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Mirrors > Home > MPE Home > Th. List > dprdfid | Structured version Unicode version |
Description: A function mapping all but one arguments to zero sums to the value of this argument in a direct product. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.) |
Ref | Expression |
---|---|
eldprdi.0 |
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eldprdi.w |
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eldprdi.1 |
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eldprdi.2 |
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dprdfid.3 |
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dprdfid.4 |
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dprdfid.f |
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Ref | Expression |
---|---|
dprdfid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dprdfid.f |
. . 3
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2 | eldprdi.w |
. . . 4
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3 | eldprdi.1 |
. . . 4
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4 | eldprdi.2 |
. . . 4
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5 | dprdfid.4 |
. . . . . . 7
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6 | 5 | ad2antrr 725 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | simpr 461 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 7 | fveq2d 5802 |
. . . . . 6
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9 | 6, 8 | eleqtrrd 2545 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | 3, 4 | dprdf2 16612 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 10 | ffvelrnda 5951 |
. . . . . . 7
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12 | eldprdi.0 |
. . . . . . . 8
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13 | 12 | subg0cl 15807 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 11, 13 | syl 16 |
. . . . . 6
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15 | 14 | adantr 465 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 9, 15 | ifclda 3928 |
. . . 4
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17 | 3, 4 | dprddomcld 16604 |
. . . . 5
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18 | fvex 5808 |
. . . . . . 7
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19 | 12, 18 | eqeltri 2538 |
. . . . . 6
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20 | 19 | a1i 11 |
. . . . 5
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21 | eqid 2454 |
. . . . 5
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22 | 17, 20, 21 | sniffsupp 7769 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 2, 3, 4, 16, 22 | dprdwd 16616 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 1, 23 | syl5eqel 2546 |
. 2
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25 | eqid 2454 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | dprdgrp 16610 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | grpmnd 15668 |
. . . . 5
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28 | 3, 26, 27 | 3syl 20 |
. . . 4
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29 | dprdfid.3 |
. . . 4
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30 | 2, 3, 4, 24, 25 | dprdff 16617 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 1 | oveq1i 6209 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | eldifsni 4108 |
. . . . . . . 8
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33 | 32 | adantl 466 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | ifnefalse 3908 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
35 | 33, 34 | syl 16 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | 35, 17 | suppss2 6832 |
. . . . 5
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37 | 31, 36 | syl5eqss 3507 |
. . . 4
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38 | 25, 12, 28, 17, 29, 30, 37 | gsumpt 16575 |
. . 3
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39 | iftrue 3904 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
40 | 39, 1 | fvmptg 5880 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 29, 5, 40 | syl2anc 661 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
42 | 38, 41 | eqtrd 2495 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | 24, 42 | jca 532 |
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 1955 ax-ext 2432 ax-rep 4510 ax-sep 4520 ax-nul 4528 ax-pow 4577 ax-pr 4638 ax-un 6481 ax-inf2 7957 ax-cnex 9448 ax-resscn 9449 ax-1cn 9450 ax-icn 9451 ax-addcl 9452 ax-addrcl 9453 ax-mulcl 9454 ax-mulrcl 9455 ax-mulcom 9456 ax-addass 9457 ax-mulass 9458 ax-distr 9459 ax-i2m1 9460 ax-1ne0 9461 ax-1rid 9462 ax-rnegex 9463 ax-rrecex 9464 ax-cnre 9465 ax-pre-lttri 9466 ax-pre-lttrn 9467 ax-pre-ltadd 9468 ax-pre-mulgt0 9469 |
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 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2649 df-nel 2650 df-ral 2803 df-rex 2804 df-reu 2805 df-rmo 2806 df-rab 2807 df-v 3078 df-sbc 3293 df-csb 3395 df-dif 3438 df-un 3440 df-in 3442 df-ss 3449 df-pss 3451 df-nul 3745 df-if 3899 df-pw 3969 df-sn 3985 df-pr 3987 df-tp 3989 df-op 3991 df-uni 4199 df-int 4236 df-iun 4280 df-iin 4281 df-br 4400 df-opab 4458 df-mpt 4459 df-tr 4493 df-eprel 4739 df-id 4743 df-po 4748 df-so 4749 df-fr 4786 df-se 4787 df-we 4788 df-ord 4829 df-on 4830 df-lim 4831 df-suc 4832 df-xp 4953 df-rel 4954 df-cnv 4955 df-co 4956 df-dm 4957 df-rn 4958 df-res 4959 df-ima 4960 df-iota 5488 df-fun 5527 df-fn 5528 df-f 5529 df-f1 5530 df-fo 5531 df-f1o 5532 df-fv 5533 df-isom 5534 df-riota 6160 df-ov 6202 df-oprab 6203 df-mpt2 6204 df-om 6586 df-1st 6686 df-2nd 6687 df-supp 6800 df-recs 6941 df-rdg 6975 df-1o 7029 df-oadd 7033 df-er 7210 df-ixp 7373 df-en 7420 df-dom 7421 df-sdom 7422 df-fin 7423 df-fsupp 7731 df-oi 7834 df-card 8219 df-pnf 9530 df-mnf 9531 df-xr 9532 df-ltxr 9533 df-le 9534 df-sub 9707 df-neg 9708 df-nn 10433 df-2 10490 df-n0 10690 df-z 10757 df-uz 10972 df-fz 11554 df-fzo 11665 df-seq 11923 df-hash 12220 df-ndx 14294 df-slot 14295 df-base 14296 df-sets 14297 df-ress 14298 df-plusg 14369 df-0g 14498 df-gsum 14499 df-mre 14642 df-mrc 14643 df-acs 14645 df-mnd 15533 df-submnd 15583 df-grp 15663 df-mulg 15666 df-subg 15796 df-cntz 15953 df-cmn 16399 df-dprd 16598 |
This theorem is referenced by: dprdfeq0 16633 dprdub 16643 dpjrid 16682 |
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