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Mirrors > Home > ILE Home > Th. List > distrsrg | Unicode version |
Description: Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.) |
Ref | Expression |
---|---|
distrsrg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 7701 | . 2 | |
2 | addsrpr 7719 | . 2 | |
3 | mulsrpr 7720 | . 2 | |
4 | mulsrpr 7720 | . 2 | |
5 | mulsrpr 7720 | . 2 | |
6 | addsrpr 7719 | . 2 | |
7 | addclpr 7511 | . . . 4 | |
8 | 7 | ad2ant2r 509 | . . 3 |
9 | addclpr 7511 | . . . 4 | |
10 | 9 | ad2ant2l 508 | . . 3 |
11 | 8, 10 | jca 306 | . 2 |
12 | mulclpr 7546 | . . . . 5 | |
13 | 12 | ad2ant2r 509 | . . . 4 |
14 | mulclpr 7546 | . . . . 5 | |
15 | 14 | ad2ant2l 508 | . . . 4 |
16 | addclpr 7511 | . . . 4 | |
17 | 13, 15, 16 | syl2anc 411 | . . 3 |
18 | mulclpr 7546 | . . . . 5 | |
19 | 18 | ad2ant2rl 511 | . . . 4 |
20 | mulclpr 7546 | . . . . 5 | |
21 | 20 | ad2ant2lr 510 | . . . 4 |
22 | addclpr 7511 | . . . 4 | |
23 | 19, 21, 22 | syl2anc 411 | . . 3 |
24 | 17, 23 | jca 306 | . 2 |
25 | mulclpr 7546 | . . . . 5 | |
26 | 25 | ad2ant2r 509 | . . . 4 |
27 | mulclpr 7546 | . . . . 5 | |
28 | 27 | ad2ant2l 508 | . . . 4 |
29 | addclpr 7511 | . . . 4 | |
30 | 26, 28, 29 | syl2anc 411 | . . 3 |
31 | mulclpr 7546 | . . . . 5 | |
32 | 31 | ad2ant2rl 511 | . . . 4 |
33 | mulclpr 7546 | . . . . 5 | |
34 | 33 | ad2ant2lr 510 | . . . 4 |
35 | addclpr 7511 | . . . 4 | |
36 | 32, 34, 35 | syl2anc 411 | . . 3 |
37 | 30, 36 | jca 306 | . 2 |
38 | simp1l 1021 | . . . . 5 | |
39 | simp2l 1023 | . . . . 5 | |
40 | simp3l 1025 | . . . . 5 | |
41 | distrprg 7562 | . . . . 5 | |
42 | 38, 39, 40, 41 | syl3anc 1238 | . . . 4 |
43 | simp1r 1022 | . . . . 5 | |
44 | simp2r 1024 | . . . . 5 | |
45 | simp3r 1026 | . . . . 5 | |
46 | distrprg 7562 | . . . . 5 | |
47 | 43, 44, 45, 46 | syl3anc 1238 | . . . 4 |
48 | 42, 47 | oveq12d 5883 | . . 3 |
49 | 38, 39, 12 | syl2anc 411 | . . . 4 |
50 | 38, 40, 25 | syl2anc 411 | . . . 4 |
51 | 43, 44, 14 | syl2anc 411 | . . . 4 |
52 | addcomprg 7552 | . . . . 5 | |
53 | 52 | adantl 277 | . . . 4 |
54 | addassprg 7553 | . . . . 5 | |
55 | 54 | adantl 277 | . . . 4 |
56 | 43, 45, 27 | syl2anc 411 | . . . 4 |
57 | addclpr 7511 | . . . . 5 | |
58 | 57 | adantl 277 | . . . 4 |
59 | 49, 50, 51, 53, 55, 56, 58 | caov4d 6049 | . . 3 |
60 | 48, 59 | eqtrd 2208 | . 2 |
61 | distrprg 7562 | . . . . 5 | |
62 | 38, 44, 45, 61 | syl3anc 1238 | . . . 4 |
63 | distrprg 7562 | . . . . 5 | |
64 | 43, 39, 40, 63 | syl3anc 1238 | . . . 4 |
65 | 62, 64 | oveq12d 5883 | . . 3 |
66 | 38, 44, 18 | syl2anc 411 | . . . 4 |
67 | 38, 45, 31 | syl2anc 411 | . . . 4 |
68 | 43, 39, 20 | syl2anc 411 | . . . 4 |
69 | 43, 40, 33 | syl2anc 411 | . . . 4 |
70 | 66, 67, 68, 53, 55, 69, 58 | caov4d 6049 | . . 3 |
71 | 65, 70 | eqtrd 2208 | . 2 |
72 | 1, 2, 3, 4, 5, 6, 11, 24, 37, 60, 71 | ecovidi 6637 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 w3a 978 wceq 1353 wcel 2146 (class class class)co 5865 cnp 7265 cpp 7267 cmp 7268 cer 7270 cnr 7271 cplr 7275 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-plr 7702 df-mr 7703 |
This theorem is referenced by: pn0sr 7745 axmulass 7847 axdistr 7848 |
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