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| 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 2737 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 2 | gsum0.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
| 3 | eqid 2737 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 4 | eqid 2737 | . . 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 5307 | . . . 4 ⊢ ∅ ∈ V | |
| 7 | 6 | a1i 11 | . . 3 ⊢ (𝐺 ∈ V → ∅ ∈ V) |
| 8 | f0 6789 | . . . 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 18696 | . 2 ⊢ (𝐺 ∈ V → (𝐺 Σg ∅) = 0 ) |
| 11 | df-gsum 17487 | . . . . 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 7567 | . . . 4 ⊢ Rel dom Σg |
| 13 | 12 | ovprc1 7470 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝐺 Σg ∅) = ∅) |
| 14 | fvprc 6898 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (0g‘𝐺) = ∅) | |
| 15 | 2, 14 | eqtrid 2789 | . . 3 ⊢ (¬ 𝐺 ∈ V → 0 = ∅) |
| 16 | 13, 15 | eqtr4d 2780 | . 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 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 [wsbc 3788 ⦋csb 3899 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 ifcif 4525 ◡ccnv 5684 dom cdm 5685 ran crn 5686 “ cima 5688 ∘ ccom 5689 ℩cio 6512 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 1c1 11156 ℤ≥cuz 12878 ...cfz 13547 seqcseq 14042 ♯chash 14369 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Σg cgsu 17485 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-seq 14043 df-gsum 17487 |
| This theorem is referenced by: gsumwsubmcl 18850 gsumccat 18854 gsumwmhm 18858 gsumwspan 18859 frmdgsum 18875 frmdup1 18877 mulgnn0gsum 19098 gsumwrev 19385 gsmsymgrfix 19446 gsmsymgreq 19450 psgnunilem2 19513 psgn0fv0 19529 psgnsn 19538 psgnprfval1 19540 gsumconst 19952 gsumfsum 21452 mplmonmul 22054 mplcoe1 22055 mplcoe5 22058 coe1fzgsumd 22308 evl1gsumd 22361 mdet0pr 22598 madugsum 22649 tmdgsum 24103 xrge0gsumle 24855 xrge0tsms 24856 jensen 27032 xrge0tsmsd 33065 gsumwun 33068 gsumle 33101 cyc3genpmlem 33171 gsumvsca1 33232 gsumvsca2 33233 elrgspnlem2 33247 elrgspnlem4 33249 domnprodn0 33279 unitprodclb 33417 rprmdvdsprod 33562 1arithidom 33565 1arithufdlem3 33574 1arithufdlem4 33575 dfufd2lem 33577 zarcmplem 33880 esumnul 34049 esumsnf 34065 sitg0 34348 mrsub0 35521 matunitlindflem1 37623 evl1gprodd 42118 idomnnzgmulnz 42134 deg1gprod 42141 lincval0 48332 |
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