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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 2821 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | gsum0.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | eqid 2821 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | eqid 2821 | . . 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 5211 | . . . 4 ⊢ ∅ ∈ V | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝐺 ∈ V → ∅ ∈ V) |
8 | f0 6560 | . . . 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 17893 | . 2 ⊢ (𝐺 ∈ V → (𝐺 Σg ∅) = 0 ) |
11 | df-gsum 16716 | . . . . 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 7285 | . . . 4 ⊢ Rel dom Σg |
13 | 12 | ovprc1 7195 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝐺 Σg ∅) = ∅) |
14 | fvprc 6663 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (0g‘𝐺) = ∅) | |
15 | 2, 14 | syl5eq 2868 | . . 3 ⊢ (¬ 𝐺 ∈ V → 0 = ∅) |
16 | 13, 15 | eqtr4d 2859 | . 2 ⊢ (¬ 𝐺 ∈ V → (𝐺 Σg ∅) = 0 ) |
17 | 10, 16 | pm2.61i 184 | 1 ⊢ (𝐺 Σg ∅) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 {crab 3142 Vcvv 3494 [wsbc 3772 ⦋csb 3883 ∖ cdif 3933 ⊆ wss 3936 ∅c0 4291 ifcif 4467 ◡ccnv 5554 dom cdm 5555 ran crn 5556 “ cima 5558 ∘ ccom 5559 ℩cio 6312 ⟶wf 6351 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 1c1 10538 ℤ≥cuz 12244 ...cfz 12893 seqcseq 13370 ♯chash 13691 Basecbs 16483 +gcplusg 16565 0gc0g 16713 Σg cgsu 16714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-seq 13371 df-gsum 16716 |
This theorem is referenced by: gsumwsubmcl 18001 gsumccatOLD 18005 gsumccat 18006 gsumwmhm 18010 gsumwspan 18011 frmdgsum 18027 frmdup1 18029 mulgnn0gsum 18234 gsumwrev 18494 gsmsymgrfix 18556 gsmsymgreq 18560 psgnunilem2 18623 psgn0fv0 18639 psgnsn 18648 psgnprfval1 18650 gsumconst 19054 mplmonmul 20245 mplcoe1 20246 mplcoe5 20249 coe1fzgsumd 20470 evl1gsumd 20520 gsumfsum 20612 mdet0pr 21201 madugsum 21252 tmdgsum 22703 xrge0gsumle 23441 xrge0tsms 23442 jensen 25566 xrge0tsmsd 30692 gsumle 30725 cyc3genpmlem 30793 gsumvsca1 30854 gsumvsca2 30855 esumnul 31307 esumsnf 31323 sitg0 31604 mrsub0 32763 matunitlindflem1 34903 lincval0 44490 |
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