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Mirrors > Home > MPE Home > Th. List > gsum0 | Structured version Visualization version Unicode version |
Description: Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.) |
Ref | Expression |
---|---|
gsum0.z |
Ref | Expression |
---|---|
gsum0 | g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . 3 | |
2 | gsum0.z | . . 3 | |
3 | eqid 2622 | . . 3 | |
4 | eqid 2622 | . . 3 | |
5 | id 22 | . . 3 | |
6 | 0ex 4790 | . . . 4 | |
7 | 6 | a1i 11 | . . 3 |
8 | f0 6086 | . . . 4 | |
9 | 8 | a1i 11 | . . 3 |
10 | 1, 2, 3, 4, 5, 7, 9 | gsumval1 17277 | . 2 g |
11 | df-gsum 16103 | . . . . 5 g | |
12 | 11 | reldmmpt2 6771 | . . . 4 g |
13 | 12 | ovprc1 6684 | . . 3 g |
14 | fvprc 6185 | . . . 4 | |
15 | 2, 14 | syl5eq 2668 | . . 3 |
16 | 13, 15 | eqtr4d 2659 | . 2 g |
17 | 10, 16 | pm2.61i 176 | 1 g |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wa 384 wceq 1483 wex 1704 wcel 1990 wral 2912 wrex 2913 crab 2916 cvv 3200 wsbc 3435 csb 3533 cdif 3571 wss 3574 c0 3915 cif 4086 ccnv 5113 cdm 5114 crn 5115 cima 5117 ccom 5118 cio 5849 wf 5884 wf1o 5887 cfv 5888 (class class class)co 6650 c1 9937 cuz 11687 cfz 12326 cseq 12801 chash 13117 cbs 15857 cplusg 15941 c0g 16100 g cgsu 16101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-seq 12802 df-gsum 16103 |
This theorem is referenced by: gsumwsubmcl 17375 gsumccat 17378 gsumwmhm 17382 gsumwspan 17383 frmdgsum 17399 frmdup1 17401 gsumwrev 17796 gsmsymgrfix 17848 gsmsymgreq 17852 psgnunilem2 17915 psgn0fv0 17931 psgnsn 17940 psgnprfval1 17942 gsumconst 18334 mplmonmul 19464 mplcoe1 19465 mplcoe5 19468 coe1fzgsumd 19672 evl1gsumd 19721 gsumfsum 19813 mdet0pr 20398 madugsum 20449 tmdgsum 21899 xrge0gsumle 22636 xrge0tsms 22637 jensen 24715 gsumle 29779 gsumvsca1 29782 gsumvsca2 29783 xrge0tsmsd 29785 esumnul 30110 esumsnf 30126 sitg0 30408 mrsub0 31413 matunitlindflem1 33405 lincval0 42204 |
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