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Mirrors > Home > NFE Home > Th. List > qrpprod | Unicode version |
Description: A quadratic relationship over a parallel product. (Contributed by SF, 24-Feb-2015.) |
Ref | Expression |
---|---|
qrpprod |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brex 4689 |
. . 3
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2 | opexb 4603 |
. . . 4
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3 | opexb 4603 |
. . . 4
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4 | 2, 3 | anbi12i 678 |
. . 3
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5 | 1, 4 | sylib 188 |
. 2
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6 | brex 4689 |
. . . 4
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7 | brex 4689 |
. . . 4
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8 | 6, 7 | anim12i 549 |
. . 3
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9 | an4 797 |
. . 3
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10 | 8, 9 | sylibr 203 |
. 2
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11 | opeq1 4578 |
. . . . . . 7
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12 | 11 | breq1d 4649 |
. . . . . 6
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13 | breq1 4642 |
. . . . . . 7
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14 | 13 | anbi1d 685 |
. . . . . 6
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15 | 12, 14 | bibi12d 312 |
. . . . 5
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16 | 15 | imbi2d 307 |
. . . 4
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17 | opeq2 4579 |
. . . . . . 7
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18 | 17 | breq1d 4649 |
. . . . . 6
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19 | breq1 4642 |
. . . . . . 7
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20 | 19 | anbi2d 684 |
. . . . . 6
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21 | 18, 20 | bibi12d 312 |
. . . . 5
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22 | 21 | imbi2d 307 |
. . . 4
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23 | opeq1 4578 |
. . . . . . 7
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24 | 23 | breq2d 4651 |
. . . . . 6
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25 | breq2 4643 |
. . . . . . 7
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26 | 25 | anbi1d 685 |
. . . . . 6
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27 | 24, 26 | bibi12d 312 |
. . . . 5
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28 | opeq2 4579 |
. . . . . . 7
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29 | 28 | breq2d 4651 |
. . . . . 6
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30 | breq2 4643 |
. . . . . . 7
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31 | 30 | anbi2d 684 |
. . . . . 6
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32 | 29, 31 | bibi12d 312 |
. . . . 5
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33 | df-pprod 5738 |
. . . . . . . 8
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34 | 33 | breqi 4645 |
. . . . . . 7
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35 | trtxp 5781 |
. . . . . . 7
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36 | 34, 35 | bitri 240 |
. . . . . 6
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37 | brco 4883 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
38 | vex 2862 |
. . . . . . . . . . . . 13
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39 | vex 2862 |
. . . . . . . . . . . . 13
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40 | 38, 39 | opbr1st 5501 |
. . . . . . . . . . . 12
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41 | eqcom 2355 |
. . . . . . . . . . . 12
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42 | 40, 41 | bitri 240 |
. . . . . . . . . . 11
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43 | 42 | anbi1i 676 |
. . . . . . . . . 10
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44 | 43 | exbii 1582 |
. . . . . . . . 9
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45 | 37, 44 | bitri 240 |
. . . . . . . 8
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46 | breq1 4642 |
. . . . . . . . 9
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47 | 38, 46 | ceqsexv 2894 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | 45, 47 | bitri 240 |
. . . . . . 7
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49 | brco 4883 |
. . . . . . . . 9
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50 | 38, 39 | opbr2nd 5502 |
. . . . . . . . . . . 12
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51 | eqcom 2355 |
. . . . . . . . . . . 12
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52 | 50, 51 | bitri 240 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | 52 | anbi1i 676 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
54 | 53 | exbii 1582 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
55 | 49, 54 | bitri 240 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
56 | breq1 4642 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
57 | 39, 56 | ceqsexv 2894 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
58 | 55, 57 | bitri 240 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
59 | 48, 58 | anbi12i 678 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
60 | 36, 59 | bitri 240 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
61 | 27, 32, 60 | vtocl2g 2918 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
62 | 16, 22, 61 | vtocl2g 2918 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
63 | 62 | imp 418 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
64 | 5, 10, 63 | pm5.21nii 342 |
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 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-co 4726 df-cnv 4785 df-2nd 4797 df-txp 5736 df-pprod 5738 |
This theorem is referenced by: dmfrec 6316 fnfreclem2 6318 fnfreclem3 6319 frecsuc 6322 |
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