Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > axmulcl | Unicode version |
Description: Closure law for multiplication of complex numbers. 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-mulcl 7074 be used later. Instead, in most cases use mulcl 7100. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axmulcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxpi 4379 | . . . . 5 | |
2 | df-c 6987 | . . . . 5 | |
3 | 1, 2 | eleq2s 2173 | . . . 4 |
4 | elxpi 4379 | . . . . 5 | |
5 | 4, 2 | eleq2s 2173 | . . . 4 |
6 | 3, 5 | anim12i 331 | . . 3 |
7 | ee4anv 1850 | . . 3 | |
8 | 6, 7 | sylibr 132 | . 2 |
9 | simpll 495 | . . . . . . 7 | |
10 | simprl 497 | . . . . . . 7 | |
11 | 9, 10 | oveq12d 5550 | . . . . . 6 |
12 | mulcnsr 7003 | . . . . . . 7 | |
13 | 12 | ad2ant2l 491 | . . . . . 6 |
14 | 11, 13 | eqtrd 2113 | . . . . 5 |
15 | simplrl 501 | . . . . . . . . 9 | |
16 | simprrl 505 | . . . . . . . . 9 | |
17 | mulclsr 6931 | . . . . . . . . 9 | |
18 | 15, 16, 17 | syl2anc 403 | . . . . . . . 8 |
19 | m1r 6929 | . . . . . . . . . 10 | |
20 | 19 | a1i 9 | . . . . . . . . 9 |
21 | simplrr 502 | . . . . . . . . . 10 | |
22 | simprrr 506 | . . . . . . . . . 10 | |
23 | mulclsr 6931 | . . . . . . . . . 10 | |
24 | 21, 22, 23 | syl2anc 403 | . . . . . . . . 9 |
25 | mulclsr 6931 | . . . . . . . . 9 | |
26 | 20, 24, 25 | syl2anc 403 | . . . . . . . 8 |
27 | addclsr 6930 | . . . . . . . 8 | |
28 | 18, 26, 27 | syl2anc 403 | . . . . . . 7 |
29 | mulclsr 6931 | . . . . . . . . 9 | |
30 | 21, 16, 29 | syl2anc 403 | . . . . . . . 8 |
31 | mulclsr 6931 | . . . . . . . . 9 | |
32 | 15, 22, 31 | syl2anc 403 | . . . . . . . 8 |
33 | addclsr 6930 | . . . . . . . 8 | |
34 | 30, 32, 33 | syl2anc 403 | . . . . . . 7 |
35 | opelxpi 4394 | . . . . . . 7 | |
36 | 28, 34, 35 | syl2anc 403 | . . . . . 6 |
37 | 36, 2 | syl6eleqr 2172 | . . . . 5 |
38 | 14, 37 | eqeltrd 2155 | . . . 4 |
39 | 38 | exlimivv 1817 | . . 3 |
40 | 39 | exlimivv 1817 | . 2 |
41 | 8, 40 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wex 1421 wcel 1433 cop 3401 cxp 4361 (class class class)co 5532 cnr 6487 cm1r 6490 cplr 6491 cmr 6492 cc 6979 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-mul 6993 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |