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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 7393 | . 2 Q0 ~Q0 | |
2 | df-0nq0 7377 | . . . . . 6 0Q0 ~Q0 | |
3 | oveq12 5860 | . . . . . 6 0Q0 ~Q0 ~Q0 0Q0 ·Q0 ~Q0 ·Q0 ~Q0 | |
4 | 2, 3 | mpan 422 | . . . . 5 ~Q0 0Q0 ·Q0 ~Q0 ·Q0 ~Q0 |
5 | peano1 4576 | . . . . . 6 | |
6 | 1pi 7266 | . . . . . 6 | |
7 | mulnnnq0 7401 | . . . . . 6 ~Q0 ·Q0 ~Q0 ~Q0 | |
8 | 5, 6, 7 | mpanl12 434 | . . . . 5 ~Q0 ·Q0 ~Q0 ~Q0 |
9 | 4, 8 | sylan9eqr 2225 | . . . 4 ~Q0 0Q0 ·Q0 ~Q0 |
10 | nnm0r 6456 | . . . . . . . . . . 11 | |
11 | 10 | oveq1d 5866 | . . . . . . . . . 10 |
12 | 1onn 6497 | . . . . . . . . . . 11 | |
13 | nnm0r 6456 | . . . . . . . . . . 11 | |
14 | 12, 13 | ax-mp 5 | . . . . . . . . . 10 |
15 | 11, 14 | eqtrdi 2219 | . . . . . . . . 9 |
16 | 15 | adantr 274 | . . . . . . . 8 |
17 | mulpiord 7268 | . . . . . . . . . . . 12 | |
18 | mulclpi 7279 | . . . . . . . . . . . 12 | |
19 | 17, 18 | eqeltrrd 2248 | . . . . . . . . . . 11 |
20 | 6, 19 | mpan 422 | . . . . . . . . . 10 |
21 | pinn 7260 | . . . . . . . . . 10 | |
22 | nnm0 6452 | . . . . . . . . . 10 | |
23 | 20, 21, 22 | 3syl 17 | . . . . . . . . 9 |
24 | 23 | adantl 275 | . . . . . . . 8 |
25 | 16, 24 | eqtr4d 2206 | . . . . . . 7 |
26 | 10, 5 | eqeltrdi 2261 | . . . . . . . 8 |
27 | enq0eceq 7388 | . . . . . . . . 9 ~Q0 ~Q0 | |
28 | 5, 6, 27 | mpanr12 437 | . . . . . . . 8 ~Q0 ~Q0 |
29 | 26, 20, 28 | syl2an 287 | . . . . . . 7 ~Q0 ~Q0 |
30 | 25, 29 | mpbird 166 | . . . . . 6 ~Q0 ~Q0 |
31 | 30, 2 | eqtr4di 2221 | . . . . 5 ~Q0 0Q0 |
32 | 31 | adantr 274 | . . . 4 ~Q0 ~Q0 0Q0 |
33 | 9, 32 | eqtrd 2203 | . . 3 ~Q0 0Q0 ·Q0 0Q0 |
34 | 33 | exlimivv 1889 | . 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 1348 wex 1485 wcel 2141 c0 3414 cop 3584 com 4572 (class class class)co 5851 c1o 6386 comu 6391 cec 6508 cnpi 7223 cmi 7225 ~Q0 ceq0 7237 Q0cnq0 7238 0Q0c0q0 7239 ·Q0 cmq0 7241 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-irdg 6347 df-1o 6393 df-oadd 6397 df-omul 6398 df-er 6510 df-ec 6512 df-qs 6516 df-ni 7255 df-mi 7257 df-enq0 7375 df-nq0 7376 df-0nq0 7377 df-mq0 7379 |
This theorem is referenced by: prarloclem5 7451 |
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