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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 2818 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | gsum0.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | eqid 2818 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | eqid 2818 | . . 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 5202 | . . . 4 ⊢ ∅ ∈ V | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝐺 ∈ V → ∅ ∈ V) |
8 | f0 6553 | . . . 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 17881 | . 2 ⊢ (𝐺 ∈ V → (𝐺 Σg ∅) = 0 ) |
11 | df-gsum 16704 | . . . . 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 7274 | . . . 4 ⊢ Rel dom Σg |
13 | 12 | ovprc1 7184 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝐺 Σg ∅) = ∅) |
14 | fvprc 6656 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (0g‘𝐺) = ∅) | |
15 | 2, 14 | syl5eq 2865 | . . 3 ⊢ (¬ 𝐺 ∈ V → 0 = ∅) |
16 | 13, 15 | eqtr4d 2856 | . 2 ⊢ (¬ 𝐺 ∈ V → (𝐺 Σg ∅) = 0 ) |
17 | 10, 16 | pm2.61i 183 | 1 ⊢ (𝐺 Σg ∅) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 {crab 3139 Vcvv 3492 [wsbc 3769 ⦋csb 3880 ∖ cdif 3930 ⊆ wss 3933 ∅c0 4288 ifcif 4463 ◡ccnv 5547 dom cdm 5548 ran crn 5549 “ cima 5551 ∘ ccom 5552 ℩cio 6305 ⟶wf 6344 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 1c1 10526 ℤ≥cuz 12231 ...cfz 12880 seqcseq 13357 ♯chash 13678 Basecbs 16471 +gcplusg 16553 0gc0g 16701 Σg cgsu 16702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-seq 13358 df-gsum 16704 |
This theorem is referenced by: gsumwsubmcl 17989 gsumccatOLD 17993 gsumccat 17994 gsumwmhm 17998 gsumwspan 17999 frmdgsum 18015 frmdup1 18017 mulgnn0gsum 18172 gsumwrev 18432 gsmsymgrfix 18485 gsmsymgreq 18489 psgnunilem2 18552 psgn0fv0 18568 psgnsn 18577 psgnprfval1 18579 gsumconst 18983 mplmonmul 20173 mplcoe1 20174 mplcoe5 20177 coe1fzgsumd 20398 evl1gsumd 20448 gsumfsum 20540 mdet0pr 21129 madugsum 21180 tmdgsum 22631 xrge0gsumle 23368 xrge0tsms 23369 jensen 25493 xrge0tsmsd 30619 gsumle 30652 cyc3genpmlem 30720 gsumvsca1 30781 gsumvsca2 30782 esumnul 31206 esumsnf 31222 sitg0 31503 mrsub0 32660 matunitlindflem1 34769 lincval0 44398 |
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