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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 7333 | . . . . . . 7 | |
2 | 1 | adantl 275 | . . . . . 6 |
3 | simp1 986 | . . . . . . 7 | |
4 | simpll 519 | . . . . . . 7 | |
5 | prop 7407 | . . . . . . . 8 | |
6 | elprnqu 7414 | . . . . . . . 8 | |
7 | 5, 6 | sylan 281 | . . . . . . 7 |
8 | 3, 4, 7 | syl2an 287 | . . . . . 6 |
9 | simprl 521 | . . . . . . 7 | |
10 | elprnqu 7414 | . . . . . . . 8 | |
11 | 5, 10 | sylan 281 | . . . . . . 7 |
12 | 3, 9, 11 | syl2an 287 | . . . . . 6 |
13 | simpl3 991 | . . . . . . 7 | |
14 | simprrr 530 | . . . . . . 7 | |
15 | prop 7407 | . . . . . . . 8 | |
16 | elprnqu 7414 | . . . . . . . 8 | |
17 | 15, 16 | sylan 281 | . . . . . . 7 |
18 | 13, 14, 17 | syl2anc 409 | . . . . . 6 |
19 | mulcomnqg 7315 | . . . . . . 7 | |
20 | 19 | adantl 275 | . . . . . 6 |
21 | 2, 8, 12, 18, 20 | caovord2d 6002 | . . . . 5 |
22 | mulclnq 7308 | . . . . . . 7 | |
23 | 8, 18, 22 | syl2anc 409 | . . . . . 6 |
24 | mulclnq 7308 | . . . . . . 7 | |
25 | 12, 18, 24 | syl2anc 409 | . . . . . 6 |
26 | simpl2 990 | . . . . . . . 8 | |
27 | simprlr 528 | . . . . . . . 8 | |
28 | prop 7407 | . . . . . . . . 9 | |
29 | elprnqu 7414 | . . . . . . . . 9 | |
30 | 28, 29 | sylan 281 | . . . . . . . 8 |
31 | 26, 27, 30 | syl2anc 409 | . . . . . . 7 |
32 | mulclnq 7308 | . . . . . . 7 | |
33 | 8, 31, 32 | syl2anc 409 | . . . . . 6 |
34 | ltanqg 7332 | . . . . . 6 | |
35 | 23, 25, 33, 34 | syl3anc 1227 | . . . . 5 |
36 | 21, 35 | bitrd 187 | . . . 4 |
37 | simpl1 989 | . . . . . 6 | |
38 | addclpr 7469 | . . . . . . . 8 | |
39 | 38 | 3adant1 1004 | . . . . . . 7 |
40 | 39 | adantr 274 | . . . . . 6 |
41 | mulclpr 7504 | . . . . . 6 | |
42 | 37, 40, 41 | syl2anc 409 | . . . . 5 |
43 | distrnqg 7319 | . . . . . . 7 | |
44 | 8, 31, 18, 43 | syl3anc 1227 | . . . . . 6 |
45 | simprll 527 | . . . . . . 7 | |
46 | df-iplp 7400 | . . . . . . . . . 10 | |
47 | addclnq 7307 | . . . . . . . . . 10 | |
48 | 46, 47 | genppreclu 7447 | . . . . . . . . 9 |
49 | 48 | imp 123 | . . . . . . . 8 |
50 | 26, 13, 27, 14, 49 | syl22anc 1228 | . . . . . . 7 |
51 | df-imp 7401 | . . . . . . . . 9 | |
52 | mulclnq 7308 | . . . . . . . . 9 | |
53 | 51, 52 | genppreclu 7447 | . . . . . . . 8 |
54 | 53 | imp 123 | . . . . . . 7 |
55 | 37, 40, 45, 50, 54 | syl22anc 1228 | . . . . . 6 |
56 | 44, 55 | eqeltrrd 2242 | . . . . 5 |
57 | prop 7407 | . . . . . 6 | |
58 | prcunqu 7417 | . . . . . 6 | |
59 | 57, 58 | sylan 281 | . . . . 5 |
60 | 42, 56, 59 | syl2anc 409 | . . . 4 |
61 | 36, 60 | sylbid 149 | . . 3 |
62 | 2, 12, 8, 31, 20 | caovord2d 6002 | . . . . 5 |
63 | ltanqg 7332 | . . . . . . 7 | |
64 | 63 | adantl 275 | . . . . . 6 |
65 | mulclnq 7308 | . . . . . . 7 | |
66 | 12, 31, 65 | syl2anc 409 | . . . . . 6 |
67 | addcomnqg 7313 | . . . . . . 7 | |
68 | 67 | adantl 275 | . . . . . 6 |
69 | 64, 66, 33, 25, 68 | caovord2d 6002 | . . . . 5 |
70 | 62, 69 | bitrd 187 | . . . 4 |
71 | distrnqg 7319 | . . . . . . 7 | |
72 | 12, 31, 18, 71 | syl3anc 1227 | . . . . . 6 |
73 | simprrl 529 | . . . . . . 7 | |
74 | 51, 52 | genppreclu 7447 | . . . . . . . 8 |
75 | 74 | imp 123 | . . . . . . 7 |
76 | 37, 40, 73, 50, 75 | syl22anc 1228 | . . . . . 6 |
77 | 72, 76 | eqeltrrd 2242 | . . . . 5 |
78 | prcunqu 7417 | . . . . . 6 | |
79 | 57, 78 | sylan 281 | . . . . 5 |
80 | 42, 77, 79 | syl2anc 409 | . . . 4 |
81 | 70, 80 | sylbid 149 | . . 3 |
82 | 61, 81 | jaod 707 | . 2 |
83 | ltsonq 7330 | . . . . 5 | |
84 | nqtri3or 7328 | . . . . 5 | |
85 | 83, 84 | sotritrieq 4297 | . . . 4 |
86 | 8, 12, 85 | syl2anc 409 | . . 3 |
87 | oveq1 5843 | . . . . . . 7 | |
88 | 87 | oveq2d 5852 | . . . . . 6 |
89 | 44, 88 | sylan9eq 2217 | . . . . 5 |
90 | 55 | adantr 274 | . . . . 5 |
91 | 89, 90 | eqeltrrd 2242 | . . . 4 |
92 | 91 | ex 114 | . . 3 |
93 | 86, 92 | sylbird 169 | . 2 |
94 | ltdcnq 7329 | . . . . 5 DECID | |
95 | ltdcnq 7329 | . . . . . 6 DECID | |
96 | 95 | ancoms 266 | . . . . 5 DECID |
97 | dcor 924 | . . . . 5 DECID DECID DECID | |
98 | 94, 96, 97 | sylc 62 | . . . 4 DECID |
99 | 8, 12, 98 | syl2anc 409 | . . 3 DECID |
100 | df-dc 825 | . . 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 824 w3a 967 wceq 1342 wcel 2135 cop 3573 class class class wbr 3976 cfv 5182 (class class class)co 5836 c1st 6098 c2nd 6099 cnq 7212 cplq 7214 cmq 7215 cltq 7217 cnp 7223 cpp 7225 cmp 7226 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-2o 6376 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 df-enq0 7356 df-nq0 7357 df-0nq0 7358 df-plq0 7359 df-mq0 7360 df-inp 7398 df-iplp 7400 df-imp 7401 |
This theorem is referenced by: distrlem5pru 7519 |
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