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| Mirrors > Home > ILE Home > Th. List > dvidrelem | Unicode version | ||
| Description: Lemma for dvidre 15674 and dvconstre 15673. Analogue of dvidlemap 15668 for real numbers rather than complex numbers. (Contributed by Jim Kingdon, 3-Oct-2025.) |
| Ref | Expression |
|---|---|
| dvidrelem.1 |
|
| dvidrelem.2 |
|
| dvidrelem.3 |
|
| Ref | Expression |
|---|---|
| dvidrelem |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dvidrelem.1 |
. . . . . 6
| |
| 2 | reex 8277 |
. . . . . . 7
| |
| 3 | cnex 8267 |
. . . . . . 7
| |
| 4 | 2, 3 | fpm 6928 |
. . . . . 6
|
| 5 | 1, 4 | syl 14 |
. . . . 5
|
| 6 | dvfpm 15666 |
. . . . 5
| |
| 7 | 5, 6 | syl 14 |
. . . 4
|
| 8 | ax-resscn 8235 |
. . . . . . . 8
| |
| 9 | 8 | a1i 9 |
. . . . . . 7
|
| 10 | ssidd 3263 |
. . . . . . 7
| |
| 11 | 9, 1, 10 | dvbss 15662 |
. . . . . 6
|
| 12 | reldvg 15656 |
. . . . . . . . 9
| |
| 13 | 9, 5, 12 | syl2anc 411 |
. . . . . . . 8
|
| 14 | 13 | adantr 276 |
. . . . . . 7
|
| 15 | simpr 110 |
. . . . . . . . 9
| |
| 16 | retop 15501 |
. . . . . . . . . 10
| |
| 17 | uniretop 15502 |
. . . . . . . . . . 11
| |
| 18 | 17 | ntrtop 15105 |
. . . . . . . . . 10
|
| 19 | 16, 18 | ax-mp 5 |
. . . . . . . . 9
|
| 20 | 15, 19 | eleqtrrdi 2328 |
. . . . . . . 8
|
| 21 | limcresi 15643 |
. . . . . . . . . 10
| |
| 22 | dvidrelem.3 |
. . . . . . . . . . . 12
| |
| 23 | ssidd 3263 |
. . . . . . . . . . . 12
| |
| 24 | cncfmptc 15573 |
. . . . . . . . . . . 12
| |
| 25 | 22, 8, 23, 24 | mp3an12i 1378 |
. . . . . . . . . . 11
|
| 26 | eqidd 2235 |
. . . . . . . . . . 11
| |
| 27 | 25, 15, 26 | cnmptlimc 15651 |
. . . . . . . . . 10
|
| 28 | 21, 27 | sselid 3240 |
. . . . . . . . 9
|
| 29 | breq1 4117 |
. . . . . . . . . . . . . 14
| |
| 30 | 29 | elrab 2976 |
. . . . . . . . . . . . 13
|
| 31 | dvidrelem.2 |
. . . . . . . . . . . . . . 15
| |
| 32 | 31 | 3exp2 1252 |
. . . . . . . . . . . . . 14
|
| 33 | 32 | imp43 355 |
. . . . . . . . . . . . 13
|
| 34 | 30, 33 | sylan2b 287 |
. . . . . . . . . . . 12
|
| 35 | 34 | mpteq2dva 4205 |
. . . . . . . . . . 11
|
| 36 | ssrab2 3327 |
. . . . . . . . . . . 12
| |
| 37 | resmpt 5091 |
. . . . . . . . . . . 12
| |
| 38 | 36, 37 | ax-mp 5 |
. . . . . . . . . . 11
|
| 39 | 35, 38 | eqtr4di 2285 |
. . . . . . . . . 10
|
| 40 | 39 | oveq1d 6073 |
. . . . . . . . 9
|
| 41 | 28, 40 | eleqtrrd 2314 |
. . . . . . . 8
|
| 42 | eqid 2234 |
. . . . . . . . . 10
| |
| 43 | 42 | tgioo2cntop 15534 |
. . . . . . . . 9
|
| 44 | eqid 2234 |
. . . . . . . . 9
| |
| 45 | 8 | a1i 9 |
. . . . . . . . 9
|
| 46 | 1 | adantr 276 |
. . . . . . . . 9
|
| 47 | ssidd 3263 |
. . . . . . . . 9
| |
| 48 | 43, 42, 44, 45, 46, 47 | eldvap 15659 |
. . . . . . . 8
|
| 49 | 20, 41, 48 | mpbir2and 953 |
. . . . . . 7
|
| 50 | releldm 4997 |
. . . . . . 7
| |
| 51 | 14, 49, 50 | syl2anc 411 |
. . . . . 6
|
| 52 | 11, 51 | eqelssd 3261 |
. . . . 5
|
| 53 | 52 | feq2d 5501 |
. . . 4
|
| 54 | 7, 53 | mpbid 147 |
. . 3
|
| 55 | 54 | ffnd 5514 |
. 2
|
| 56 | fnconstg 5570 |
. . 3
| |
| 57 | 22, 56 | mp1i 10 |
. 2
|
| 58 | 7 | adantr 276 |
. . . . . 6
|
| 59 | 58 | ffund 5517 |
. . . . 5
|
| 60 | funbrfvb 5722 |
. . . . 5
| |
| 61 | 59, 51, 60 | syl2anc 411 |
. . . 4
|
| 62 | 49, 61 | mpbird 167 |
. . 3
|
| 63 | 22 | a1i 9 |
. . . 4
|
| 64 | fvconst2g 5903 |
. . . 4
| |
| 65 | 63, 64 | sylan 283 |
. . 3
|
| 66 | 62, 65 | eqtr4d 2270 |
. 2
|
| 67 | 55, 57, 66 | eqfnfvd 5783 |
1
|
| 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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263 |
| This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-map 6897 df-pm 6898 df-sup 7288 df-inf 7289 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-n0 9514 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-xneg 10124 df-xadd 10125 df-ioo 10244 df-seqfrec 10834 df-exp 10925 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-rest 13538 df-topgen 13557 df-psmet 14803 df-xmet 14804 df-met 14805 df-bl 14806 df-mopn 14807 df-top 14975 df-topon 14988 df-bases 15020 df-ntr 15073 df-cn 15165 df-cnp 15166 df-cncf 15548 df-limced 15633 df-dvap 15634 |
| This theorem is referenced by: dvconstre 15673 dvidre 15674 |
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