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Mirrors > Home > MPE Home > Th. List > gsum0 | Structured version Unicode version |
Description: Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.) |
Ref | Expression |
---|---|
gsum0.z |
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Ref | Expression |
---|---|
gsum0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2454 |
. . 3
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2 | gsum0.z |
. . 3
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3 | eqid 2454 |
. . 3
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4 | eqid 2454 |
. . 3
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5 | id 22 |
. . 3
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6 | 0ex 4533 |
. . . 4
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7 | 6 | a1i 11 |
. . 3
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8 | f0 5703 |
. . . 4
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9 | 8 | a1i 11 |
. . 3
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10 | 1, 2, 3, 4, 5, 7, 9 | gsumval1 15631 |
. 2
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11 | df-gsum 14503 |
. . . . 5
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12 | 11 | reldmmpt2 6314 |
. . . 4
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13 | 12 | ovprc1 6231 |
. . 3
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14 | fvprc 5796 |
. . . 4
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15 | 2, 14 | syl5eq 2507 |
. . 3
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16 | 13, 15 | eqtr4d 2498 |
. 2
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17 | 10, 16 | pm2.61i 164 |
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-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 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-ral 2804 df-rex 2805 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-op 3995 df-uni 4203 df-br 4404 df-opab 4462 df-mpt 4463 df-id 4747 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-recs 6945 df-rdg 6979 df-seq 11927 df-gsum 14503 |
This theorem is referenced by: gsumwsubmcl 15638 gsumccat 15641 gsumwmhm 15645 gsumwspan 15646 frmdgsum 15662 frmdup1 15664 gsumwrev 16003 gsmsymgrfix 16055 gsmsymgreq 16059 psgnunilem2 16123 psgn0fv0 16139 psgnsn 16148 psgnprfval1 16150 gsumconst 16552 mplmonmul 17670 mplcoe1 17671 mplcoe5 17675 mplcoe2OLD 17677 evl1gsumd 17919 gsumfsum 18007 mdet0pr 18533 madugsum 18584 tmdgsum 19801 xrge0gsumle 20545 xrge0tsms 20546 jensen 22518 gsumle 26411 gsumvsca1 26416 gsumvsca2 26417 xrge0tsmsd 26418 esumnul 26667 esumsn 26680 sitg0 26896 coe1fzgsumd 31011 lincval0 31101 |
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