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| Mirrors > Home > ILE Home > Th. List > reeff1oleme | Unicode version | ||
| Description: Lemma for reeff1o 15245. (Contributed by Jim Kingdon, 15-May-2024.) |
| Ref | Expression |
|---|---|
| reeff1oleme |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ere 11981 |
. . . . 5
| |
| 2 | 1 | a1i 9 |
. . . 4
|
| 3 | elioore 10034 |
. . . 4
| |
| 4 | 0xr 8119 |
. . . . . . 7
| |
| 5 | 1 | rexri 8130 |
. . . . . . 7
|
| 6 | elioo2 10043 |
. . . . . . 7
| |
| 7 | 4, 5, 6 | mp2an 426 |
. . . . . 6
|
| 8 | 7 | simp2bi 1016 |
. . . . 5
|
| 9 | 3, 8 | gt0ap0d 8702 |
. . . 4
|
| 10 | 2, 3, 9 | redivclapd 8908 |
. . 3
|
| 11 | 3 | recnd 8101 |
. . . . . 6
|
| 12 | 11 | mulid2d 8091 |
. . . . 5
|
| 13 | 7 | simp3bi 1017 |
. . . . 5
|
| 14 | 12, 13 | eqbrtrd 4066 |
. . . 4
|
| 15 | 1red 8087 |
. . . . 5
| |
| 16 | ltmuldiv 8947 |
. . . . 5
| |
| 17 | 15, 2, 3, 8, 16 | syl112anc 1254 |
. . . 4
|
| 18 | 14, 17 | mpbid 147 |
. . 3
|
| 19 | reeff1olem 15243 |
. . 3
| |
| 20 | 10, 18, 19 | syl2anc 411 |
. 2
|
| 21 | 1red 8087 |
. . . 4
| |
| 22 | simprl 529 |
. . . 4
| |
| 23 | 21, 22 | resubcld 8453 |
. . 3
|
| 24 | 1cnd 8088 |
. . . . 5
| |
| 25 | 22 | recnd 8101 |
. . . . 5
|
| 26 | efsub 11992 |
. . . . 5
| |
| 27 | 24, 25, 26 | syl2anc 411 |
. . . 4
|
| 28 | simprr 531 |
. . . . . . 7
| |
| 29 | df-e 11960 |
. . . . . . . 8
| |
| 30 | 29 | oveq1i 5954 |
. . . . . . 7
|
| 31 | 28, 30 | eqtr2di 2255 |
. . . . . 6
|
| 32 | efcl 11975 |
. . . . . . . 8
| |
| 33 | 24, 32 | syl 14 |
. . . . . . 7
|
| 34 | efcl 11975 |
. . . . . . . 8
| |
| 35 | 25, 34 | syl 14 |
. . . . . . 7
|
| 36 | 11 | adantr 276 |
. . . . . . 7
|
| 37 | 9 | adantr 276 |
. . . . . . 7
|
| 38 | 33, 35, 36, 37 | divmulap2d 8897 |
. . . . . 6
|
| 39 | 31, 38 | mpbid 147 |
. . . . 5
|
| 40 | 22 | rpefcld 11997 |
. . . . . . 7
|
| 41 | 40 | rpap0d 9824 |
. . . . . 6
|
| 42 | 33, 36, 35, 41 | divmulap3d 8898 |
. . . . 5
|
| 43 | 39, 42 | mpbird 167 |
. . . 4
|
| 44 | 27, 43 | eqtrd 2238 |
. . 3
|
| 45 | fveqeq2 5585 |
. . . 4
| |
| 46 | 45 | rspcev 2877 |
. . 3
|
| 47 | 23, 44, 46 | syl2anc 411 |
. 2
|
| 48 | 20, 47 | rexlimddv 2628 |
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 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-mulext 8043 ax-arch 8044 ax-caucvg 8045 ax-pre-suploc 8046 ax-addf 8047 ax-mulf 8048 |
| This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-disj 4022 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-po 4343 df-iso 4344 df-iord 4413 df-on 4415 df-ilim 4416 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-isom 5280 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-of 6158 df-1st 6226 df-2nd 6227 df-recs 6391 df-irdg 6456 df-frec 6477 df-1o 6502 df-oadd 6506 df-er 6620 df-map 6737 df-pm 6738 df-en 6828 df-dom 6829 df-fin 6830 df-sup 7086 df-inf 7087 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 df-div 8746 df-inn 9037 df-2 9095 df-3 9096 df-4 9097 df-n0 9296 df-z 9373 df-uz 9649 df-q 9741 df-rp 9776 df-xneg 9894 df-xadd 9895 df-ioo 10014 df-ico 10016 df-icc 10017 df-fz 10131 df-fzo 10265 df-seqfrec 10593 df-exp 10684 df-fac 10871 df-bc 10893 df-ihash 10921 df-shft 11126 df-cj 11153 df-re 11154 df-im 11155 df-rsqrt 11309 df-abs 11310 df-clim 11590 df-sumdc 11665 df-ef 11959 df-e 11960 df-rest 13073 df-topgen 13092 df-psmet 14305 df-xmet 14306 df-met 14307 df-bl 14308 df-mopn 14309 df-top 14470 df-topon 14483 df-bases 14515 df-ntr 14568 df-cn 14660 df-cnp 14661 df-tx 14725 df-cncf 15043 df-limced 15128 df-dvap 15129 |
| This theorem is referenced by: reeff1o 15245 |
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