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Mirrors > Home > ILE Home > Th. List > mulcomsrg | Unicode version |
Description: Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Ref | Expression |
---|---|
mulcomsrg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 7649 | . 2 | |
2 | mulsrpr 7668 | . 2 | |
3 | mulsrpr 7668 | . 2 | |
4 | mulcomprg 7502 | . . . 4 | |
5 | 4 | ad2ant2r 501 | . . 3 |
6 | mulcomprg 7502 | . . . 4 | |
7 | 6 | ad2ant2l 500 | . . 3 |
8 | 5, 7 | oveq12d 5844 | . 2 |
9 | mulcomprg 7502 | . . . . 5 | |
10 | 9 | ad2ant2rl 503 | . . . 4 |
11 | mulcomprg 7502 | . . . . 5 | |
12 | 11 | ad2ant2lr 502 | . . . 4 |
13 | 10, 12 | oveq12d 5844 | . . 3 |
14 | mulclpr 7494 | . . . . . 6 | |
15 | 14 | ancoms 266 | . . . . 5 |
16 | 15 | ad2ant2rl 503 | . . . 4 |
17 | mulclpr 7494 | . . . . . 6 | |
18 | 17 | ancoms 266 | . . . . 5 |
19 | 18 | ad2ant2lr 502 | . . . 4 |
20 | addcomprg 7500 | . . . 4 | |
21 | 16, 19, 20 | syl2anc 409 | . . 3 |
22 | 13, 21 | eqtrd 2190 | . 2 |
23 | 1, 2, 3, 8, 22 | ecovicom 6590 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 (class class class)co 5826 cnp 7213 cpp 7215 cmp 7216 cer 7218 cnr 7219 cmr 7224 |
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 4081 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-iinf 4549 |
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 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-tr 4065 df-eprel 4251 df-id 4255 df-po 4258 df-iso 4259 df-iord 4328 df-on 4330 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-ov 5829 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-recs 6254 df-irdg 6319 df-1o 6365 df-2o 6366 df-oadd 6369 df-omul 6370 df-er 6482 df-ec 6484 df-qs 6488 df-ni 7226 df-pli 7227 df-mi 7228 df-lti 7229 df-plpq 7266 df-mpq 7267 df-enq 7269 df-nqqs 7270 df-plqqs 7271 df-mqqs 7272 df-1nqqs 7273 df-rq 7274 df-ltnqqs 7275 df-enq0 7346 df-nq0 7347 df-0nq0 7348 df-plq0 7349 df-mq0 7350 df-inp 7388 df-iplp 7390 df-imp 7391 df-enr 7648 df-nr 7649 df-mr 7651 |
This theorem is referenced by: mulresr 7760 axmulcom 7793 axmulass 7795 axcnre 7803 |
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