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Mirrors > Home > MPE Home > Th. List > 2sqlem8a | Structured version Unicode version |
Description: Lemma for 2sqlem8 22829. (Contributed by Mario Carneiro, 4-Jun-2016.) |
Ref | Expression |
---|---|
2sq.1 |
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2sqlem7.2 |
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2sqlem9.5 |
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2sqlem9.7 |
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2sqlem8.n |
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2sqlem8.m |
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2sqlem8.1 |
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2sqlem8.2 |
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2sqlem8.3 |
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2sqlem8.4 |
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2sqlem8.c |
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2sqlem8.d |
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Ref | Expression |
---|---|
2sqlem8a |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sqlem8.1 |
. . . 4
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2 | 2sqlem8.m |
. . . . . 6
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3 | eluz2b3 11031 |
. . . . . 6
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4 | 2, 3 | sylib 196 |
. . . . 5
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5 | 4 | simpld 459 |
. . . 4
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6 | 2sqlem8.c |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 1, 5, 6 | 4sqlem5 14107 |
. . 3
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8 | 7 | simpld 459 |
. 2
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9 | 2sqlem8.2 |
. . . 4
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10 | 2sqlem8.d |
. . . 4
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11 | 9, 5, 10 | 4sqlem5 14107 |
. . 3
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12 | 11 | simpld 459 |
. 2
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13 | 4 | simprd 463 |
. . . 4
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14 | simpr 461 |
. . . . . . . . . 10
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15 | 1, 5, 6, 14 | 4sqlem9 14111 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 15 | ex 434 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | eluzelz 10973 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 2, 17 | syl 16 |
. . . . . . . . 9
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19 | dvdssq 13848 |
. . . . . . . . 9
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20 | 18, 1, 19 | syl2anc 661 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 16, 20 | sylibrd 234 |
. . . . . . 7
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22 | simpr 461 |
. . . . . . . . . 10
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23 | 9, 5, 10, 22 | 4sqlem9 14111 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 23 | ex 434 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | dvdssq 13848 |
. . . . . . . . 9
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26 | 18, 9, 25 | syl2anc 661 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 24, 26 | sylibrd 234 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 2sqlem8.3 |
. . . . . . . . . . 11
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29 | ax-1ne0 9454 |
. . . . . . . . . . . 12
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30 | 29 | a1i 11 |
. . . . . . . . . . 11
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31 | 28, 30 | eqnetrd 2741 |
. . . . . . . . . 10
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32 | 31 | neneqd 2651 |
. . . . . . . . 9
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33 | gcdeq0 13809 |
. . . . . . . . . 10
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34 | 1, 9, 33 | syl2anc 661 |
. . . . . . . . 9
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35 | 32, 34 | mtbid 300 |
. . . . . . . 8
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36 | dvdslegcd 13804 |
. . . . . . . 8
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37 | 18, 1, 9, 35, 36 | syl31anc 1222 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | 21, 27, 37 | syl2and 483 |
. . . . . 6
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39 | 28 | breq2d 4404 |
. . . . . . 7
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40 | nnle1eq1 10453 |
. . . . . . . 8
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41 | 5, 40 | syl 16 |
. . . . . . 7
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42 | 39, 41 | bitrd 253 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | 38, 42 | sylibd 214 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
44 | 43 | necon3ad 2658 |
. . . 4
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45 | 13, 44 | mpd 15 |
. . 3
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46 | 8 | zcnd 10851 |
. . . . 5
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47 | sqeq0 12033 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
48 | 46, 47 | syl 16 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 12 | zcnd 10851 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
50 | sqeq0 12033 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
51 | 49, 50 | syl 16 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
52 | 48, 51 | anbi12d 710 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | 45, 52 | mtbid 300 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
54 | gcdn0cl 13802 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
55 | 8, 12, 53, 54 | syl21anc 1218 |
1
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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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4513 ax-nul 4521 ax-pow 4570 ax-pr 4631 ax-un 6474 ax-cnex 9441 ax-resscn 9442 ax-1cn 9443 ax-icn 9444 ax-addcl 9445 ax-addrcl 9446 ax-mulcl 9447 ax-mulrcl 9448 ax-mulcom 9449 ax-addass 9450 ax-mulass 9451 ax-distr 9452 ax-i2m1 9453 ax-1ne0 9454 ax-1rid 9455 ax-rnegex 9456 ax-rrecex 9457 ax-cnre 9458 ax-pre-lttri 9459 ax-pre-lttrn 9460 ax-pre-ltadd 9461 ax-pre-mulgt0 9462 ax-pre-sup 9463 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rmo 2803 df-rab 2804 df-v 3072 df-sbc 3287 df-csb 3389 df-dif 3431 df-un 3433 df-in 3435 df-ss 3442 df-pss 3444 df-nul 3738 df-if 3892 df-pw 3962 df-sn 3978 df-pr 3980 df-tp 3982 df-op 3984 df-uni 4192 df-iun 4273 df-br 4393 df-opab 4451 df-mpt 4452 df-tr 4486 df-eprel 4732 df-id 4736 df-po 4741 df-so 4742 df-fr 4779 df-we 4781 df-ord 4822 df-on 4823 df-lim 4824 df-suc 4825 df-xp 4946 df-rel 4947 df-cnv 4948 df-co 4949 df-dm 4950 df-rn 4951 df-res 4952 df-ima 4953 df-iota 5481 df-fun 5520 df-fn 5521 df-f 5522 df-f1 5523 df-fo 5524 df-f1o 5525 df-fv 5526 df-riota 6153 df-ov 6195 df-oprab 6196 df-mpt2 6197 df-om 6579 df-2nd 6680 df-recs 6934 df-rdg 6968 df-er 7203 df-en 7413 df-dom 7414 df-sdom 7415 df-sup 7794 df-pnf 9523 df-mnf 9524 df-xr 9525 df-ltxr 9526 df-le 9527 df-sub 9700 df-neg 9701 df-div 10097 df-nn 10426 df-2 10483 df-3 10484 df-n0 10683 df-z 10750 df-uz 10965 df-rp 11095 df-fl 11745 df-mod 11812 df-seq 11910 df-exp 11969 df-cj 12692 df-re 12693 df-im 12694 df-sqr 12828 df-abs 12829 df-dvds 13640 df-gcd 13795 |
This theorem is referenced by: 2sqlem8 22829 |
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