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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 7659 | . 2 | |
2 | addsrpr 7677 | . 2 | |
3 | mulsrpr 7678 | . 2 | |
4 | mulsrpr 7678 | . 2 | |
5 | mulsrpr 7678 | . 2 | |
6 | addsrpr 7677 | . 2 | |
7 | addclpr 7469 | . . . 4 | |
8 | 7 | ad2ant2r 501 | . . 3 |
9 | addclpr 7469 | . . . 4 | |
10 | 9 | ad2ant2l 500 | . . 3 |
11 | 8, 10 | jca 304 | . 2 |
12 | mulclpr 7504 | . . . . 5 | |
13 | 12 | ad2ant2r 501 | . . . 4 |
14 | mulclpr 7504 | . . . . 5 | |
15 | 14 | ad2ant2l 500 | . . . 4 |
16 | addclpr 7469 | . . . 4 | |
17 | 13, 15, 16 | syl2anc 409 | . . 3 |
18 | mulclpr 7504 | . . . . 5 | |
19 | 18 | ad2ant2rl 503 | . . . 4 |
20 | mulclpr 7504 | . . . . 5 | |
21 | 20 | ad2ant2lr 502 | . . . 4 |
22 | addclpr 7469 | . . . 4 | |
23 | 19, 21, 22 | syl2anc 409 | . . 3 |
24 | 17, 23 | jca 304 | . 2 |
25 | mulclpr 7504 | . . . . 5 | |
26 | 25 | ad2ant2r 501 | . . . 4 |
27 | mulclpr 7504 | . . . . 5 | |
28 | 27 | ad2ant2l 500 | . . . 4 |
29 | addclpr 7469 | . . . 4 | |
30 | 26, 28, 29 | syl2anc 409 | . . 3 |
31 | mulclpr 7504 | . . . . 5 | |
32 | 31 | ad2ant2rl 503 | . . . 4 |
33 | mulclpr 7504 | . . . . 5 | |
34 | 33 | ad2ant2lr 502 | . . . 4 |
35 | addclpr 7469 | . . . 4 | |
36 | 32, 34, 35 | syl2anc 409 | . . 3 |
37 | 30, 36 | jca 304 | . 2 |
38 | simp1l 1010 | . . . . 5 | |
39 | simp2l 1012 | . . . . 5 | |
40 | simp3l 1014 | . . . . 5 | |
41 | distrprg 7520 | . . . . 5 | |
42 | 38, 39, 40, 41 | syl3anc 1227 | . . . 4 |
43 | simp1r 1011 | . . . . 5 | |
44 | simp2r 1013 | . . . . 5 | |
45 | simp3r 1015 | . . . . 5 | |
46 | distrprg 7520 | . . . . 5 | |
47 | 43, 44, 45, 46 | syl3anc 1227 | . . . 4 |
48 | 42, 47 | oveq12d 5854 | . . 3 |
49 | 38, 39, 12 | syl2anc 409 | . . . 4 |
50 | 38, 40, 25 | syl2anc 409 | . . . 4 |
51 | 43, 44, 14 | syl2anc 409 | . . . 4 |
52 | addcomprg 7510 | . . . . 5 | |
53 | 52 | adantl 275 | . . . 4 |
54 | addassprg 7511 | . . . . 5 | |
55 | 54 | adantl 275 | . . . 4 |
56 | 43, 45, 27 | syl2anc 409 | . . . 4 |
57 | addclpr 7469 | . . . . 5 | |
58 | 57 | adantl 275 | . . . 4 |
59 | 49, 50, 51, 53, 55, 56, 58 | caov4d 6017 | . . 3 |
60 | 48, 59 | eqtrd 2197 | . 2 |
61 | distrprg 7520 | . . . . 5 | |
62 | 38, 44, 45, 61 | syl3anc 1227 | . . . 4 |
63 | distrprg 7520 | . . . . 5 | |
64 | 43, 39, 40, 63 | syl3anc 1227 | . . . 4 |
65 | 62, 64 | oveq12d 5854 | . . 3 |
66 | 38, 44, 18 | syl2anc 409 | . . . 4 |
67 | 38, 45, 31 | syl2anc 409 | . . . 4 |
68 | 43, 39, 20 | syl2anc 409 | . . . 4 |
69 | 43, 40, 33 | syl2anc 409 | . . . 4 |
70 | 66, 67, 68, 53, 55, 69, 58 | caov4d 6017 | . . 3 |
71 | 65, 70 | eqtrd 2197 | . 2 |
72 | 1, 2, 3, 4, 5, 6, 11, 24, 37, 60, 71 | ecovidi 6604 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 967 wceq 1342 wcel 2135 (class class class)co 5836 cnp 7223 cpp 7225 cmp 7226 cer 7228 cnr 7229 cplr 7233 cmr 7234 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-2o 6376 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 df-enq0 7356 df-nq0 7357 df-0nq0 7358 df-plq0 7359 df-mq0 7360 df-inp 7398 df-iplp 7400 df-imp 7401 df-enr 7658 df-nr 7659 df-plr 7660 df-mr 7661 |
This theorem is referenced by: pn0sr 7703 axmulass 7805 axdistr 7806 |
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