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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 | PProd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brex 4689 | . . 3 PProd | |
2 | opexb 4603 | . . . 4 | |
3 | opexb 4603 | . . . 4 | |
4 | 2, 3 | anbi12i 678 | . . 3 |
5 | 1, 4 | sylib 188 | . 2 PProd |
6 | brex 4689 | . . . 4 | |
7 | brex 4689 | . . . 4 | |
8 | 6, 7 | anim12i 549 | . . 3 |
9 | an4 797 | . . 3 | |
10 | 8, 9 | sylibr 203 | . 2 |
11 | opeq1 4578 | . . . . . . 7 | |
12 | 11 | breq1d 4649 | . . . . . 6 PProd PProd |
13 | breq1 4642 | . . . . . . 7 | |
14 | 13 | anbi1d 685 | . . . . . 6 |
15 | 12, 14 | bibi12d 312 | . . . . 5 PProd PProd |
16 | 15 | imbi2d 307 | . . . 4 PProd PProd |
17 | opeq2 4579 | . . . . . . 7 | |
18 | 17 | breq1d 4649 | . . . . . 6 PProd PProd |
19 | breq1 4642 | . . . . . . 7 | |
20 | 19 | anbi2d 684 | . . . . . 6 |
21 | 18, 20 | bibi12d 312 | . . . . 5 PProd PProd |
22 | 21 | imbi2d 307 | . . . 4 PProd PProd |
23 | opeq1 4578 | . . . . . . 7 | |
24 | 23 | breq2d 4651 | . . . . . 6 PProd PProd |
25 | breq2 4643 | . . . . . . 7 | |
26 | 25 | anbi1d 685 | . . . . . 6 |
27 | 24, 26 | bibi12d 312 | . . . . 5 PProd PProd |
28 | opeq2 4579 | . . . . . . 7 | |
29 | 28 | breq2d 4651 | . . . . . 6 PProd PProd |
30 | breq2 4643 | . . . . . . 7 | |
31 | 30 | anbi2d 684 | . . . . . 6 |
32 | 29, 31 | bibi12d 312 | . . . . 5 PProd PProd |
33 | df-pprod 5738 | . . . . . . . 8 PProd | |
34 | 33 | breqi 4645 | . . . . . . 7 PProd |
35 | trtxp 5781 | . . . . . . 7 | |
36 | 34, 35 | bitri 240 | . . . . . 6 PProd |
37 | brco 4883 | . . . . . . . . 9 | |
38 | vex 2862 | . . . . . . . . . . . . 13 | |
39 | vex 2862 | . . . . . . . . . . . . 13 | |
40 | 38, 39 | opbr1st 5501 | . . . . . . . . . . . 12 |
41 | eqcom 2355 | . . . . . . . . . . . 12 | |
42 | 40, 41 | bitri 240 | . . . . . . . . . . 11 |
43 | 42 | anbi1i 676 | . . . . . . . . . 10 |
44 | 43 | exbii 1582 | . . . . . . . . 9 |
45 | 37, 44 | bitri 240 | . . . . . . . 8 |
46 | breq1 4642 | . . . . . . . . 9 | |
47 | 38, 46 | ceqsexv 2894 | . . . . . . . 8 |
48 | 45, 47 | bitri 240 | . . . . . . 7 |
49 | brco 4883 | . . . . . . . . 9 | |
50 | 38, 39 | opbr2nd 5502 | . . . . . . . . . . . 12 |
51 | eqcom 2355 | . . . . . . . . . . . 12 | |
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 PProd |
61 | 27, 32, 60 | vtocl2g 2918 | . . . 4 PProd |
62 | 16, 22, 61 | vtocl2g 2918 | . . 3 PProd |
63 | 62 | imp 418 | . 2 PProd |
64 | 5, 10, 63 | pm5.21nii 342 | 1 PProd |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 wex 1541 wceq 1642 wcel 1710 cvv 2859 cop 4561 class class class wbr 4639 c1st 4717 ccom 4721 c2nd 4783 ctxp 5735 PProd cpprod 5737 |
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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