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Mirrors > Home > ILE Home > Th. List > qmulcl | Unicode version |
Description: Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.) |
Ref | Expression |
---|---|
qmulcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elq 8707 | . 2 | |
2 | elq 8707 | . 2 | |
3 | zmulcl 8404 | . . . . . . . . . . 11 | |
4 | nnmulcl 8060 | . . . . . . . . . . 11 | |
5 | 3, 4 | anim12i 331 | . . . . . . . . . 10 |
6 | 5 | an4s 552 | . . . . . . . . 9 |
7 | 6 | adantr 270 | . . . . . . . 8 |
8 | oveq12 5541 | . . . . . . . . 9 | |
9 | zcn 8356 | . . . . . . . . . . . 12 | |
10 | zcn 8356 | . . . . . . . . . . . 12 | |
11 | 9, 10 | anim12i 331 | . . . . . . . . . . 11 |
12 | 11 | ad2ant2r 492 | . . . . . . . . . 10 |
13 | nncn 8047 | . . . . . . . . . . . . 13 | |
14 | nnap0 8068 | . . . . . . . . . . . . 13 # | |
15 | 13, 14 | jca 300 | . . . . . . . . . . . 12 # |
16 | nncn 8047 | . . . . . . . . . . . . 13 | |
17 | nnap0 8068 | . . . . . . . . . . . . 13 # | |
18 | 16, 17 | jca 300 | . . . . . . . . . . . 12 # |
19 | 15, 18 | anim12i 331 | . . . . . . . . . . 11 # # |
20 | 19 | ad2ant2l 491 | . . . . . . . . . 10 # # |
21 | divmuldivap 7800 | . . . . . . . . . 10 # # | |
22 | 12, 20, 21 | syl2anc 403 | . . . . . . . . 9 |
23 | 8, 22 | sylan9eqr 2135 | . . . . . . . 8 |
24 | rspceov 5567 | . . . . . . . . . 10 | |
25 | 24 | 3expa 1138 | . . . . . . . . 9 |
26 | elq 8707 | . . . . . . . . 9 | |
27 | 25, 26 | sylibr 132 | . . . . . . . 8 |
28 | 7, 23, 27 | syl2anc 403 | . . . . . . 7 |
29 | 28 | an4s 552 | . . . . . 6 |
30 | 29 | exp43 364 | . . . . 5 |
31 | 30 | rexlimivv 2482 | . . . 4 |
32 | 31 | rexlimdvv 2483 | . . 3 |
33 | 32 | imp 122 | . 2 |
34 | 1, 2, 33 | syl2anb 285 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 wrex 2349 class class class wbr 3785 (class class class)co 5532 cc 6979 cc0 6981 cmul 6986 # cap 7681 cdiv 7760 cn 8039 cz 8351 cq 8704 |
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-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 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-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 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-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-id 4048 df-po 4051 df-iso 4052 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-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-n0 8289 df-z 8352 df-q 8705 |
This theorem is referenced by: qdivcl 8728 flqmulnn0 9301 modqcl 9328 mulqmod0 9332 modqmulnn 9344 modqcyc 9361 mulp1mod1 9367 modqmul1 9379 q2txmodxeq0 9386 modqaddmulmod 9393 modqdi 9394 modqsubdir 9395 qexpcl 9492 qexpclz 9497 qsqcl 9547 dvdslelemd 10243 |
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