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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 7375 | . . . . . . 7 | |
2 | 1 | adantl 277 | . . . . . 6 |
3 | simp1 997 | . . . . . . 7 | |
4 | simpll 527 | . . . . . . 7 | |
5 | prop 7449 | . . . . . . . 8 | |
6 | elprnqu 7456 | . . . . . . . 8 | |
7 | 5, 6 | sylan 283 | . . . . . . 7 |
8 | 3, 4, 7 | syl2an 289 | . . . . . 6 |
9 | simprl 529 | . . . . . . 7 | |
10 | elprnqu 7456 | . . . . . . . 8 | |
11 | 5, 10 | sylan 283 | . . . . . . 7 |
12 | 3, 9, 11 | syl2an 289 | . . . . . 6 |
13 | simpl3 1002 | . . . . . . 7 | |
14 | simprrr 540 | . . . . . . 7 | |
15 | prop 7449 | . . . . . . . 8 | |
16 | elprnqu 7456 | . . . . . . . 8 | |
17 | 15, 16 | sylan 283 | . . . . . . 7 |
18 | 13, 14, 17 | syl2anc 411 | . . . . . 6 |
19 | mulcomnqg 7357 | . . . . . . 7 | |
20 | 19 | adantl 277 | . . . . . 6 |
21 | 2, 8, 12, 18, 20 | caovord2d 6034 | . . . . 5 |
22 | mulclnq 7350 | . . . . . . 7 | |
23 | 8, 18, 22 | syl2anc 411 | . . . . . 6 |
24 | mulclnq 7350 | . . . . . . 7 | |
25 | 12, 18, 24 | syl2anc 411 | . . . . . 6 |
26 | simpl2 1001 | . . . . . . . 8 | |
27 | simprlr 538 | . . . . . . . 8 | |
28 | prop 7449 | . . . . . . . . 9 | |
29 | elprnqu 7456 | . . . . . . . . 9 | |
30 | 28, 29 | sylan 283 | . . . . . . . 8 |
31 | 26, 27, 30 | syl2anc 411 | . . . . . . 7 |
32 | mulclnq 7350 | . . . . . . 7 | |
33 | 8, 31, 32 | syl2anc 411 | . . . . . 6 |
34 | ltanqg 7374 | . . . . . 6 | |
35 | 23, 25, 33, 34 | syl3anc 1238 | . . . . 5 |
36 | 21, 35 | bitrd 188 | . . . 4 |
37 | simpl1 1000 | . . . . . 6 | |
38 | addclpr 7511 | . . . . . . . 8 | |
39 | 38 | 3adant1 1015 | . . . . . . 7 |
40 | 39 | adantr 276 | . . . . . 6 |
41 | mulclpr 7546 | . . . . . 6 | |
42 | 37, 40, 41 | syl2anc 411 | . . . . 5 |
43 | distrnqg 7361 | . . . . . . 7 | |
44 | 8, 31, 18, 43 | syl3anc 1238 | . . . . . 6 |
45 | simprll 537 | . . . . . . 7 | |
46 | df-iplp 7442 | . . . . . . . . . 10 | |
47 | addclnq 7349 | . . . . . . . . . 10 | |
48 | 46, 47 | genppreclu 7489 | . . . . . . . . 9 |
49 | 48 | imp 124 | . . . . . . . 8 |
50 | 26, 13, 27, 14, 49 | syl22anc 1239 | . . . . . . 7 |
51 | df-imp 7443 | . . . . . . . . 9 | |
52 | mulclnq 7350 | . . . . . . . . 9 | |
53 | 51, 52 | genppreclu 7489 | . . . . . . . 8 |
54 | 53 | imp 124 | . . . . . . 7 |
55 | 37, 40, 45, 50, 54 | syl22anc 1239 | . . . . . 6 |
56 | 44, 55 | eqeltrrd 2253 | . . . . 5 |
57 | prop 7449 | . . . . . 6 | |
58 | prcunqu 7459 | . . . . . 6 | |
59 | 57, 58 | sylan 283 | . . . . 5 |
60 | 42, 56, 59 | syl2anc 411 | . . . 4 |
61 | 36, 60 | sylbid 150 | . . 3 |
62 | 2, 12, 8, 31, 20 | caovord2d 6034 | . . . . 5 |
63 | ltanqg 7374 | . . . . . . 7 | |
64 | 63 | adantl 277 | . . . . . 6 |
65 | mulclnq 7350 | . . . . . . 7 | |
66 | 12, 31, 65 | syl2anc 411 | . . . . . 6 |
67 | addcomnqg 7355 | . . . . . . 7 | |
68 | 67 | adantl 277 | . . . . . 6 |
69 | 64, 66, 33, 25, 68 | caovord2d 6034 | . . . . 5 |
70 | 62, 69 | bitrd 188 | . . . 4 |
71 | distrnqg 7361 | . . . . . . 7 | |
72 | 12, 31, 18, 71 | syl3anc 1238 | . . . . . 6 |
73 | simprrl 539 | . . . . . . 7 | |
74 | 51, 52 | genppreclu 7489 | . . . . . . . 8 |
75 | 74 | imp 124 | . . . . . . 7 |
76 | 37, 40, 73, 50, 75 | syl22anc 1239 | . . . . . 6 |
77 | 72, 76 | eqeltrrd 2253 | . . . . 5 |
78 | prcunqu 7459 | . . . . . 6 | |
79 | 57, 78 | sylan 283 | . . . . 5 |
80 | 42, 77, 79 | syl2anc 411 | . . . 4 |
81 | 70, 80 | sylbid 150 | . . 3 |
82 | 61, 81 | jaod 717 | . 2 |
83 | ltsonq 7372 | . . . . 5 | |
84 | nqtri3or 7370 | . . . . 5 | |
85 | 83, 84 | sotritrieq 4319 | . . . 4 |
86 | 8, 12, 85 | syl2anc 411 | . . 3 |
87 | oveq1 5872 | . . . . . . 7 | |
88 | 87 | oveq2d 5881 | . . . . . 6 |
89 | 44, 88 | sylan9eq 2228 | . . . . 5 |
90 | 55 | adantr 276 | . . . . 5 |
91 | 89, 90 | eqeltrrd 2253 | . . . 4 |
92 | 91 | ex 115 | . . 3 |
93 | 86, 92 | sylbird 170 | . 2 |
94 | ltdcnq 7371 | . . . . 5 DECID | |
95 | ltdcnq 7371 | . . . . . 6 DECID | |
96 | 95 | ancoms 268 | . . . . 5 DECID |
97 | dcor 935 | . . . . 5 DECID DECID DECID | |
98 | 94, 96, 97 | sylc 62 | . . . 4 DECID |
99 | 8, 12, 98 | syl2anc 411 | . . 3 DECID |
100 | df-dc 835 | . . 3 DECID | |
101 | 99, 100 | sylib 122 | . 2 |
102 | 82, 93, 101 | mpjaod 718 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 104 wb 105 wo 708 DECID wdc 834 w3a 978 wceq 1353 wcel 2146 cop 3592 class class class wbr 3998 cfv 5208 (class class class)co 5865 c1st 6129 c2nd 6130 cnq 7254 cplq 7256 cmq 7257 cltq 7259 cnp 7265 cpp 7267 cmp 7268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-eprel 4283 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-irdg 6361 df-1o 6407 df-2o 6408 df-oadd 6411 df-omul 6412 df-er 6525 df-ec 6527 df-qs 6531 df-ni 7278 df-pli 7279 df-mi 7280 df-lti 7281 df-plpq 7318 df-mpq 7319 df-enq 7321 df-nqqs 7322 df-plqqs 7323 df-mqqs 7324 df-1nqqs 7325 df-rq 7326 df-ltnqqs 7327 df-enq0 7398 df-nq0 7399 df-0nq0 7400 df-plq0 7401 df-mq0 7402 df-inp 7440 df-iplp 7442 df-imp 7443 |
This theorem is referenced by: distrlem5pru 7561 |
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