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Mirrors > Home > ILE Home > Th. List > nq0m0r | Unicode version |
Description: Multiplication with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.) |
Ref | Expression |
---|---|
nq0m0r | Q0 0Q0 ·Q0 0Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nq0nn 7243 | . 2 Q0 ~Q0 | |
2 | df-0nq0 7227 | . . . . . 6 0Q0 ~Q0 | |
3 | oveq12 5776 | . . . . . 6 0Q0 ~Q0 ~Q0 0Q0 ·Q0 ~Q0 ·Q0 ~Q0 | |
4 | 2, 3 | mpan 420 | . . . . 5 ~Q0 0Q0 ·Q0 ~Q0 ·Q0 ~Q0 |
5 | peano1 4503 | . . . . . 6 | |
6 | 1pi 7116 | . . . . . 6 | |
7 | mulnnnq0 7251 | . . . . . 6 ~Q0 ·Q0 ~Q0 ~Q0 | |
8 | 5, 6, 7 | mpanl12 432 | . . . . 5 ~Q0 ·Q0 ~Q0 ~Q0 |
9 | 4, 8 | sylan9eqr 2192 | . . . 4 ~Q0 0Q0 ·Q0 ~Q0 |
10 | nnm0r 6368 | . . . . . . . . . . 11 | |
11 | 10 | oveq1d 5782 | . . . . . . . . . 10 |
12 | 1onn 6409 | . . . . . . . . . . 11 | |
13 | nnm0r 6368 | . . . . . . . . . . 11 | |
14 | 12, 13 | ax-mp 5 | . . . . . . . . . 10 |
15 | 11, 14 | syl6eq 2186 | . . . . . . . . 9 |
16 | 15 | adantr 274 | . . . . . . . 8 |
17 | mulpiord 7118 | . . . . . . . . . . . 12 | |
18 | mulclpi 7129 | . . . . . . . . . . . 12 | |
19 | 17, 18 | eqeltrrd 2215 | . . . . . . . . . . 11 |
20 | 6, 19 | mpan 420 | . . . . . . . . . 10 |
21 | pinn 7110 | . . . . . . . . . 10 | |
22 | nnm0 6364 | . . . . . . . . . 10 | |
23 | 20, 21, 22 | 3syl 17 | . . . . . . . . 9 |
24 | 23 | adantl 275 | . . . . . . . 8 |
25 | 16, 24 | eqtr4d 2173 | . . . . . . 7 |
26 | 10, 5 | eqeltrdi 2228 | . . . . . . . 8 |
27 | enq0eceq 7238 | . . . . . . . . 9 ~Q0 ~Q0 | |
28 | 5, 6, 27 | mpanr12 435 | . . . . . . . 8 ~Q0 ~Q0 |
29 | 26, 20, 28 | syl2an 287 | . . . . . . 7 ~Q0 ~Q0 |
30 | 25, 29 | mpbird 166 | . . . . . 6 ~Q0 ~Q0 |
31 | 30, 2 | syl6eqr 2188 | . . . . 5 ~Q0 0Q0 |
32 | 31 | adantr 274 | . . . 4 ~Q0 ~Q0 0Q0 |
33 | 9, 32 | eqtrd 2170 | . . 3 ~Q0 0Q0 ·Q0 0Q0 |
34 | 33 | exlimivv 1868 | . 2 ~Q0 0Q0 ·Q0 0Q0 |
35 | 1, 34 | syl 14 | 1 Q0 0Q0 ·Q0 0Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 c0 3358 cop 3525 com 4499 (class class class)co 5767 c1o 6299 comu 6304 cec 6420 cnpi 7073 cmi 7075 ~Q0 ceq0 7087 Q0cnq0 7088 0Q0c0q0 7089 ·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-1o 6306 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-0nq0 7227 df-mq0 7229 |
This theorem is referenced by: prarloclem5 7301 |
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