![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > gsum0 | Structured version Visualization version GIF version |
Description: Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.) |
Ref | Expression |
---|---|
gsum0.z | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
gsum0 | ⊢ (𝐺 Σg ∅) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | gsum0.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | eqid 2726 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | eqid 2726 | . . 3 ⊢ {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)} = {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)} | |
5 | id 22 | . . 3 ⊢ (𝐺 ∈ V → 𝐺 ∈ V) | |
6 | 0ex 5312 | . . . 4 ⊢ ∅ ∈ V | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝐺 ∈ V → ∅ ∈ V) |
8 | f0 6783 | . . . 4 ⊢ ∅:∅⟶{𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)} | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝐺 ∈ V → ∅:∅⟶{𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)}) |
10 | 1, 2, 3, 4, 5, 7, 9 | gsumval1 18676 | . 2 ⊢ (𝐺 ∈ V → (𝐺 Σg ∅) = 0 ) |
11 | df-gsum 17457 | . . . . 5 ⊢ Σg = (𝑤 ∈ V, 𝑓 ∈ V ↦ ⦋{𝑥 ∈ (Base‘𝑤) ∣ ∀𝑦 ∈ (Base‘𝑤)((𝑥(+g‘𝑤)𝑦) = 𝑦 ∧ (𝑦(+g‘𝑤)𝑥) = 𝑦)} / 𝑜⦌if(ran 𝑓 ⊆ 𝑜, (0g‘𝑤), if(dom 𝑓 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))), (℩𝑥∃𝑔[(◡𝑓 “ (V ∖ 𝑜)) / 𝑦](𝑔:(1...(♯‘𝑦))–1-1-onto→𝑦 ∧ 𝑥 = (seq1((+g‘𝑤), (𝑓 ∘ 𝑔))‘(♯‘𝑦))))))) | |
12 | 11 | reldmmpo 7560 | . . . 4 ⊢ Rel dom Σg |
13 | 12 | ovprc1 7463 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝐺 Σg ∅) = ∅) |
14 | fvprc 6893 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (0g‘𝐺) = ∅) | |
15 | 2, 14 | eqtrid 2778 | . . 3 ⊢ (¬ 𝐺 ∈ V → 0 = ∅) |
16 | 13, 15 | eqtr4d 2769 | . 2 ⊢ (¬ 𝐺 ∈ V → (𝐺 Σg ∅) = 0 ) |
17 | 10, 16 | pm2.61i 182 | 1 ⊢ (𝐺 Σg ∅) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 394 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 {crab 3419 Vcvv 3462 [wsbc 3776 ⦋csb 3892 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4325 ifcif 4533 ◡ccnv 5681 dom cdm 5682 ran crn 5683 “ cima 5685 ∘ ccom 5686 ℩cio 6504 ⟶wf 6550 –1-1-onto→wf1o 6553 ‘cfv 6554 (class class class)co 7424 1c1 11159 ℤ≥cuz 12874 ...cfz 13538 seqcseq 14021 ♯chash 14347 Basecbs 17213 +gcplusg 17266 0gc0g 17454 Σg cgsu 17455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-seq 14022 df-gsum 17457 |
This theorem is referenced by: gsumwsubmcl 18827 gsumccat 18831 gsumwmhm 18835 gsumwspan 18836 frmdgsum 18852 frmdup1 18854 mulgnn0gsum 19074 gsumwrev 19363 gsmsymgrfix 19426 gsmsymgreq 19430 psgnunilem2 19493 psgn0fv0 19509 psgnsn 19518 psgnprfval1 19520 gsumconst 19932 gsumfsum 21431 mplmonmul 22043 mplcoe1 22044 mplcoe5 22047 coe1fzgsumd 22295 evl1gsumd 22348 mdet0pr 22585 madugsum 22636 tmdgsum 24090 xrge0gsumle 24840 xrge0tsms 24841 jensen 27017 xrge0tsmsd 32926 gsumle 32959 cyc3genpmlem 33029 gsumvsca1 33090 gsumvsca2 33091 domnprodn0 33130 unitprodclb 33264 rprmdvdsprod 33409 1arithidom 33412 1arithufdlem3 33421 1arithufdlem4 33422 dfufd2lem 33424 zarcmplem 33696 esumnul 33881 esumsnf 33897 sitg0 34180 mrsub0 35344 matunitlindflem1 37317 evl1gprodd 41815 idomnnzgmulnz 41831 deg1gprod 41838 lincval0 47798 |
Copyright terms: Public domain | W3C validator |