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Mirrors > Home > ILE Home > Th. List > resqrexlemfp1 | Unicode version |
Description: Lemma for resqrex 10638. Recursion rule. This sequence is the ancient method for computing square roots, often known as the babylonian method, although known to many ancient cultures. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.) |
Ref | Expression |
---|---|
resqrexlemex.seq |
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resqrexlemex.a |
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resqrexlemex.agt0 |
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Ref | Expression |
---|---|
resqrexlemfp1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnnuz 9212 |
. . . . . 6
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2 | 1 | biimpi 119 |
. . . . 5
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3 | 2 | adantl 273 |
. . . 4
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4 | elnnuz 9212 |
. . . . . 6
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5 | resqrexlemex.a |
. . . . . . 7
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6 | resqrexlemex.agt0 |
. . . . . . 7
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7 | 5, 6 | resqrexlem1arp 10617 |
. . . . . 6
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8 | 4, 7 | sylan2br 284 |
. . . . 5
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9 | 8 | adantlr 464 |
. . . 4
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10 | 5, 6 | resqrexlemp1rp 10618 |
. . . . 5
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11 | 10 | adantlr 464 |
. . . 4
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12 | 3, 9, 11 | seq3p1 10076 |
. . 3
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13 | resqrexlemex.seq |
. . . 4
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14 | 13 | fveq1i 5354 |
. . 3
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15 | 13 | fveq1i 5354 |
. . . 4
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16 | 15 | oveq1i 5716 |
. . 3
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17 | 12, 14, 16 | 3eqtr4g 2157 |
. 2
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18 | id 19 |
. . . . . . 7
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19 | oveq2 5714 |
. . . . . . 7
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20 | 18, 19 | oveq12d 5724 |
. . . . . 6
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21 | 20 | oveq1d 5721 |
. . . . 5
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22 | eqidd 2101 |
. . . . 5
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23 | 21, 22 | cbvmpov 5783 |
. . . 4
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24 | 23 | a1i 9 |
. . 3
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25 | id 19 |
. . . . . 6
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26 | oveq2 5714 |
. . . . . 6
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27 | 25, 26 | oveq12d 5724 |
. . . . 5
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28 | 27 | oveq1d 5721 |
. . . 4
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29 | 28 | ad2antrl 477 |
. . 3
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30 | 13, 5, 6 | resqrexlemf 10619 |
. . . 4
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31 | 30 | ffvelrnda 5487 |
. . 3
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32 | peano2nn 8590 |
. . . 4
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33 | 5, 6 | resqrexlem1arp 10617 |
. . . 4
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34 | 32, 33 | sylan2 282 |
. . 3
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35 | 31 | rpred 9330 |
. . . . 5
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36 | 5 | adantr 272 |
. . . . . 6
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37 | 36, 31 | rerpdivcld 9362 |
. . . . 5
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38 | 35, 37 | readdcld 7667 |
. . . 4
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39 | 38 | rehalfcld 8818 |
. . 3
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40 | 24, 29, 31, 34, 39 | ovmpod 5830 |
. 2
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41 | 17, 40 | eqtrd 2132 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-ilim 4229 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-frec 6218 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-inn 8579 df-2 8637 df-n0 8830 df-z 8907 df-uz 9177 df-rp 9292 df-seqfrec 10060 |
This theorem is referenced by: resqrexlemover 10622 resqrexlemdec 10623 resqrexlemlo 10625 resqrexlemcalc1 10626 |
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