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Mirrors > Home > ILE Home > Th. List > distrlem4pru | Unicode version |
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
Ref | Expression |
---|---|
distrlem4pru |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmnqg 7177 | . . . . . . 7 | |
2 | 1 | adantl 275 | . . . . . 6 |
3 | simp1 966 | . . . . . . 7 | |
4 | simpll 503 | . . . . . . 7 | |
5 | prop 7251 | . . . . . . . 8 | |
6 | elprnqu 7258 | . . . . . . . 8 | |
7 | 5, 6 | sylan 281 | . . . . . . 7 |
8 | 3, 4, 7 | syl2an 287 | . . . . . 6 |
9 | simprl 505 | . . . . . . 7 | |
10 | elprnqu 7258 | . . . . . . . 8 | |
11 | 5, 10 | sylan 281 | . . . . . . 7 |
12 | 3, 9, 11 | syl2an 287 | . . . . . 6 |
13 | simpl3 971 | . . . . . . 7 | |
14 | simprrr 514 | . . . . . . 7 | |
15 | prop 7251 | . . . . . . . 8 | |
16 | elprnqu 7258 | . . . . . . . 8 | |
17 | 15, 16 | sylan 281 | . . . . . . 7 |
18 | 13, 14, 17 | syl2anc 408 | . . . . . 6 |
19 | mulcomnqg 7159 | . . . . . . 7 | |
20 | 19 | adantl 275 | . . . . . 6 |
21 | 2, 8, 12, 18, 20 | caovord2d 5908 | . . . . 5 |
22 | mulclnq 7152 | . . . . . . 7 | |
23 | 8, 18, 22 | syl2anc 408 | . . . . . 6 |
24 | mulclnq 7152 | . . . . . . 7 | |
25 | 12, 18, 24 | syl2anc 408 | . . . . . 6 |
26 | simpl2 970 | . . . . . . . 8 | |
27 | simprlr 512 | . . . . . . . 8 | |
28 | prop 7251 | . . . . . . . . 9 | |
29 | elprnqu 7258 | . . . . . . . . 9 | |
30 | 28, 29 | sylan 281 | . . . . . . . 8 |
31 | 26, 27, 30 | syl2anc 408 | . . . . . . 7 |
32 | mulclnq 7152 | . . . . . . 7 | |
33 | 8, 31, 32 | syl2anc 408 | . . . . . 6 |
34 | ltanqg 7176 | . . . . . 6 | |
35 | 23, 25, 33, 34 | syl3anc 1201 | . . . . 5 |
36 | 21, 35 | bitrd 187 | . . . 4 |
37 | simpl1 969 | . . . . . 6 | |
38 | addclpr 7313 | . . . . . . . 8 | |
39 | 38 | 3adant1 984 | . . . . . . 7 |
40 | 39 | adantr 274 | . . . . . 6 |
41 | mulclpr 7348 | . . . . . 6 | |
42 | 37, 40, 41 | syl2anc 408 | . . . . 5 |
43 | distrnqg 7163 | . . . . . . 7 | |
44 | 8, 31, 18, 43 | syl3anc 1201 | . . . . . 6 |
45 | simprll 511 | . . . . . . 7 | |
46 | df-iplp 7244 | . . . . . . . . . 10 | |
47 | addclnq 7151 | . . . . . . . . . 10 | |
48 | 46, 47 | genppreclu 7291 | . . . . . . . . 9 |
49 | 48 | imp 123 | . . . . . . . 8 |
50 | 26, 13, 27, 14, 49 | syl22anc 1202 | . . . . . . 7 |
51 | df-imp 7245 | . . . . . . . . 9 | |
52 | mulclnq 7152 | . . . . . . . . 9 | |
53 | 51, 52 | genppreclu 7291 | . . . . . . . 8 |
54 | 53 | imp 123 | . . . . . . 7 |
55 | 37, 40, 45, 50, 54 | syl22anc 1202 | . . . . . 6 |
56 | 44, 55 | eqeltrrd 2195 | . . . . 5 |
57 | prop 7251 | . . . . . 6 | |
58 | prcunqu 7261 | . . . . . 6 | |
59 | 57, 58 | sylan 281 | . . . . 5 |
60 | 42, 56, 59 | syl2anc 408 | . . . 4 |
61 | 36, 60 | sylbid 149 | . . 3 |
62 | 2, 12, 8, 31, 20 | caovord2d 5908 | . . . . 5 |
63 | ltanqg 7176 | . . . . . . 7 | |
64 | 63 | adantl 275 | . . . . . 6 |
65 | mulclnq 7152 | . . . . . . 7 | |
66 | 12, 31, 65 | syl2anc 408 | . . . . . 6 |
67 | addcomnqg 7157 | . . . . . . 7 | |
68 | 67 | adantl 275 | . . . . . 6 |
69 | 64, 66, 33, 25, 68 | caovord2d 5908 | . . . . 5 |
70 | 62, 69 | bitrd 187 | . . . 4 |
71 | distrnqg 7163 | . . . . . . 7 | |
72 | 12, 31, 18, 71 | syl3anc 1201 | . . . . . 6 |
73 | simprrl 513 | . . . . . . 7 | |
74 | 51, 52 | genppreclu 7291 | . . . . . . . 8 |
75 | 74 | imp 123 | . . . . . . 7 |
76 | 37, 40, 73, 50, 75 | syl22anc 1202 | . . . . . 6 |
77 | 72, 76 | eqeltrrd 2195 | . . . . 5 |
78 | prcunqu 7261 | . . . . . 6 | |
79 | 57, 78 | sylan 281 | . . . . 5 |
80 | 42, 77, 79 | syl2anc 408 | . . . 4 |
81 | 70, 80 | sylbid 149 | . . 3 |
82 | 61, 81 | jaod 691 | . 2 |
83 | ltsonq 7174 | . . . . 5 | |
84 | nqtri3or 7172 | . . . . 5 | |
85 | 83, 84 | sotritrieq 4217 | . . . 4 |
86 | 8, 12, 85 | syl2anc 408 | . . 3 |
87 | oveq1 5749 | . . . . . . 7 | |
88 | 87 | oveq2d 5758 | . . . . . 6 |
89 | 44, 88 | sylan9eq 2170 | . . . . 5 |
90 | 55 | adantr 274 | . . . . 5 |
91 | 89, 90 | eqeltrrd 2195 | . . . 4 |
92 | 91 | ex 114 | . . 3 |
93 | 86, 92 | sylbird 169 | . 2 |
94 | ltdcnq 7173 | . . . . 5 DECID | |
95 | ltdcnq 7173 | . . . . . 6 DECID | |
96 | 95 | ancoms 266 | . . . . 5 DECID |
97 | dcor 904 | . . . . 5 DECID DECID DECID | |
98 | 94, 96, 97 | sylc 62 | . . . 4 DECID |
99 | 8, 12, 98 | syl2anc 408 | . . 3 DECID |
100 | df-dc 805 | . . 3 DECID | |
101 | 99, 100 | sylib 121 | . 2 |
102 | 82, 93, 101 | mpjaod 692 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 DECID wdc 804 w3a 947 wceq 1316 wcel 1465 cop 3500 class class class wbr 3899 cfv 5093 (class class class)co 5742 c1st 6004 c2nd 6005 cnq 7056 cplq 7058 cmq 7059 cltq 7061 cnp 7067 cpp 7069 cmp 7070 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-eprel 4181 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-2o 6282 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-pli 7081 df-mi 7082 df-lti 7083 df-plpq 7120 df-mpq 7121 df-enq 7123 df-nqqs 7124 df-plqqs 7125 df-mqqs 7126 df-1nqqs 7127 df-rq 7128 df-ltnqqs 7129 df-enq0 7200 df-nq0 7201 df-0nq0 7202 df-plq0 7203 df-mq0 7204 df-inp 7242 df-iplp 7244 df-imp 7245 |
This theorem is referenced by: distrlem5pru 7363 |
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