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Mirrors > Home > ILE Home > Th. List > addlocprlemgt | Unicode version |
Description: Lemma for addlocpr 7351. The case. (Contributed by Jim Kingdon, 6-Dec-2019.) |
Ref | Expression |
---|---|
addlocprlem.a | |
addlocprlem.b | |
addlocprlem.qr | |
addlocprlem.p | |
addlocprlem.qppr | |
addlocprlem.dlo | |
addlocprlem.uup | |
addlocprlem.du | |
addlocprlem.elo | |
addlocprlem.tup | |
addlocprlem.et |
Ref | Expression |
---|---|
addlocprlemgt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addlocprlem.a | . . . . . . 7 | |
2 | addlocprlem.b | . . . . . . 7 | |
3 | addlocprlem.qr | . . . . . . 7 | |
4 | addlocprlem.p | . . . . . . 7 | |
5 | addlocprlem.qppr | . . . . . . 7 | |
6 | addlocprlem.dlo | . . . . . . 7 | |
7 | addlocprlem.uup | . . . . . . 7 | |
8 | addlocprlem.du | . . . . . . 7 | |
9 | addlocprlem.elo | . . . . . . 7 | |
10 | addlocprlem.tup | . . . . . . 7 | |
11 | addlocprlem.et | . . . . . . 7 | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | addlocprlemeqgt 7347 | . . . . . 6 |
13 | 12 | adantr 274 | . . . . 5 |
14 | prop 7290 | . . . . . . . . . . . 12 | |
15 | 1, 14 | syl 14 | . . . . . . . . . . 11 |
16 | elprnql 7296 | . . . . . . . . . . 11 | |
17 | 15, 6, 16 | syl2anc 408 | . . . . . . . . . 10 |
18 | prop 7290 | . . . . . . . . . . . 12 | |
19 | 2, 18 | syl 14 | . . . . . . . . . . 11 |
20 | elprnql 7296 | . . . . . . . . . . 11 | |
21 | 19, 9, 20 | syl2anc 408 | . . . . . . . . . 10 |
22 | addclnq 7190 | . . . . . . . . . 10 | |
23 | 17, 21, 22 | syl2anc 408 | . . . . . . . . 9 |
24 | ltrelnq 7180 | . . . . . . . . . . . 12 | |
25 | 24 | brel 4591 | . . . . . . . . . . 11 |
26 | 3, 25 | syl 14 | . . . . . . . . . 10 |
27 | 26 | simpld 111 | . . . . . . . . 9 |
28 | addclnq 7190 | . . . . . . . . . 10 | |
29 | 4, 4, 28 | syl2anc 408 | . . . . . . . . 9 |
30 | ltanqg 7215 | . . . . . . . . 9 | |
31 | 23, 27, 29, 30 | syl3anc 1216 | . . . . . . . 8 |
32 | addcomnqg 7196 | . . . . . . . . . 10 | |
33 | 29, 23, 32 | syl2anc 408 | . . . . . . . . 9 |
34 | addcomnqg 7196 | . . . . . . . . . 10 | |
35 | 29, 27, 34 | syl2anc 408 | . . . . . . . . 9 |
36 | 33, 35 | breq12d 3942 | . . . . . . . 8 |
37 | 31, 36 | bitrd 187 | . . . . . . 7 |
38 | 37 | biimpa 294 | . . . . . 6 |
39 | 5 | breq2d 3941 | . . . . . . 7 |
40 | 39 | adantr 274 | . . . . . 6 |
41 | 38, 40 | mpbid 146 | . . . . 5 |
42 | 13, 41 | jca 304 | . . . 4 |
43 | ltsonq 7213 | . . . . 5 | |
44 | 43, 24 | sotri 4934 | . . . 4 |
45 | 42, 44 | syl 14 | . . 3 |
46 | 1, 7 | jca 304 | . . . . 5 |
47 | 2, 10 | jca 304 | . . . . 5 |
48 | 26 | simprd 113 | . . . . 5 |
49 | addnqpru 7345 | . . . . 5 | |
50 | 46, 47, 48, 49 | syl21anc 1215 | . . . 4 |
51 | 50 | adantr 274 | . . 3 |
52 | 45, 51 | mpd 13 | . 2 |
53 | 52 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cop 3530 class class class wbr 3929 cfv 5123 (class class class)co 5774 c1st 6036 c2nd 6037 cnq 7095 cplq 7097 cltq 7100 cnp 7106 cpp 7108 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7119 df-pli 7120 df-mi 7121 df-lti 7122 df-plpq 7159 df-mpq 7160 df-enq 7162 df-nqqs 7163 df-plqqs 7164 df-mqqs 7165 df-1nqqs 7166 df-rq 7167 df-ltnqqs 7168 df-inp 7281 df-iplp 7283 |
This theorem is referenced by: addlocprlem 7350 |
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