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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 7080 be used later. Instead, use adddi 7105. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axdistr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcnqs 7009 | . 2 | |
2 | addcnsrec 7010 | . 2 | |
3 | mulcnsrec 7011 | . 2 | |
4 | mulcnsrec 7011 | . 2 | |
5 | mulcnsrec 7011 | . 2 | |
6 | addcnsrec 7010 | . 2 | |
7 | addclsr 6930 | . . . 4 | |
8 | addclsr 6930 | . . . 4 | |
9 | 7, 8 | anim12i 331 | . . 3 |
10 | 9 | an4s 552 | . 2 |
11 | mulclsr 6931 | . . . . 5 | |
12 | m1r 6929 | . . . . . 6 | |
13 | mulclsr 6931 | . . . . . 6 | |
14 | mulclsr 6931 | . . . . . 6 | |
15 | 12, 13, 14 | sylancr 405 | . . . . 5 |
16 | addclsr 6930 | . . . . 5 | |
17 | 11, 15, 16 | syl2an 283 | . . . 4 |
18 | 17 | an4s 552 | . . 3 |
19 | mulclsr 6931 | . . . . 5 | |
20 | mulclsr 6931 | . . . . 5 | |
21 | addclsr 6930 | . . . . 5 | |
22 | 19, 20, 21 | syl2anr 284 | . . . 4 |
23 | 22 | an42s 553 | . . 3 |
24 | 18, 23 | jca 300 | . 2 |
25 | mulclsr 6931 | . . . . 5 | |
26 | mulclsr 6931 | . . . . . 6 | |
27 | mulclsr 6931 | . . . . . 6 | |
28 | 12, 26, 27 | sylancr 405 | . . . . 5 |
29 | addclsr 6930 | . . . . 5 | |
30 | 25, 28, 29 | syl2an 283 | . . . 4 |
31 | 30 | an4s 552 | . . 3 |
32 | mulclsr 6931 | . . . . 5 | |
33 | mulclsr 6931 | . . . . 5 | |
34 | addclsr 6930 | . . . . 5 | |
35 | 32, 33, 34 | syl2anr 284 | . . . 4 |
36 | 35 | an42s 553 | . . 3 |
37 | 31, 36 | jca 300 | . 2 |
38 | simp1l 962 | . . . . 5 | |
39 | simp2l 964 | . . . . 5 | |
40 | simp3l 966 | . . . . 5 | |
41 | distrsrg 6936 | . . . . 5 | |
42 | 38, 39, 40, 41 | syl3anc 1169 | . . . 4 |
43 | simp1r 963 | . . . . . . 7 | |
44 | simp2r 965 | . . . . . . 7 | |
45 | simp3r 967 | . . . . . . 7 | |
46 | distrsrg 6936 | . . . . . . 7 | |
47 | 43, 44, 45, 46 | syl3anc 1169 | . . . . . 6 |
48 | 47 | oveq2d 5548 | . . . . 5 |
49 | 12 | a1i 9 | . . . . . 6 |
50 | 43, 44, 13 | syl2anc 403 | . . . . . 6 |
51 | 43, 45, 26 | syl2anc 403 | . . . . . 6 |
52 | distrsrg 6936 | . . . . . 6 | |
53 | 49, 50, 51, 52 | syl3anc 1169 | . . . . 5 |
54 | 48, 53 | eqtrd 2113 | . . . 4 |
55 | 42, 54 | oveq12d 5550 | . . 3 |
56 | 38, 39, 11 | syl2anc 403 | . . . 4 |
57 | 38, 40, 25 | syl2anc 403 | . . . 4 |
58 | 12, 50, 14 | sylancr 405 | . . . 4 |
59 | addcomsrg 6932 | . . . . 5 | |
60 | 59 | adantl 271 | . . . 4 |
61 | addasssrg 6933 | . . . . 5 | |
62 | 61 | adantl 271 | . . . 4 |
63 | 12, 51, 27 | sylancr 405 | . . . 4 |
64 | addclsr 6930 | . . . . 5 | |
65 | 64 | adantl 271 | . . . 4 |
66 | 56, 57, 58, 60, 62, 63, 65 | caov4d 5705 | . . 3 |
67 | 55, 66 | eqtrd 2113 | . 2 |
68 | distrsrg 6936 | . . . . 5 | |
69 | 43, 39, 40, 68 | syl3anc 1169 | . . . 4 |
70 | distrsrg 6936 | . . . . 5 | |
71 | 38, 44, 45, 70 | syl3anc 1169 | . . . 4 |
72 | 69, 71 | oveq12d 5550 | . . 3 |
73 | 43, 39, 19 | syl2anc 403 | . . . 4 |
74 | 43, 40, 32 | syl2anc 403 | . . . 4 |
75 | 38, 44, 20 | syl2anc 403 | . . . 4 |
76 | 38, 45, 33 | syl2anc 403 | . . . 4 |
77 | 73, 74, 75, 60, 62, 76, 65 | caov4d 5705 | . . 3 |
78 | 72, 77 | eqtrd 2113 | . 2 |
79 | 1, 2, 3, 4, 5, 6, 10, 24, 37, 67, 78 | ecovidi 6241 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wceq 1284 wcel 1433 cep 4042 ccnv 4362 (class class class)co 5532 cnr 6487 cm1r 6490 cplr 6491 cmr 6492 cc 6979 caddc 6984 cmul 6986 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-i1p 6657 df-iplp 6658 df-imp 6659 df-enr 6903 df-nr 6904 df-plr 6905 df-mr 6906 df-m1r 6910 df-c 6987 df-add 6992 df-mul 6993 |
This theorem is referenced by: (None) |
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