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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 7535 | . 2 | |
2 | addsrpr 7553 | . 2 | |
3 | mulsrpr 7554 | . 2 | |
4 | mulsrpr 7554 | . 2 | |
5 | mulsrpr 7554 | . 2 | |
6 | addsrpr 7553 | . 2 | |
7 | addclpr 7345 | . . . 4 | |
8 | 7 | ad2ant2r 500 | . . 3 |
9 | addclpr 7345 | . . . 4 | |
10 | 9 | ad2ant2l 499 | . . 3 |
11 | 8, 10 | jca 304 | . 2 |
12 | mulclpr 7380 | . . . . 5 | |
13 | 12 | ad2ant2r 500 | . . . 4 |
14 | mulclpr 7380 | . . . . 5 | |
15 | 14 | ad2ant2l 499 | . . . 4 |
16 | addclpr 7345 | . . . 4 | |
17 | 13, 15, 16 | syl2anc 408 | . . 3 |
18 | mulclpr 7380 | . . . . 5 | |
19 | 18 | ad2ant2rl 502 | . . . 4 |
20 | mulclpr 7380 | . . . . 5 | |
21 | 20 | ad2ant2lr 501 | . . . 4 |
22 | addclpr 7345 | . . . 4 | |
23 | 19, 21, 22 | syl2anc 408 | . . 3 |
24 | 17, 23 | jca 304 | . 2 |
25 | mulclpr 7380 | . . . . 5 | |
26 | 25 | ad2ant2r 500 | . . . 4 |
27 | mulclpr 7380 | . . . . 5 | |
28 | 27 | ad2ant2l 499 | . . . 4 |
29 | addclpr 7345 | . . . 4 | |
30 | 26, 28, 29 | syl2anc 408 | . . 3 |
31 | mulclpr 7380 | . . . . 5 | |
32 | 31 | ad2ant2rl 502 | . . . 4 |
33 | mulclpr 7380 | . . . . 5 | |
34 | 33 | ad2ant2lr 501 | . . . 4 |
35 | addclpr 7345 | . . . 4 | |
36 | 32, 34, 35 | syl2anc 408 | . . 3 |
37 | 30, 36 | jca 304 | . 2 |
38 | simp1l 1005 | . . . . 5 | |
39 | simp2l 1007 | . . . . 5 | |
40 | simp3l 1009 | . . . . 5 | |
41 | distrprg 7396 | . . . . 5 | |
42 | 38, 39, 40, 41 | syl3anc 1216 | . . . 4 |
43 | simp1r 1006 | . . . . 5 | |
44 | simp2r 1008 | . . . . 5 | |
45 | simp3r 1010 | . . . . 5 | |
46 | distrprg 7396 | . . . . 5 | |
47 | 43, 44, 45, 46 | syl3anc 1216 | . . . 4 |
48 | 42, 47 | oveq12d 5792 | . . 3 |
49 | 38, 39, 12 | syl2anc 408 | . . . 4 |
50 | 38, 40, 25 | syl2anc 408 | . . . 4 |
51 | 43, 44, 14 | syl2anc 408 | . . . 4 |
52 | addcomprg 7386 | . . . . 5 | |
53 | 52 | adantl 275 | . . . 4 |
54 | addassprg 7387 | . . . . 5 | |
55 | 54 | adantl 275 | . . . 4 |
56 | 43, 45, 27 | syl2anc 408 | . . . 4 |
57 | addclpr 7345 | . . . . 5 | |
58 | 57 | adantl 275 | . . . 4 |
59 | 49, 50, 51, 53, 55, 56, 58 | caov4d 5955 | . . 3 |
60 | 48, 59 | eqtrd 2172 | . 2 |
61 | distrprg 7396 | . . . . 5 | |
62 | 38, 44, 45, 61 | syl3anc 1216 | . . . 4 |
63 | distrprg 7396 | . . . . 5 | |
64 | 43, 39, 40, 63 | syl3anc 1216 | . . . 4 |
65 | 62, 64 | oveq12d 5792 | . . 3 |
66 | 38, 44, 18 | syl2anc 408 | . . . 4 |
67 | 38, 45, 31 | syl2anc 408 | . . . 4 |
68 | 43, 39, 20 | syl2anc 408 | . . . 4 |
69 | 43, 40, 33 | syl2anc 408 | . . . 4 |
70 | 66, 67, 68, 53, 55, 69, 58 | caov4d 5955 | . . 3 |
71 | 65, 70 | eqtrd 2172 | . 2 |
72 | 1, 2, 3, 4, 5, 6, 11, 24, 37, 60, 71 | ecovidi 6541 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 (class class class)co 5774 cnp 7099 cpp 7101 cmp 7102 cer 7104 cnr 7105 cplr 7109 cmr 7110 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-2o 6314 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-enq0 7232 df-nq0 7233 df-0nq0 7234 df-plq0 7235 df-mq0 7236 df-inp 7274 df-iplp 7276 df-imp 7277 df-enr 7534 df-nr 7535 df-plr 7536 df-mr 7537 |
This theorem is referenced by: pn0sr 7579 axmulass 7681 axdistr 7682 |
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