Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nqnq0m | Unicode version |
Description: Multiplication of positive fractions is equal with or ·Q0. (Contributed by Jim Kingdon, 10-Nov-2019.) |
Ref | Expression |
---|---|
nqnq0m | ·Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nqpi 7298 | . . . 4 | |
2 | nqpi 7298 | . . . 4 | |
3 | 1, 2 | anim12i 336 | . . 3 |
4 | ee4anv 1914 | . . 3 | |
5 | 3, 4 | sylibr 133 | . 2 |
6 | oveq12 5833 | . . . . . . 7 | |
7 | mulpiord 7237 | . . . . . . . . . . 11 | |
8 | 7 | ad2ant2r 501 | . . . . . . . . . 10 |
9 | mulpiord 7237 | . . . . . . . . . . 11 | |
10 | 9 | ad2ant2l 500 | . . . . . . . . . 10 |
11 | 8, 10 | opeq12d 3749 | . . . . . . . . 9 |
12 | 11 | eceq1d 6516 | . . . . . . . 8 ~Q0 ~Q0 |
13 | mulpipqqs 7293 | . . . . . . . . 9 | |
14 | mulclpi 7248 | . . . . . . . . . . 11 | |
15 | 14 | ad2ant2r 501 | . . . . . . . . . 10 |
16 | mulclpi 7248 | . . . . . . . . . . 11 | |
17 | 16 | ad2ant2l 500 | . . . . . . . . . 10 |
18 | nqnq0pi 7358 | . . . . . . . . . 10 ~Q0 | |
19 | 15, 17, 18 | syl2anc 409 | . . . . . . . . 9 ~Q0 |
20 | 13, 19 | eqtr4d 2193 | . . . . . . . 8 ~Q0 |
21 | pinn 7229 | . . . . . . . . . 10 | |
22 | 21 | anim1i 338 | . . . . . . . . 9 |
23 | pinn 7229 | . . . . . . . . . 10 | |
24 | 23 | anim1i 338 | . . . . . . . . 9 |
25 | mulnnnq0 7370 | . . . . . . . . 9 ~Q0 ·Q0 ~Q0 ~Q0 | |
26 | 22, 24, 25 | syl2an 287 | . . . . . . . 8 ~Q0 ·Q0 ~Q0 ~Q0 |
27 | 12, 20, 26 | 3eqtr4d 2200 | . . . . . . 7 ~Q0 ·Q0 ~Q0 |
28 | 6, 27 | sylan9eqr 2212 | . . . . . 6 ~Q0 ·Q0 ~Q0 |
29 | nqnq0pi 7358 | . . . . . . . . . . 11 ~Q0 | |
30 | 29 | adantr 274 | . . . . . . . . . 10 ~Q0 |
31 | 30 | eqeq2d 2169 | . . . . . . . . 9 ~Q0 |
32 | nqnq0pi 7358 | . . . . . . . . . . 11 ~Q0 | |
33 | 32 | adantl 275 | . . . . . . . . . 10 ~Q0 |
34 | 33 | eqeq2d 2169 | . . . . . . . . 9 ~Q0 |
35 | 31, 34 | anbi12d 465 | . . . . . . . 8 ~Q0 ~Q0 |
36 | 35 | pm5.32i 450 | . . . . . . 7 ~Q0 ~Q0 |
37 | oveq12 5833 | . . . . . . . 8 ~Q0 ~Q0 ·Q0 ~Q0 ·Q0 ~Q0 | |
38 | 37 | adantl 275 | . . . . . . 7 ~Q0 ~Q0 ·Q0 ~Q0 ·Q0 ~Q0 |
39 | 36, 38 | sylbir 134 | . . . . . 6 ·Q0 ~Q0 ·Q0 ~Q0 |
40 | 28, 39 | eqtr4d 2193 | . . . . 5 ·Q0 |
41 | 40 | an4s 578 | . . . 4 ·Q0 |
42 | 41 | exlimivv 1876 | . . 3 ·Q0 |
43 | 42 | exlimivv 1876 | . 2 ·Q0 |
44 | 5, 43 | syl 14 | 1 ·Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wex 1472 wcel 2128 cop 3563 com 4549 (class class class)co 5824 comu 6361 cec 6478 cnpi 7192 cmi 7194 ceq 7199 cnq 7200 cmq 7203 ~Q0 ceq0 7206 ·Q0 cmq0 7210 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-oadd 6367 df-omul 6368 df-er 6480 df-ec 6482 df-qs 6486 df-ni 7224 df-mi 7226 df-mpq 7265 df-enq 7267 df-nqqs 7268 df-mqqs 7270 df-enq0 7344 df-nq0 7345 df-mq0 7348 |
This theorem is referenced by: prarloclemlo 7414 prarloclemcalc 7422 |
Copyright terms: Public domain | W3C validator |