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Mirrors > Home > ILE Home > Th. List > distrlem4prl | Unicode version |
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
Ref | Expression |
---|---|
distrlem4prl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmnqg 7316 | . . . . . . 7 | |
2 | 1 | adantl 275 | . . . . . 6 |
3 | simp1 982 | . . . . . . 7 | |
4 | simpll 519 | . . . . . . 7 | |
5 | prop 7390 | . . . . . . . 8 | |
6 | elprnql 7396 | . . . . . . . 8 | |
7 | 5, 6 | sylan 281 | . . . . . . 7 |
8 | 3, 4, 7 | syl2an 287 | . . . . . 6 |
9 | simprl 521 | . . . . . . 7 | |
10 | elprnql 7396 | . . . . . . . 8 | |
11 | 5, 10 | sylan 281 | . . . . . . 7 |
12 | 3, 9, 11 | syl2an 287 | . . . . . 6 |
13 | simpl2 986 | . . . . . . 7 | |
14 | simprlr 528 | . . . . . . 7 | |
15 | prop 7390 | . . . . . . . 8 | |
16 | elprnql 7396 | . . . . . . . 8 | |
17 | 15, 16 | sylan 281 | . . . . . . 7 |
18 | 13, 14, 17 | syl2anc 409 | . . . . . 6 |
19 | mulcomnqg 7298 | . . . . . . 7 | |
20 | 19 | adantl 275 | . . . . . 6 |
21 | 2, 8, 12, 18, 20 | caovord2d 5987 | . . . . 5 |
22 | ltanqg 7315 | . . . . . . 7 | |
23 | 22 | adantl 275 | . . . . . 6 |
24 | mulclnq 7291 | . . . . . . 7 | |
25 | 8, 18, 24 | syl2anc 409 | . . . . . 6 |
26 | mulclnq 7291 | . . . . . . 7 | |
27 | 12, 18, 26 | syl2anc 409 | . . . . . 6 |
28 | simpl3 987 | . . . . . . . 8 | |
29 | simprrr 530 | . . . . . . . 8 | |
30 | prop 7390 | . . . . . . . . 9 | |
31 | elprnql 7396 | . . . . . . . . 9 | |
32 | 30, 31 | sylan 281 | . . . . . . . 8 |
33 | 28, 29, 32 | syl2anc 409 | . . . . . . 7 |
34 | mulclnq 7291 | . . . . . . 7 | |
35 | 12, 33, 34 | syl2anc 409 | . . . . . 6 |
36 | addcomnqg 7296 | . . . . . . 7 | |
37 | 36 | adantl 275 | . . . . . 6 |
38 | 23, 25, 27, 35, 37 | caovord2d 5987 | . . . . 5 |
39 | 21, 38 | bitrd 187 | . . . 4 |
40 | simpl1 985 | . . . . . 6 | |
41 | addclpr 7452 | . . . . . . . 8 | |
42 | 41 | 3adant1 1000 | . . . . . . 7 |
43 | 42 | adantr 274 | . . . . . 6 |
44 | mulclpr 7487 | . . . . . 6 | |
45 | 40, 43, 44 | syl2anc 409 | . . . . 5 |
46 | distrnqg 7302 | . . . . . . 7 | |
47 | 12, 18, 33, 46 | syl3anc 1220 | . . . . . 6 |
48 | simprrl 529 | . . . . . . 7 | |
49 | df-iplp 7383 | . . . . . . . . . 10 | |
50 | addclnq 7290 | . . . . . . . . . 10 | |
51 | 49, 50 | genpprecll 7429 | . . . . . . . . 9 |
52 | 51 | imp 123 | . . . . . . . 8 |
53 | 13, 28, 14, 29, 52 | syl22anc 1221 | . . . . . . 7 |
54 | df-imp 7384 | . . . . . . . . 9 | |
55 | mulclnq 7291 | . . . . . . . . 9 | |
56 | 54, 55 | genpprecll 7429 | . . . . . . . 8 |
57 | 56 | imp 123 | . . . . . . 7 |
58 | 40, 43, 48, 53, 57 | syl22anc 1221 | . . . . . 6 |
59 | 47, 58 | eqeltrrd 2235 | . . . . 5 |
60 | prop 7390 | . . . . . 6 | |
61 | prcdnql 7399 | . . . . . 6 | |
62 | 60, 61 | sylan 281 | . . . . 5 |
63 | 45, 59, 62 | syl2anc 409 | . . . 4 |
64 | 39, 63 | sylbid 149 | . . 3 |
65 | 2, 12, 8, 33, 20 | caovord2d 5987 | . . . . 5 |
66 | mulclnq 7291 | . . . . . . 7 | |
67 | 8, 33, 66 | syl2anc 409 | . . . . . 6 |
68 | ltanqg 7315 | . . . . . 6 | |
69 | 35, 67, 25, 68 | syl3anc 1220 | . . . . 5 |
70 | 65, 69 | bitrd 187 | . . . 4 |
71 | distrnqg 7302 | . . . . . . 7 | |
72 | 8, 18, 33, 71 | syl3anc 1220 | . . . . . 6 |
73 | simprll 527 | . . . . . . 7 | |
74 | 54, 55 | genpprecll 7429 | . . . . . . . 8 |
75 | 74 | imp 123 | . . . . . . 7 |
76 | 40, 43, 73, 53, 75 | syl22anc 1221 | . . . . . 6 |
77 | 72, 76 | eqeltrrd 2235 | . . . . 5 |
78 | prcdnql 7399 | . . . . . 6 | |
79 | 60, 78 | sylan 281 | . . . . 5 |
80 | 45, 77, 79 | syl2anc 409 | . . . 4 |
81 | 70, 80 | sylbid 149 | . . 3 |
82 | 64, 81 | jaod 707 | . 2 |
83 | ltsonq 7313 | . . . . 5 | |
84 | nqtri3or 7311 | . . . . 5 | |
85 | 83, 84 | sotritrieq 4285 | . . . 4 |
86 | 8, 12, 85 | syl2anc 409 | . . 3 |
87 | oveq1 5828 | . . . . . . 7 | |
88 | 87 | oveq2d 5837 | . . . . . 6 |
89 | 72, 88 | sylan9eq 2210 | . . . . 5 |
90 | 76 | adantr 274 | . . . . 5 |
91 | 89, 90 | eqeltrrd 2235 | . . . 4 |
92 | 91 | ex 114 | . . 3 |
93 | 86, 92 | sylbird 169 | . 2 |
94 | ltdcnq 7312 | . . . . 5 DECID | |
95 | ltdcnq 7312 | . . . . . 6 DECID | |
96 | 95 | ancoms 266 | . . . . 5 DECID |
97 | dcor 920 | . . . . 5 DECID DECID DECID | |
98 | 94, 96, 97 | sylc 62 | . . . 4 DECID |
99 | 8, 12, 98 | syl2anc 409 | . . 3 DECID |
100 | df-dc 821 | . . 3 DECID | |
101 | 99, 100 | sylib 121 | . 2 |
102 | 82, 93, 101 | mpjaod 708 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 DECID wdc 820 w3a 963 wceq 1335 wcel 2128 cop 3563 class class class wbr 3965 cfv 5169 (class class class)co 5821 c1st 6083 c2nd 6084 cnq 7195 cplq 7197 cmq 7198 cltq 7200 cnp 7206 cpp 7208 cmp 7209 |
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 4495 ax-iinf 4546 |
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-eprel 4249 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-irdg 6314 df-1o 6360 df-2o 6361 df-oadd 6364 df-omul 6365 df-er 6477 df-ec 6479 df-qs 6483 df-ni 7219 df-pli 7220 df-mi 7221 df-lti 7222 df-plpq 7259 df-mpq 7260 df-enq 7262 df-nqqs 7263 df-plqqs 7264 df-mqqs 7265 df-1nqqs 7266 df-rq 7267 df-ltnqqs 7268 df-enq0 7339 df-nq0 7340 df-0nq0 7341 df-plq0 7342 df-mq0 7343 df-inp 7381 df-iplp 7383 df-imp 7384 |
This theorem is referenced by: distrlem5prl 7501 |
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