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Mirrors > Home > MPE Home > Th. List > gsumz | Structured version Visualization version Unicode version |
Description: Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.) |
Ref | Expression |
---|---|
gsumz.z |
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Ref | Expression |
---|---|
gsumz |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2451 |
. 2
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2 | gsumz.z |
. 2
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3 | eqid 2451 |
. 2
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4 | eqid 2451 |
. 2
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5 | simpl 459 |
. 2
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6 | simpr 463 |
. 2
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7 | fvex 5875 |
. . . . . . 7
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8 | 2, 7 | eqeltri 2525 |
. . . . . 6
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9 | 8 | snid 3996 |
. . . . 5
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10 | 1, 2, 3, 4 | gsumvallem2 16619 |
. . . . 5
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11 | 9, 10 | syl5eleqr 2536 |
. . . 4
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12 | 11 | ad2antrr 732 |
. . 3
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13 | eqid 2451 |
. . 3
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14 | 12, 13 | fmptd 6046 |
. 2
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15 | 1, 2, 3, 4, 5, 6, 14 | gsumval1 16520 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-reu 2744 df-rmo 2745 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-br 4403 df-opab 4462 df-mpt 4463 df-id 4749 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-seq 12214 df-0g 15340 df-gsum 15341 df-mgm 16488 df-sgrp 16527 df-mnd 16537 |
This theorem is referenced by: gsumval3 17541 gsumzres 17543 gsumzcl2 17544 gsumzf1o 17546 gsumzaddlem 17554 gsumzmhm 17570 gsumzoppg 17577 gsum2d 17604 dprdfeq0 17655 dprddisj2 17672 mplsubrglem 18663 evlslem1 18738 coe1tmmul2 18869 coe1tmmul 18870 cply1mul 18887 gsummoncoe1 18898 dmatmul 19522 smadiadetlem1a 19688 cpmatmcllem 19742 mp2pm2mplem4 19833 chfacfscmulgsum 19884 chfacfpmmulgsum 19888 tsms0 21156 tgptsmscls 21164 tdeglem4 23009 mdegmullem 23027 dchrptlem3 24194 gsummptres 28547 esum0 28870 ply1mulgsumlem2 40232 lincvalsc0 40267 linc0scn0 40269 |
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