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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 7340 | . . . 4 | |
2 | nqpi 7340 | . . . 4 | |
3 | 1, 2 | anim12i 336 | . . 3 |
4 | ee4anv 1927 | . . 3 | |
5 | 3, 4 | sylibr 133 | . 2 |
6 | oveq12 5862 | . . . . . . 7 | |
7 | mulpiord 7279 | . . . . . . . . . . 11 | |
8 | 7 | ad2ant2r 506 | . . . . . . . . . 10 |
9 | mulpiord 7279 | . . . . . . . . . . 11 | |
10 | 9 | ad2ant2l 505 | . . . . . . . . . 10 |
11 | 8, 10 | opeq12d 3773 | . . . . . . . . 9 |
12 | 11 | eceq1d 6549 | . . . . . . . 8 ~Q0 ~Q0 |
13 | mulpipqqs 7335 | . . . . . . . . 9 | |
14 | mulclpi 7290 | . . . . . . . . . . 11 | |
15 | 14 | ad2ant2r 506 | . . . . . . . . . 10 |
16 | mulclpi 7290 | . . . . . . . . . . 11 | |
17 | 16 | ad2ant2l 505 | . . . . . . . . . 10 |
18 | nqnq0pi 7400 | . . . . . . . . . 10 ~Q0 | |
19 | 15, 17, 18 | syl2anc 409 | . . . . . . . . 9 ~Q0 |
20 | 13, 19 | eqtr4d 2206 | . . . . . . . 8 ~Q0 |
21 | pinn 7271 | . . . . . . . . . 10 | |
22 | 21 | anim1i 338 | . . . . . . . . 9 |
23 | pinn 7271 | . . . . . . . . . 10 | |
24 | 23 | anim1i 338 | . . . . . . . . 9 |
25 | mulnnnq0 7412 | . . . . . . . . 9 ~Q0 ·Q0 ~Q0 ~Q0 | |
26 | 22, 24, 25 | syl2an 287 | . . . . . . . 8 ~Q0 ·Q0 ~Q0 ~Q0 |
27 | 12, 20, 26 | 3eqtr4d 2213 | . . . . . . 7 ~Q0 ·Q0 ~Q0 |
28 | 6, 27 | sylan9eqr 2225 | . . . . . 6 ~Q0 ·Q0 ~Q0 |
29 | nqnq0pi 7400 | . . . . . . . . . . 11 ~Q0 | |
30 | 29 | adantr 274 | . . . . . . . . . 10 ~Q0 |
31 | 30 | eqeq2d 2182 | . . . . . . . . 9 ~Q0 |
32 | nqnq0pi 7400 | . . . . . . . . . . 11 ~Q0 | |
33 | 32 | adantl 275 | . . . . . . . . . 10 ~Q0 |
34 | 33 | eqeq2d 2182 | . . . . . . . . 9 ~Q0 |
35 | 31, 34 | anbi12d 470 | . . . . . . . 8 ~Q0 ~Q0 |
36 | 35 | pm5.32i 451 | . . . . . . 7 ~Q0 ~Q0 |
37 | oveq12 5862 | . . . . . . . 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 2206 | . . . . 5 ·Q0 |
41 | 40 | an4s 583 | . . . 4 ·Q0 |
42 | 41 | exlimivv 1889 | . . 3 ·Q0 |
43 | 42 | exlimivv 1889 | . 2 ·Q0 |
44 | 5, 43 | syl 14 | 1 ·Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wex 1485 wcel 2141 cop 3586 com 4574 (class class class)co 5853 comu 6393 cec 6511 cnpi 7234 cmi 7236 ceq 7241 cnq 7242 cmq 7245 ~Q0 ceq0 7248 ·Q0 cmq0 7252 |
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 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
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 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-oadd 6399 df-omul 6400 df-er 6513 df-ec 6515 df-qs 6519 df-ni 7266 df-mi 7268 df-mpq 7307 df-enq 7309 df-nqqs 7310 df-mqqs 7312 df-enq0 7386 df-nq0 7387 df-mq0 7390 |
This theorem is referenced by: prarloclemlo 7456 prarloclemcalc 7464 |
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