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Mirrors > Home > ILE Home > Th. List > mulclnq0 | Unicode version |
Description: Closure of multiplication on nonnegative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.) |
Ref | Expression |
---|---|
mulclnq0 | Q0 Q0 ·Q0 Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nq0 7226 | . . 3 Q0 ~Q0 | |
2 | oveq1 5774 | . . . 4 ~Q0 ~Q0 ·Q0 ~Q0 ·Q0 ~Q0 | |
3 | 2 | eleq1d 2206 | . . 3 ~Q0 ~Q0 ·Q0 ~Q0 ~Q0 ·Q0 ~Q0 ~Q0 |
4 | oveq2 5775 | . . . 4 ~Q0 ·Q0 ~Q0 ·Q0 | |
5 | 4 | eleq1d 2206 | . . 3 ~Q0 ·Q0 ~Q0 ~Q0 ·Q0 ~Q0 |
6 | mulnnnq0 7251 | . . . 4 ~Q0 ·Q0 ~Q0 ~Q0 | |
7 | nnmcl 6370 | . . . . . . 7 | |
8 | mulpiord 7118 | . . . . . . . 8 | |
9 | mulclpi 7129 | . . . . . . . 8 | |
10 | 8, 9 | eqeltrrd 2215 | . . . . . . 7 |
11 | 7, 10 | anim12i 336 | . . . . . 6 |
12 | 11 | an4s 577 | . . . . 5 |
13 | opelxpi 4566 | . . . . 5 | |
14 | enq0ex 7240 | . . . . . 6 ~Q0 | |
15 | 14 | ecelqsi 6476 | . . . . 5 ~Q0 ~Q0 |
16 | 12, 13, 15 | 3syl 17 | . . . 4 ~Q0 ~Q0 |
17 | 6, 16 | eqeltrd 2214 | . . 3 ~Q0 ·Q0 ~Q0 ~Q0 |
18 | 1, 3, 5, 17 | 2ecoptocl 6510 | . 2 Q0 Q0 ·Q0 ~Q0 |
19 | 18, 1 | eleqtrrdi 2231 | 1 Q0 Q0 ·Q0 Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cop 3525 com 4499 cxp 4532 (class class class)co 5767 comu 6304 cec 6420 cqs 6421 cnpi 7073 cmi 7075 ~Q0 ceq0 7087 Q0cnq0 7088 ·Q0 cmq0 7091 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-mi 7107 df-enq0 7225 df-nq0 7226 df-mq0 7229 |
This theorem is referenced by: prarloclemcalc 7303 |
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