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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 7528 | . 2 | |
2 | addsrpr 7546 | . 2 | |
3 | mulsrpr 7547 | . 2 | |
4 | mulsrpr 7547 | . 2 | |
5 | mulsrpr 7547 | . 2 | |
6 | addsrpr 7546 | . 2 | |
7 | addclpr 7338 | . . . 4 | |
8 | 7 | ad2ant2r 500 | . . 3 |
9 | addclpr 7338 | . . . 4 | |
10 | 9 | ad2ant2l 499 | . . 3 |
11 | 8, 10 | jca 304 | . 2 |
12 | mulclpr 7373 | . . . . 5 | |
13 | 12 | ad2ant2r 500 | . . . 4 |
14 | mulclpr 7373 | . . . . 5 | |
15 | 14 | ad2ant2l 499 | . . . 4 |
16 | addclpr 7338 | . . . 4 | |
17 | 13, 15, 16 | syl2anc 408 | . . 3 |
18 | mulclpr 7373 | . . . . 5 | |
19 | 18 | ad2ant2rl 502 | . . . 4 |
20 | mulclpr 7373 | . . . . 5 | |
21 | 20 | ad2ant2lr 501 | . . . 4 |
22 | addclpr 7338 | . . . 4 | |
23 | 19, 21, 22 | syl2anc 408 | . . 3 |
24 | 17, 23 | jca 304 | . 2 |
25 | mulclpr 7373 | . . . . 5 | |
26 | 25 | ad2ant2r 500 | . . . 4 |
27 | mulclpr 7373 | . . . . 5 | |
28 | 27 | ad2ant2l 499 | . . . 4 |
29 | addclpr 7338 | . . . 4 | |
30 | 26, 28, 29 | syl2anc 408 | . . 3 |
31 | mulclpr 7373 | . . . . 5 | |
32 | 31 | ad2ant2rl 502 | . . . 4 |
33 | mulclpr 7373 | . . . . 5 | |
34 | 33 | ad2ant2lr 501 | . . . 4 |
35 | addclpr 7338 | . . . 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 7389 | . . . . 5 | |
42 | 38, 39, 40, 41 | syl3anc 1216 | . . . 4 |
43 | simp1r 1006 | . . . . 5 | |
44 | simp2r 1008 | . . . . 5 | |
45 | simp3r 1010 | . . . . 5 | |
46 | distrprg 7389 | . . . . 5 | |
47 | 43, 44, 45, 46 | syl3anc 1216 | . . . 4 |
48 | 42, 47 | oveq12d 5785 | . . 3 |
49 | 38, 39, 12 | syl2anc 408 | . . . 4 |
50 | 38, 40, 25 | syl2anc 408 | . . . 4 |
51 | 43, 44, 14 | syl2anc 408 | . . . 4 |
52 | addcomprg 7379 | . . . . 5 | |
53 | 52 | adantl 275 | . . . 4 |
54 | addassprg 7380 | . . . . 5 | |
55 | 54 | adantl 275 | . . . 4 |
56 | 43, 45, 27 | syl2anc 408 | . . . 4 |
57 | addclpr 7338 | . . . . 5 | |
58 | 57 | adantl 275 | . . . 4 |
59 | 49, 50, 51, 53, 55, 56, 58 | caov4d 5948 | . . 3 |
60 | 48, 59 | eqtrd 2170 | . 2 |
61 | distrprg 7389 | . . . . 5 | |
62 | 38, 44, 45, 61 | syl3anc 1216 | . . . 4 |
63 | distrprg 7389 | . . . . 5 | |
64 | 43, 39, 40, 63 | syl3anc 1216 | . . . 4 |
65 | 62, 64 | oveq12d 5785 | . . 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 5948 | . . 3 |
71 | 65, 70 | eqtrd 2170 | . 2 |
72 | 1, 2, 3, 4, 5, 6, 11, 24, 37, 60, 71 | ecovidi 6534 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 (class class class)co 5767 cnp 7092 cpp 7094 cmp 7095 cer 7097 cnr 7098 cplr 7102 cmr 7103 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-eprel 4206 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-2o 6307 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-pli 7106 df-mi 7107 df-lti 7108 df-plpq 7145 df-mpq 7146 df-enq 7148 df-nqqs 7149 df-plqqs 7150 df-mqqs 7151 df-1nqqs 7152 df-rq 7153 df-ltnqqs 7154 df-enq0 7225 df-nq0 7226 df-0nq0 7227 df-plq0 7228 df-mq0 7229 df-inp 7267 df-iplp 7269 df-imp 7270 df-enr 7527 df-nr 7528 df-plr 7529 df-mr 7530 |
This theorem is referenced by: pn0sr 7572 axmulass 7674 axdistr 7675 |
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