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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 7374 | . 2 Q0 ~Q0 | |
2 | df-0nq0 7358 | . . . . . 6 0Q0 ~Q0 | |
3 | oveq12 5845 | . . . . . 6 0Q0 ~Q0 ~Q0 0Q0 ·Q0 ~Q0 ·Q0 ~Q0 | |
4 | 2, 3 | mpan 421 | . . . . 5 ~Q0 0Q0 ·Q0 ~Q0 ·Q0 ~Q0 |
5 | peano1 4565 | . . . . . 6 | |
6 | 1pi 7247 | . . . . . 6 | |
7 | mulnnnq0 7382 | . . . . . 6 ~Q0 ·Q0 ~Q0 ~Q0 | |
8 | 5, 6, 7 | mpanl12 433 | . . . . 5 ~Q0 ·Q0 ~Q0 ~Q0 |
9 | 4, 8 | sylan9eqr 2219 | . . . 4 ~Q0 0Q0 ·Q0 ~Q0 |
10 | nnm0r 6438 | . . . . . . . . . . 11 | |
11 | 10 | oveq1d 5851 | . . . . . . . . . 10 |
12 | 1onn 6479 | . . . . . . . . . . 11 | |
13 | nnm0r 6438 | . . . . . . . . . . 11 | |
14 | 12, 13 | ax-mp 5 | . . . . . . . . . 10 |
15 | 11, 14 | eqtrdi 2213 | . . . . . . . . 9 |
16 | 15 | adantr 274 | . . . . . . . 8 |
17 | mulpiord 7249 | . . . . . . . . . . . 12 | |
18 | mulclpi 7260 | . . . . . . . . . . . 12 | |
19 | 17, 18 | eqeltrrd 2242 | . . . . . . . . . . 11 |
20 | 6, 19 | mpan 421 | . . . . . . . . . 10 |
21 | pinn 7241 | . . . . . . . . . 10 | |
22 | nnm0 6434 | . . . . . . . . . 10 | |
23 | 20, 21, 22 | 3syl 17 | . . . . . . . . 9 |
24 | 23 | adantl 275 | . . . . . . . 8 |
25 | 16, 24 | eqtr4d 2200 | . . . . . . 7 |
26 | 10, 5 | eqeltrdi 2255 | . . . . . . . 8 |
27 | enq0eceq 7369 | . . . . . . . . 9 ~Q0 ~Q0 | |
28 | 5, 6, 27 | mpanr12 436 | . . . . . . . 8 ~Q0 ~Q0 |
29 | 26, 20, 28 | syl2an 287 | . . . . . . 7 ~Q0 ~Q0 |
30 | 25, 29 | mpbird 166 | . . . . . 6 ~Q0 ~Q0 |
31 | 30, 2 | eqtr4di 2215 | . . . . 5 ~Q0 0Q0 |
32 | 31 | adantr 274 | . . . 4 ~Q0 ~Q0 0Q0 |
33 | 9, 32 | eqtrd 2197 | . . 3 ~Q0 0Q0 ·Q0 0Q0 |
34 | 33 | exlimivv 1883 | . 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 1342 wex 1479 wcel 2135 c0 3404 cop 3573 com 4561 (class class class)co 5836 c1o 6368 comu 6373 cec 6490 cnpi 7204 cmi 7206 ~Q0 ceq0 7218 Q0cnq0 7219 0Q0c0q0 7220 ·Q0 cmq0 7222 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-mi 7238 df-enq0 7356 df-nq0 7357 df-0nq0 7358 df-mq0 7360 |
This theorem is referenced by: prarloclem5 7432 |
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