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Mirrors > Home > ILE Home > Th. List > recvguniqlem | GIF version |
Description: Lemma for recvguniq 11006. Some of the rearrangements of the expressions. (Contributed by Jim Kingdon, 8-Aug-2021.) |
Ref | Expression |
---|---|
recvguniqlem.f | β’ (π β πΉ:ββΆβ) |
recvguniqlem.a | β’ (π β π΄ β β) |
recvguniqlem.b | β’ (π β π΅ β β) |
recvguniqlem.k | β’ (π β πΎ β β) |
recvguniqlem.lt1 | β’ (π β π΄ < ((πΉβπΎ) + ((π΄ β π΅) / 2))) |
recvguniqlem.lt2 | β’ (π β (πΉβπΎ) < (π΅ + ((π΄ β π΅) / 2))) |
Ref | Expression |
---|---|
recvguniqlem | β’ (π β β₯) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recvguniqlem.a | . . 3 β’ (π β π΄ β β) | |
2 | recvguniqlem.f | . . . . 5 β’ (π β πΉ:ββΆβ) | |
3 | recvguniqlem.k | . . . . 5 β’ (π β πΎ β β) | |
4 | 2, 3 | ffvelcdmd 5654 | . . . 4 β’ (π β (πΉβπΎ) β β) |
5 | recvguniqlem.b | . . . . . 6 β’ (π β π΅ β β) | |
6 | 1, 5 | resubcld 8340 | . . . . 5 β’ (π β (π΄ β π΅) β β) |
7 | 6 | rehalfcld 9167 | . . . 4 β’ (π β ((π΄ β π΅) / 2) β β) |
8 | 4, 7 | readdcld 7989 | . . 3 β’ (π β ((πΉβπΎ) + ((π΄ β π΅) / 2)) β β) |
9 | recvguniqlem.lt1 | . . 3 β’ (π β π΄ < ((πΉβπΎ) + ((π΄ β π΅) / 2))) | |
10 | 5, 7 | readdcld 7989 | . . . . 5 β’ (π β (π΅ + ((π΄ β π΅) / 2)) β β) |
11 | recvguniqlem.lt2 | . . . . 5 β’ (π β (πΉβπΎ) < (π΅ + ((π΄ β π΅) / 2))) | |
12 | 4, 10, 7, 11 | ltadd1dd 8515 | . . . 4 β’ (π β ((πΉβπΎ) + ((π΄ β π΅) / 2)) < ((π΅ + ((π΄ β π΅) / 2)) + ((π΄ β π΅) / 2))) |
13 | 5 | recnd 7988 | . . . . . 6 β’ (π β π΅ β β) |
14 | 7 | recnd 7988 | . . . . . 6 β’ (π β ((π΄ β π΅) / 2) β β) |
15 | 13, 14, 14 | addassd 7982 | . . . . 5 β’ (π β ((π΅ + ((π΄ β π΅) / 2)) + ((π΄ β π΅) / 2)) = (π΅ + (((π΄ β π΅) / 2) + ((π΄ β π΅) / 2)))) |
16 | 6 | recnd 7988 | . . . . . . 7 β’ (π β (π΄ β π΅) β β) |
17 | 16 | 2halvesd 9166 | . . . . . 6 β’ (π β (((π΄ β π΅) / 2) + ((π΄ β π΅) / 2)) = (π΄ β π΅)) |
18 | 17 | oveq2d 5893 | . . . . 5 β’ (π β (π΅ + (((π΄ β π΅) / 2) + ((π΄ β π΅) / 2))) = (π΅ + (π΄ β π΅))) |
19 | 1 | recnd 7988 | . . . . . 6 β’ (π β π΄ β β) |
20 | 13, 19 | pncan3d 8273 | . . . . 5 β’ (π β (π΅ + (π΄ β π΅)) = π΄) |
21 | 15, 18, 20 | 3eqtrd 2214 | . . . 4 β’ (π β ((π΅ + ((π΄ β π΅) / 2)) + ((π΄ β π΅) / 2)) = π΄) |
22 | 12, 21 | breqtrd 4031 | . . 3 β’ (π β ((πΉβπΎ) + ((π΄ β π΅) / 2)) < π΄) |
23 | 1, 8, 1, 9, 22 | lttrd 8085 | . 2 β’ (π β π΄ < π΄) |
24 | 1 | ltnrd 8071 | . 2 β’ (π β Β¬ π΄ < π΄) |
25 | 23, 24 | pm2.21fal 1373 | 1 β’ (π β β₯) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β₯wfal 1358 β wcel 2148 class class class wbr 4005 βΆwf 5214 βcfv 5218 (class class class)co 5877 βcr 7812 + caddc 7816 < clt 7994 β cmin 8130 / cdiv 8631 βcn 8921 2c2 8972 |
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 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-id 4295 df-po 4298 df-iso 4299 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-2 8980 |
This theorem is referenced by: recvguniq 11006 |
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