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Mirrors > Home > MPE Home > Th. List > gsumzinv | Structured version Unicode version |
Description: Inverse of a group sum. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.) |
Ref | Expression |
---|---|
gsumzinv.b |
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gsumzinv.0 |
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gsumzinv.z |
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gsumzinv.i |
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gsumzinv.g |
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gsumzinv.a |
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gsumzinv.f |
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gsumzinv.c |
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gsumzinv.n |
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Ref | Expression |
---|---|
gsumzinv |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumzinv.b |
. . 3
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2 | gsumzinv.0 |
. . 3
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3 | gsumzinv.z |
. . 3
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4 | eqid 2454 |
. . 3
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5 | gsumzinv.g |
. . . 4
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6 | grpmnd 15670 |
. . . 4
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7 | 5, 6 | syl 16 |
. . 3
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8 | gsumzinv.a |
. . 3
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9 | gsumzinv.i |
. . . . . 6
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10 | 1, 9 | grpinvf 15702 |
. . . . 5
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11 | 5, 10 | syl 16 |
. . . 4
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12 | gsumzinv.f |
. . . 4
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13 | fco 5677 |
. . . 4
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14 | 11, 12, 13 | syl2anc 661 |
. . 3
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15 | 4, 9 | invoppggim 15995 |
. . . . . 6
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16 | gimghm 15912 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | ghmmhm 15877 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 5, 15, 16, 17 | 4syl 21 |
. . . . 5
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19 | gsumzinv.c |
. . . . 5
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20 | eqid 2454 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 3, 20 | cntzmhm2 15977 |
. . . . 5
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22 | 18, 19, 21 | syl2anc 661 |
. . . 4
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23 | rnco2 5454 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 23 | fveq2i 5803 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 4, 3 | oppgcntz 15999 |
. . . . 5
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26 | 24, 25 | eqtri 2483 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 22, 23, 26 | 3sstr4g 3506 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | fvex 5810 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | 2, 28 | eqeltri 2538 |
. . . . 5
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30 | 29 | a1i 11 |
. . . 4
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31 | fvex 5810 |
. . . . . 6
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32 | 1, 31 | eqeltri 2538 |
. . . . 5
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33 | 32 | a1i 11 |
. . . 4
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34 | gsumzinv.n |
. . . 4
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35 | 2, 9 | grpinvid 15709 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | 5, 35 | syl 16 |
. . . 4
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37 | 30, 12, 11, 8, 33, 34, 36 | fsuppco2 7764 |
. . 3
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38 | 1, 2, 3, 4, 7, 8, 14, 27, 37 | gsumzoppg 16563 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | 4 | oppgmnd 15989 |
. . . 4
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40 | 7, 39 | syl 16 |
. . 3
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41 | 1, 3, 7, 40, 8, 18, 12, 19, 2, 34 | gsumzmhm 16553 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
42 | 38, 41 | eqtr3d 2497 |
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 4512 ax-sep 4522 ax-nul 4530 ax-pow 4579 ax-pr 4640 ax-un 6483 ax-cnex 9450 ax-resscn 9451 ax-1cn 9452 ax-icn 9453 ax-addcl 9454 ax-addrcl 9455 ax-mulcl 9456 ax-mulrcl 9457 ax-mulcom 9458 ax-addass 9459 ax-mulass 9460 ax-distr 9461 ax-i2m1 9462 ax-1ne0 9463 ax-1rid 9464 ax-rnegex 9465 ax-rrecex 9466 ax-cnre 9467 ax-pre-lttri 9468 ax-pre-lttrn 9469 ax-pre-ltadd 9470 ax-pre-mulgt0 9471 |
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 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3397 df-dif 3440 df-un 3442 df-in 3444 df-ss 3451 df-pss 3453 df-nul 3747 df-if 3901 df-pw 3971 df-sn 3987 df-pr 3989 df-tp 3991 df-op 3993 df-uni 4201 df-int 4238 df-iun 4282 df-iin 4283 df-br 4402 df-opab 4460 df-mpt 4461 df-tr 4495 df-eprel 4741 df-id 4745 df-po 4750 df-so 4751 df-fr 4788 df-se 4789 df-we 4790 df-ord 4831 df-on 4832 df-lim 4833 df-suc 4834 df-xp 4955 df-rel 4956 df-cnv 4957 df-co 4958 df-dm 4959 df-rn 4960 df-res 4961 df-ima 4962 df-iota 5490 df-fun 5529 df-fn 5530 df-f 5531 df-f1 5532 df-fo 5533 df-f1o 5534 df-fv 5535 df-isom 5536 df-riota 6162 df-ov 6204 df-oprab 6205 df-mpt2 6206 df-om 6588 df-1st 6688 df-2nd 6689 df-supp 6802 df-tpos 6856 df-recs 6943 df-rdg 6977 df-1o 7031 df-oadd 7035 df-er 7212 df-map 7327 df-en 7422 df-dom 7423 df-sdom 7424 df-fin 7425 df-fsupp 7733 df-oi 7836 df-card 8221 df-pnf 9532 df-mnf 9533 df-xr 9534 df-ltxr 9535 df-le 9536 df-sub 9709 df-neg 9710 df-nn 10435 df-2 10492 df-n0 10692 df-z 10759 df-uz 10974 df-fz 11556 df-fzo 11667 df-seq 11925 df-hash 12222 df-ndx 14296 df-slot 14297 df-base 14298 df-sets 14299 df-ress 14300 df-plusg 14371 df-0g 14500 df-gsum 14501 df-mre 14644 df-mrc 14645 df-acs 14647 df-mnd 15535 df-mhm 15584 df-submnd 15585 df-grp 15665 df-minusg 15666 df-ghm 15865 df-gim 15907 df-cntz 15955 df-oppg 15981 df-cmn 16401 |
This theorem is referenced by: dprdfinv 16632 |
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