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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 7724 be used later. Instead, use adddi 7752. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axdistr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcnqs 7649 | . 2 | |
2 | addcnsrec 7650 | . 2 | |
3 | mulcnsrec 7651 | . 2 | |
4 | mulcnsrec 7651 | . 2 | |
5 | mulcnsrec 7651 | . 2 | |
6 | addcnsrec 7650 | . 2 | |
7 | addclsr 7561 | . . . 4 | |
8 | addclsr 7561 | . . . 4 | |
9 | 7, 8 | anim12i 336 | . . 3 |
10 | 9 | an4s 577 | . 2 |
11 | mulclsr 7562 | . . . . 5 | |
12 | m1r 7560 | . . . . . 6 | |
13 | mulclsr 7562 | . . . . . 6 | |
14 | mulclsr 7562 | . . . . . 6 | |
15 | 12, 13, 14 | sylancr 410 | . . . . 5 |
16 | addclsr 7561 | . . . . 5 | |
17 | 11, 15, 16 | syl2an 287 | . . . 4 |
18 | 17 | an4s 577 | . . 3 |
19 | mulclsr 7562 | . . . . 5 | |
20 | mulclsr 7562 | . . . . 5 | |
21 | addclsr 7561 | . . . . 5 | |
22 | 19, 20, 21 | syl2anr 288 | . . . 4 |
23 | 22 | an42s 578 | . . 3 |
24 | 18, 23 | jca 304 | . 2 |
25 | mulclsr 7562 | . . . . 5 | |
26 | mulclsr 7562 | . . . . . 6 | |
27 | mulclsr 7562 | . . . . . 6 | |
28 | 12, 26, 27 | sylancr 410 | . . . . 5 |
29 | addclsr 7561 | . . . . 5 | |
30 | 25, 28, 29 | syl2an 287 | . . . 4 |
31 | 30 | an4s 577 | . . 3 |
32 | mulclsr 7562 | . . . . 5 | |
33 | mulclsr 7562 | . . . . 5 | |
34 | addclsr 7561 | . . . . 5 | |
35 | 32, 33, 34 | syl2anr 288 | . . . 4 |
36 | 35 | an42s 578 | . . 3 |
37 | 31, 36 | jca 304 | . 2 |
38 | simp1l 1005 | . . . . 5 | |
39 | simp2l 1007 | . . . . 5 | |
40 | simp3l 1009 | . . . . 5 | |
41 | distrsrg 7567 | . . . . 5 | |
42 | 38, 39, 40, 41 | syl3anc 1216 | . . . 4 |
43 | simp1r 1006 | . . . . . . 7 | |
44 | simp2r 1008 | . . . . . . 7 | |
45 | simp3r 1010 | . . . . . . 7 | |
46 | distrsrg 7567 | . . . . . . 7 | |
47 | 43, 44, 45, 46 | syl3anc 1216 | . . . . . 6 |
48 | 47 | oveq2d 5790 | . . . . 5 |
49 | 12 | a1i 9 | . . . . . 6 |
50 | 43, 44, 13 | syl2anc 408 | . . . . . 6 |
51 | 43, 45, 26 | syl2anc 408 | . . . . . 6 |
52 | distrsrg 7567 | . . . . . 6 | |
53 | 49, 50, 51, 52 | syl3anc 1216 | . . . . 5 |
54 | 48, 53 | eqtrd 2172 | . . . 4 |
55 | 42, 54 | oveq12d 5792 | . . 3 |
56 | 38, 39, 11 | syl2anc 408 | . . . 4 |
57 | 38, 40, 25 | syl2anc 408 | . . . 4 |
58 | 12, 50, 14 | sylancr 410 | . . . 4 |
59 | addcomsrg 7563 | . . . . 5 | |
60 | 59 | adantl 275 | . . . 4 |
61 | addasssrg 7564 | . . . . 5 | |
62 | 61 | adantl 275 | . . . 4 |
63 | 12, 51, 27 | sylancr 410 | . . . 4 |
64 | addclsr 7561 | . . . . 5 | |
65 | 64 | adantl 275 | . . . 4 |
66 | 56, 57, 58, 60, 62, 63, 65 | caov4d 5955 | . . 3 |
67 | 55, 66 | eqtrd 2172 | . 2 |
68 | distrsrg 7567 | . . . . 5 | |
69 | 43, 39, 40, 68 | syl3anc 1216 | . . . 4 |
70 | distrsrg 7567 | . . . . 5 | |
71 | 38, 44, 45, 70 | syl3anc 1216 | . . . 4 |
72 | 69, 71 | oveq12d 5792 | . . 3 |
73 | 43, 39, 19 | syl2anc 408 | . . . 4 |
74 | 43, 40, 32 | syl2anc 408 | . . . 4 |
75 | 38, 44, 20 | syl2anc 408 | . . . 4 |
76 | 38, 45, 33 | syl2anc 408 | . . . 4 |
77 | 73, 74, 75, 60, 62, 76, 65 | caov4d 5955 | . . 3 |
78 | 72, 77 | eqtrd 2172 | . 2 |
79 | 1, 2, 3, 4, 5, 6, 10, 24, 37, 67, 78 | ecovidi 6541 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 cep 4209 ccnv 4538 (class class class)co 5774 cnr 7105 cm1r 7108 cplr 7109 cmr 7110 cc 7618 caddc 7623 cmul 7625 |
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-i1p 7275 df-iplp 7276 df-imp 7277 df-enr 7534 df-nr 7535 df-plr 7536 df-mr 7537 df-m1r 7541 df-c 7626 df-add 7631 df-mul 7632 |
This theorem is referenced by: (None) |
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