 |
4 | 1, 1, 3 | mpanr12 436 | . 2 |
5 | 00sr 7690 | . . . . . . . 8 | |
6 | 1, 5 | ax-mp 5 | . . . . . . 7 |
7 | 6 | oveq2i 5836 | . . . . . 6 |
8 | m1r 7673 | . . . . . . 7 | |
9 | 00sr 7690 | . . . . . . 7 | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 |
11 | 7, 10 | eqtri 2178 | . . . . 5 |
12 | 11 | oveq2i 5836 | . . . 4 |
13 | mulclsr 7675 | . . . . 5 | |
14 | 0idsr 7688 | . . . . 5 | |
15 | 13, 14 | syl 14 | . . . 4 |
16 | 12, 15 | syl5eq 2202 | . . 3 |
17 | mulcomsrg 7678 | . . . . . . 7 | |
18 | 1, 17 | mpan 421 | . . . . . 6 |
19 | 00sr 7690 | . . . . . 6 | |
20 | 18, 19 | eqtrd 2190 | . . . . 5 |
21 | 00sr 7690 | . . . . 5 | |
22 | 20, 21 | oveqan12rd 5845 | . . . 4 |
23 | 0idsr 7688 | . . . . 5 | |
24 | 1, 23 | ax-mp 5 | . . . 4 |
25 | 22, 24 | eqtrdi 2206 | . . 3 |
26 | 16, 25 | opeq12d 3750 | . 2 |
27 | 4, 26 | eqtrd 2190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 cop 3563 (class class class)co 5825 cnr 7218 c0r 7219 cm1r 7221 cplr 7222 cmr 7223 cmul 7738 |
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 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 |
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 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-eprel 4250 df-id 4254 df-po 4257 df-iso 4258 df-iord 4327 df-on 4329 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-irdg 6318 df-1o 6364 df-2o 6365 df-oadd 6368 df-omul 6369 df-er 6481 df-ec 6483 df-qs 6487 df-ni 7225 df-pli 7226 df-mi 7227 df-lti 7228 df-plpq 7265 df-mpq 7266 df-enq 7268 df-nqqs 7269 df-plqqs 7270 df-mqqs 7271 df-1nqqs 7272 df-rq 7273 df-ltnqqs 7274 df-enq0 7345 df-nq0 7346 df-0nq0 7347 df-plq0 7348 df-mq0 7349 df-inp 7387 df-i1p 7388 df-iplp 7389 df-imp 7390 df-enr 7647 df-nr 7648 df-plr 7649 df-mr 7650 df-0r 7652 df-m1r 7654 df-c 7739 df-mul 7745 |
This theorem is referenced by: recidpirq 7779 axmulrcl 7788 ax1rid 7798 axprecex 7801 axpre-mulgt0 7808 axpre-mulext 7809 |
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