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| Mirrors > Home > ILE Home > Th. List > resqrexlemlo | Unicode version | ||
| Description: Lemma for resqrex 11337. A (variable) lower bound for each term of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.) |
| Ref | Expression |
|---|---|
| resqrexlemex.seq |
|
| resqrexlemex.a |
|
| resqrexlemex.agt0 |
|
| Ref | Expression |
|---|---|
| resqrexlemlo |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oveq2 5952 |
. . . . . 6
| |
| 2 | 1 | oveq2d 5960 |
. . . . 5
|
| 3 | fveq2 5576 |
. . . . 5
| |
| 4 | 2, 3 | breq12d 4057 |
. . . 4
|
| 5 | 4 | imbi2d 230 |
. . 3
|
| 6 | oveq2 5952 |
. . . . . 6
| |
| 7 | 6 | oveq2d 5960 |
. . . . 5
|
| 8 | fveq2 5576 |
. . . . 5
| |
| 9 | 7, 8 | breq12d 4057 |
. . . 4
|
| 10 | 9 | imbi2d 230 |
. . 3
|
| 11 | oveq2 5952 |
. . . . . 6
| |
| 12 | 11 | oveq2d 5960 |
. . . . 5
|
| 13 | fveq2 5576 |
. . . . 5
| |
| 14 | 12, 13 | breq12d 4057 |
. . . 4
|
| 15 | 14 | imbi2d 230 |
. . 3
|
| 16 | oveq2 5952 |
. . . . . 6
| |
| 17 | 16 | oveq2d 5960 |
. . . . 5
|
| 18 | fveq2 5576 |
. . . . 5
| |
| 19 | 17, 18 | breq12d 4057 |
. . . 4
|
| 20 | 19 | imbi2d 230 |
. . 3
|
| 21 | 2cnd 9109 |
. . . . . . . 8
| |
| 22 | 21 | exp1d 10813 |
. . . . . . 7
|
| 23 | 2rp 9780 |
. . . . . . 7
| |
| 24 | 22, 23 | eqeltrdi 2296 |
. . . . . 6
|
| 25 | 24 | rprecred 9830 |
. . . . 5
|
| 26 | 1red 8087 |
. . . . 5
| |
| 27 | resqrexlemex.a |
. . . . . 6
| |
| 28 | 26, 27 | readdcld 8102 |
. . . . 5
|
| 29 | 22 | oveq2d 5960 |
. . . . . 6
|
| 30 | halflt1 9254 |
. . . . . 6
| |
| 31 | 29, 30 | eqbrtrdi 4083 |
. . . . 5
|
| 32 | resqrexlemex.agt0 |
. . . . . 6
| |
| 33 | 26, 27 | addge01d 8606 |
. . . . . 6
|
| 34 | 32, 33 | mpbid 147 |
. . . . 5
|
| 35 | 25, 26, 28, 31, 34 | ltletrd 8496 |
. . . 4
|
| 36 | resqrexlemex.seq |
. . . . 5
| |
| 37 | 36, 27, 32 | resqrexlemf1 11319 |
. . . 4
|
| 38 | 35, 37 | breqtrrd 4072 |
. . 3
|
| 39 | 23 | a1i 9 |
. . . . . . . . . . 11
|
| 40 | nnz 9391 |
. . . . . . . . . . . 12
| |
| 41 | 40 | ad2antlr 489 |
. . . . . . . . . . 11
|
| 42 | 39, 41 | rpexpcld 10842 |
. . . . . . . . . 10
|
| 43 | 42 | rpcnd 9820 |
. . . . . . . . 9
|
| 44 | 2cnd 9109 |
. . . . . . . . 9
| |
| 45 | 42 | rpap0d 9824 |
. . . . . . . . 9
|
| 46 | 39 | rpap0d 9824 |
. . . . . . . . 9
|
| 47 | 43, 44, 45, 46 | recdivap2d 8881 |
. . . . . . . 8
|
| 48 | nnnn0 9302 |
. . . . . . . . . . 11
| |
| 49 | 48 | ad2antlr 489 |
. . . . . . . . . 10
|
| 50 | 44, 49 | expp1d 10819 |
. . . . . . . . 9
|
| 51 | 50 | oveq2d 5960 |
. . . . . . . 8
|
| 52 | 47, 51 | eqtr4d 2241 |
. . . . . . 7
|
| 53 | 42 | rprecred 9830 |
. . . . . . . . 9
|
| 54 | 36, 27, 32 | resqrexlemf 11318 |
. . . . . . . . . . . . 13
|
| 55 | 54 | ffvelcdmda 5715 |
. . . . . . . . . . . 12
|
| 56 | 55 | rpred 9818 |
. . . . . . . . . . 11
|
| 57 | 56 | adantr 276 |
. . . . . . . . . 10
|
| 58 | 27 | adantr 276 |
. . . . . . . . . . . 12
|
| 59 | 58, 55 | rerpdivcld 9850 |
. . . . . . . . . . 11
|
| 60 | 59 | adantr 276 |
. . . . . . . . . 10
|
| 61 | 57, 60 | readdcld 8102 |
. . . . . . . . 9
|
| 62 | simpr 110 |
. . . . . . . . . 10
| |
| 63 | 32 | adantr 276 |
. . . . . . . . . . . . 13
|
| 64 | 58, 55, 63 | divge0d 9859 |
. . . . . . . . . . . 12
|
| 65 | 56, 59 | addge01d 8606 |
. . . . . . . . . . . 12
|
| 66 | 64, 65 | mpbid 147 |
. . . . . . . . . . 11
|
| 67 | 66 | adantr 276 |
. . . . . . . . . 10
|
| 68 | 53, 57, 61, 62, 67 | ltletrd 8496 |
. . . . . . . . 9
|
| 69 | 53, 61, 39, 68 | ltdiv1dd 9876 |
. . . . . . . 8
|
| 70 | 36, 27, 32 | resqrexlemfp1 11320 |
. . . . . . . . 9
|
| 71 | 70 | adantr 276 |
. . . . . . . 8
|
| 72 | 69, 71 | breqtrrd 4072 |
. . . . . . 7
|
| 73 | 52, 72 | eqbrtrrd 4068 |
. . . . . 6
|
| 74 | 73 | ex 115 |
. . . . 5
|
| 75 | 74 | expcom 116 |
. . . 4
|
| 76 | 75 | a2d 26 |
. . 3
|
| 77 | 5, 10, 15, 20, 38, 76 | nnind 9052 |
. 2
|
| 78 | 77 | impcom 125 |
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 |
| This theorem depends on definitions: df-bi 117 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-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-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-recs 6391 df-frec 6477 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-n0 9296 df-z 9373 df-uz 9649 df-rp 9776 df-seqfrec 10593 df-exp 10684 |
| This theorem is referenced by: resqrexlemnm 11329 |
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