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Mirrors > Home > ILE Home > Th. List > axdistr | Unicode version |
Description: Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-distr 6988 be used later. Instead, use adddi 7013. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axdistr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcnqs 6917 | . 2 | |
2 | addcnsrec 6918 | . 2 | |
3 | mulcnsrec 6919 | . 2 | |
4 | mulcnsrec 6919 | . 2 | |
5 | mulcnsrec 6919 | . 2 | |
6 | addcnsrec 6918 | . 2 | |
7 | addclsr 6838 | . . . 4 | |
8 | addclsr 6838 | . . . 4 | |
9 | 7, 8 | anim12i 321 | . . 3 |
10 | 9 | an4s 522 | . 2 |
11 | mulclsr 6839 | . . . . 5 | |
12 | m1r 6837 | . . . . . 6 | |
13 | mulclsr 6839 | . . . . . 6 | |
14 | mulclsr 6839 | . . . . . 6 | |
15 | 12, 13, 14 | sylancr 393 | . . . . 5 |
16 | addclsr 6838 | . . . . 5 | |
17 | 11, 15, 16 | syl2an 273 | . . . 4 |
18 | 17 | an4s 522 | . . 3 |
19 | mulclsr 6839 | . . . . 5 | |
20 | mulclsr 6839 | . . . . 5 | |
21 | addclsr 6838 | . . . . 5 | |
22 | 19, 20, 21 | syl2anr 274 | . . . 4 |
23 | 22 | an42s 523 | . . 3 |
24 | 18, 23 | jca 290 | . 2 |
25 | mulclsr 6839 | . . . . 5 | |
26 | mulclsr 6839 | . . . . . 6 | |
27 | mulclsr 6839 | . . . . . 6 | |
28 | 12, 26, 27 | sylancr 393 | . . . . 5 |
29 | addclsr 6838 | . . . . 5 | |
30 | 25, 28, 29 | syl2an 273 | . . . 4 |
31 | 30 | an4s 522 | . . 3 |
32 | mulclsr 6839 | . . . . 5 | |
33 | mulclsr 6839 | . . . . 5 | |
34 | addclsr 6838 | . . . . 5 | |
35 | 32, 33, 34 | syl2anr 274 | . . . 4 |
36 | 35 | an42s 523 | . . 3 |
37 | 31, 36 | jca 290 | . 2 |
38 | simp1l 928 | . . . . 5 | |
39 | simp2l 930 | . . . . 5 | |
40 | simp3l 932 | . . . . 5 | |
41 | distrsrg 6844 | . . . . 5 | |
42 | 38, 39, 40, 41 | syl3anc 1135 | . . . 4 |
43 | simp1r 929 | . . . . . . 7 | |
44 | simp2r 931 | . . . . . . 7 | |
45 | simp3r 933 | . . . . . . 7 | |
46 | distrsrg 6844 | . . . . . . 7 | |
47 | 43, 44, 45, 46 | syl3anc 1135 | . . . . . 6 |
48 | 47 | oveq2d 5528 | . . . . 5 |
49 | 12 | a1i 9 | . . . . . 6 |
50 | 43, 44, 13 | syl2anc 391 | . . . . . 6 |
51 | 43, 45, 26 | syl2anc 391 | . . . . . 6 |
52 | distrsrg 6844 | . . . . . 6 | |
53 | 49, 50, 51, 52 | syl3anc 1135 | . . . . 5 |
54 | 48, 53 | eqtrd 2072 | . . . 4 |
55 | 42, 54 | oveq12d 5530 | . . 3 |
56 | 38, 39, 11 | syl2anc 391 | . . . 4 |
57 | 38, 40, 25 | syl2anc 391 | . . . 4 |
58 | 12, 50, 14 | sylancr 393 | . . . 4 |
59 | addcomsrg 6840 | . . . . 5 | |
60 | 59 | adantl 262 | . . . 4 |
61 | addasssrg 6841 | . . . . 5 | |
62 | 61 | adantl 262 | . . . 4 |
63 | 12, 51, 27 | sylancr 393 | . . . 4 |
64 | addclsr 6838 | . . . . 5 | |
65 | 64 | adantl 262 | . . . 4 |
66 | 56, 57, 58, 60, 62, 63, 65 | caov4d 5685 | . . 3 |
67 | 55, 66 | eqtrd 2072 | . 2 |
68 | distrsrg 6844 | . . . . 5 | |
69 | 43, 39, 40, 68 | syl3anc 1135 | . . . 4 |
70 | distrsrg 6844 | . . . . 5 | |
71 | 38, 44, 45, 70 | syl3anc 1135 | . . . 4 |
72 | 69, 71 | oveq12d 5530 | . . 3 |
73 | 43, 39, 19 | syl2anc 391 | . . . 4 |
74 | 43, 40, 32 | syl2anc 391 | . . . 4 |
75 | 38, 44, 20 | syl2anc 391 | . . . 4 |
76 | 38, 45, 33 | syl2anc 391 | . . . 4 |
77 | 73, 74, 75, 60, 62, 76, 65 | caov4d 5685 | . . 3 |
78 | 72, 77 | eqtrd 2072 | . 2 |
79 | 1, 2, 3, 4, 5, 6, 10, 24, 37, 67, 78 | ecovidi 6218 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 cep 4024 ccnv 4344 (class class class)co 5512 cnr 6395 cm1r 6398 cplr 6399 cmr 6400 cc 6887 caddc 6892 cmul 6894 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-imp 6567 df-enr 6811 df-nr 6812 df-plr 6813 df-mr 6814 df-m1r 6818 df-c 6895 df-add 6900 df-mul 6901 |
This theorem is referenced by: (None) |
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