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Mirrors > Home > MPE Home > Th. List > 2sqb | Structured version Visualization version Unicode version |
Description: The converse to 2sq 24352. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Ref | Expression |
---|---|
2sqb |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2634 |
. . . 4
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2 | prmz 14674 |
. . . . . . . . . 10
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3 | 2 | ad3antrrr 741 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | simplrr 776 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | bezout 14558 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 3, 4, 5 | syl2anc 671 |
. . . . . . . 8
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7 | simplll 773 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | simpllr 774 |
. . . . . . . . . . 11
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9 | simplr 767 |
. . . . . . . . . . 11
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10 | simprll 777 |
. . . . . . . . . . 11
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11 | simprlr 778 |
. . . . . . . . . . 11
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12 | simprr 771 |
. . . . . . . . . . 11
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13 | 7, 8, 9, 10, 11, 12 | 2sqblem 24353 |
. . . . . . . . . 10
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14 | 13 | expr 624 |
. . . . . . . . 9
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15 | 14 | rexlimdvva 2897 |
. . . . . . . 8
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16 | 6, 15 | mpd 15 |
. . . . . . 7
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17 | 16 | ex 440 |
. . . . . 6
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18 | 17 | rexlimdvva 2897 |
. . . . 5
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19 | 18 | impancom 446 |
. . . 4
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20 | 1, 19 | syl5bir 226 |
. . 3
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21 | 20 | orrd 384 |
. 2
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22 | 1z 10995 |
. . . . 5
![]() ![]() ![]() ![]() | |
23 | oveq1 6321 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | sq1 12400 |
. . . . . . . . 9
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25 | 23, 24 | syl6eq 2511 |
. . . . . . . 8
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26 | 25 | oveq1d 6329 |
. . . . . . 7
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27 | 26 | eqeq2d 2471 |
. . . . . 6
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28 | oveq1 6321 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | 28, 24 | syl6eq 2511 |
. . . . . . . . 9
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30 | 29 | oveq2d 6330 |
. . . . . . . 8
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31 | 1p1e2 10750 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
32 | 30, 31 | syl6eq 2511 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 32 | eqeq2d 2471 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 27, 33 | rspc2ev 3172 |
. . . . 5
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35 | 22, 22, 34 | mp3an12 1363 |
. . . 4
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36 | 35 | adantl 472 |
. . 3
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37 | 2sq 24352 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
38 | 36, 37 | jaodan 799 |
. 2
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39 | 21, 38 | impbida 848 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-rep 4528 ax-sep 4538 ax-nul 4547 ax-pow 4594 ax-pr 4652 ax-un 6609 ax-inf2 8171 ax-cnex 9620 ax-resscn 9621 ax-1cn 9622 ax-icn 9623 ax-addcl 9624 ax-addrcl 9625 ax-mulcl 9626 ax-mulrcl 9627 ax-mulcom 9628 ax-addass 9629 ax-mulass 9630 ax-distr 9631 ax-i2m1 9632 ax-1ne0 9633 ax-1rid 9634 ax-rnegex 9635 ax-rrecex 9636 ax-cnre 9637 ax-pre-lttri 9638 ax-pre-lttrn 9639 ax-pre-ltadd 9640 ax-pre-mulgt0 9641 ax-pre-sup 9642 ax-addf 9643 ax-mulf 9644 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-nel 2635 df-ral 2753 df-rex 2754 df-reu 2755 df-rmo 2756 df-rab 2757 df-v 3058 df-sbc 3279 df-csb 3375 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-pss 3431 df-nul 3743 df-if 3893 df-pw 3964 df-sn 3980 df-pr 3982 df-tp 3984 df-op 3986 df-uni 4212 df-int 4248 df-iun 4293 df-iin 4294 df-br 4416 df-opab 4475 df-mpt 4476 df-tr 4511 df-eprel 4763 df-id 4767 df-po 4773 df-so 4774 df-fr 4811 df-se 4812 df-we 4813 df-xp 4858 df-rel 4859 df-cnv 4860 df-co 4861 df-dm 4862 df-rn 4863 df-res 4864 df-ima 4865 df-pred 5398 df-ord 5444 df-on 5445 df-lim 5446 df-suc 5447 df-iota 5564 df-fun 5602 df-fn 5603 df-f 5604 df-f1 5605 df-fo 5606 df-f1o 5607 df-fv 5608 df-isom 5609 df-riota 6276 df-ov 6317 df-oprab 6318 df-mpt2 6319 df-of 6557 df-ofr 6558 df-om 6719 df-1st 6819 df-2nd 6820 df-supp 6941 df-tpos 6998 df-wrecs 7053 df-recs 7115 df-rdg 7153 df-1o 7207 df-2o 7208 df-oadd 7211 df-er 7388 df-ec 7390 df-qs 7394 df-map 7499 df-pm 7500 df-ixp 7548 df-en 7595 df-dom 7596 df-sdom 7597 df-fin 7598 df-fsupp 7909 df-sup 7981 df-inf 7982 df-oi 8050 df-card 8398 df-cda 8623 df-pnf 9702 df-mnf 9703 df-xr 9704 df-ltxr 9705 df-le 9706 df-sub 9887 df-neg 9888 df-div 10297 df-nn 10637 df-2 10695 df-3 10696 df-4 10697 df-5 10698 df-6 10699 df-7 10700 df-8 10701 df-9 10702 df-10 10703 df-n0 10898 df-z 10966 df-dec 11080 df-uz 11188 df-q 11293 df-rp 11331 df-fz 11813 df-fzo 11946 df-fl 12059 df-mod 12128 df-seq 12245 df-exp 12304 df-hash 12547 df-cj 13210 df-re 13211 df-im 13212 df-sqrt 13346 df-abs 13347 df-dvds 14354 df-gcd 14517 df-prm 14671 df-phi 14762 df-pc 14835 df-gz 14922 df-struct 15171 df-ndx 15172 df-slot 15173 df-base 15174 df-sets 15175 df-ress 15176 df-plusg 15251 df-mulr 15252 df-starv 15253 df-sca 15254 df-vsca 15255 df-ip 15256 df-tset 15257 df-ple 15258 df-ds 15260 df-unif 15261 df-hom 15262 df-cco 15263 df-0g 15388 df-gsum 15389 df-prds 15394 df-pws 15396 df-imas 15455 df-qus 15457 df-mre 15540 df-mrc 15541 df-acs 15543 df-mgm 16536 df-sgrp 16575 df-mnd 16585 df-mhm 16630 df-submnd 16631 df-grp 16721 df-minusg 16722 df-sbg 16723 df-mulg 16724 df-subg 16862 df-nsg 16863 df-eqg 16864 df-ghm 16929 df-cntz 17019 df-cmn 17480 df-abl 17481 df-mgp 17772 df-ur 17784 df-srg 17788 df-ring 17830 df-cring 17831 df-oppr 17899 df-dvdsr 17917 df-unit 17918 df-invr 17948 df-dvr 17959 df-rnghom 17991 df-drng 18025 df-field 18026 df-subrg 18054 df-lmod 18141 df-lss 18204 df-lsp 18243 df-sra 18443 df-rgmod 18444 df-lidl 18445 df-rsp 18446 df-2idl 18504 df-nzr 18530 df-rlreg 18555 df-domn 18556 df-idom 18557 df-assa 18584 df-asp 18585 df-ascl 18586 df-psr 18628 df-mvr 18629 df-mpl 18630 df-opsr 18632 df-evls 18777 df-evl 18778 df-psr1 18821 df-vr1 18822 df-ply1 18823 df-coe1 18824 df-evl1 18953 df-cnfld 19019 df-zring 19088 df-zrh 19123 df-zn 19126 df-mdeg 23052 df-deg1 23053 df-mon1 23128 df-uc1p 23129 df-q1p 23130 df-r1p 23131 df-lgs 24271 |
This theorem is referenced by: (None) |
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