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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 2610 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | gsum0.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | eqid 2610 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | eqid 2610 | . . 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 4718 | . . . 4 ⊢ ∅ ∈ V | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝐺 ∈ V → ∅ ∈ V) |
8 | f0 5999 | . . . 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 17100 | . 2 ⊢ (𝐺 ∈ V → (𝐺 Σg ∅) = 0 ) |
11 | df-gsum 15926 | . . . . 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 | reldmmpt2 6669 | . . . 4 ⊢ Rel dom Σg |
13 | 12 | ovprc1 6582 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝐺 Σg ∅) = ∅) |
14 | fvprc 6097 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (0g‘𝐺) = ∅) | |
15 | 2, 14 | syl5eq 2656 | . . 3 ⊢ (¬ 𝐺 ∈ V → 0 = ∅) |
16 | 13, 15 | eqtr4d 2647 | . 2 ⊢ (¬ 𝐺 ∈ V → (𝐺 Σg ∅) = 0 ) |
17 | 10, 16 | pm2.61i 175 | 1 ⊢ (𝐺 Σg ∅) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 [wsbc 3402 ⦋csb 3499 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 ifcif 4036 ◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 ∘ ccom 5042 ℩cio 5766 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 1c1 9816 ℤ≥cuz 11563 ...cfz 12197 seqcseq 12663 #chash 12979 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Σg cgsu 15924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seq 12664 df-gsum 15926 |
This theorem is referenced by: gsumwsubmcl 17198 gsumccat 17201 gsumwmhm 17205 gsumwspan 17206 frmdgsum 17222 frmdup1 17224 gsumwrev 17619 gsmsymgrfix 17671 gsmsymgreq 17675 psgnunilem2 17738 psgn0fv0 17754 psgnsn 17763 psgnprfval1 17765 gsumconst 18157 mplmonmul 19285 mplcoe1 19286 mplcoe5 19289 coe1fzgsumd 19493 evl1gsumd 19542 gsumfsum 19632 mdet0pr 20217 madugsum 20268 tmdgsum 21709 xrge0gsumle 22444 xrge0tsms 22445 jensen 24515 gsumle 29110 gsumvsca1 29113 gsumvsca2 29114 xrge0tsmsd 29116 esumnul 29437 esumsnf 29453 sitg0 29735 mrsub0 30667 matunitlindflem1 32575 lincval0 41998 |
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