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Mirrors > Home > ILE Home > Th. List > suplocexprlemex | Unicode version |
Description: Lemma for suplocexpr 7721. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.) |
Ref | Expression |
---|---|
suplocexpr.m |
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suplocexpr.ub |
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suplocexpr.loc |
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suplocexpr.b |
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Ref | Expression |
---|---|
suplocexprlemex |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suplocexpr.b |
. . 3
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2 | suplocexpr.m |
. . . . . 6
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3 | suplocexpr.ub |
. . . . . 6
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4 | suplocexpr.loc |
. . . . . 6
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5 | 2, 3, 4 | suplocexprlemss 7711 |
. . . . 5
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6 | 1 | suplocexprlem2b 7710 |
. . . . 5
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7 | 5, 6 | syl 14 |
. . . 4
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8 | 7 | opeq2d 3785 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | 1, 8 | eqtr4id 2229 |
. 2
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10 | suplocexprlemell 7709 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 10 | biimpi 120 |
. . . . . . . 8
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12 | 11 | adantl 277 |
. . . . . . 7
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13 | 5 | ad2antrr 488 |
. . . . . . . . . 10
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14 | simprl 529 |
. . . . . . . . . 10
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15 | 13, 14 | sseldd 3156 |
. . . . . . . . 9
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16 | prop 7471 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 15, 16 | syl 14 |
. . . . . . . 8
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18 | simprr 531 |
. . . . . . . 8
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19 | elprnql 7477 |
. . . . . . . 8
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20 | 17, 18, 19 | syl2anc 411 |
. . . . . . 7
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21 | 12, 20 | rexlimddv 2599 |
. . . . . 6
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22 | 21 | ex 115 |
. . . . 5
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23 | 22 | ssrdv 3161 |
. . . 4
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24 | ssrab2 3240 |
. . . . 5
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25 | 7, 24 | eqsstrdi 3207 |
. . . 4
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26 | 2, 3, 4 | suplocexprlemml 7712 |
. . . . 5
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27 | 2, 3, 4, 1 | suplocexprlemmu 7714 |
. . . . 5
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28 | 26, 27 | jca 306 |
. . . 4
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29 | 23, 25, 28 | jca31 309 |
. . 3
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30 | 2, 3, 4 | suplocexprlemrl 7713 |
. . . . 5
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31 | 2, 3, 4, 1 | suplocexprlemru 7715 |
. . . . 5
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32 | 30, 31 | jca 306 |
. . . 4
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33 | 2, 3, 4, 1 | suplocexprlemdisj 7716 |
. . . 4
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34 | 2, 3, 4, 1 | suplocexprlemloc 7717 |
. . . 4
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35 | 32, 33, 34 | 3jca 1177 |
. . 3
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36 | elinp 7470 |
. . 3
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37 | 29, 35, 36 | sylanbrc 417 |
. 2
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38 | 9, 37 | eqeltrd 2254 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 |
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 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-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-eprel 4288 df-id 4292 df-po 4295 df-iso 4296 df-iord 4365 df-on 4367 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-recs 6303 df-irdg 6368 df-1o 6414 df-2o 6415 df-oadd 6418 df-omul 6419 df-er 6532 df-ec 6534 df-qs 6538 df-ni 7300 df-pli 7301 df-mi 7302 df-lti 7303 df-plpq 7340 df-mpq 7341 df-enq 7343 df-nqqs 7344 df-plqqs 7345 df-mqqs 7346 df-1nqqs 7347 df-rq 7348 df-ltnqqs 7349 df-enq0 7420 df-nq0 7421 df-0nq0 7422 df-plq0 7423 df-mq0 7424 df-inp 7462 df-iltp 7466 |
This theorem is referenced by: suplocexprlemub 7719 suplocexpr 7721 |
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