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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 2733 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | gsum0.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | eqid 2733 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | eqid 2733 | . . 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 5308 | . . . 4 ⊢ ∅ ∈ V | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝐺 ∈ V → ∅ ∈ V) |
8 | f0 6773 | . . . 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 18602 | . 2 ⊢ (𝐺 ∈ V → (𝐺 Σg ∅) = 0 ) |
11 | df-gsum 17388 | . . . . 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 7543 | . . . 4 ⊢ Rel dom Σg |
13 | 12 | ovprc1 7448 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝐺 Σg ∅) = ∅) |
14 | fvprc 6884 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (0g‘𝐺) = ∅) | |
15 | 2, 14 | eqtrid 2785 | . . 3 ⊢ (¬ 𝐺 ∈ V → 0 = ∅) |
16 | 13, 15 | eqtr4d 2776 | . 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 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 {crab 3433 Vcvv 3475 [wsbc 3778 ⦋csb 3894 ∖ cdif 3946 ⊆ wss 3949 ∅c0 4323 ifcif 4529 ◡ccnv 5676 dom cdm 5677 ran crn 5678 “ cima 5680 ∘ ccom 5681 ℩cio 6494 ⟶wf 6540 –1-1-onto→wf1o 6543 ‘cfv 6544 (class class class)co 7409 1c1 11111 ℤ≥cuz 12822 ...cfz 13484 seqcseq 13966 ♯chash 14290 Basecbs 17144 +gcplusg 17197 0gc0g 17385 Σg cgsu 17386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-seq 13967 df-gsum 17388 |
This theorem is referenced by: gsumwsubmcl 18718 gsumccat 18722 gsumwmhm 18726 gsumwspan 18727 frmdgsum 18743 frmdup1 18745 mulgnn0gsum 18960 gsumwrev 19233 gsmsymgrfix 19296 gsmsymgreq 19300 psgnunilem2 19363 psgn0fv0 19379 psgnsn 19388 psgnprfval1 19390 gsumconst 19802 gsumfsum 21012 mplmonmul 21591 mplcoe1 21592 mplcoe5 21595 coe1fzgsumd 21826 evl1gsumd 21876 mdet0pr 22094 madugsum 22145 tmdgsum 23599 xrge0gsumle 24349 xrge0tsms 24350 jensen 26493 xrge0tsmsd 32209 gsumle 32242 cyc3genpmlem 32310 gsumvsca1 32371 gsumvsca2 32372 zarcmplem 32861 esumnul 33046 esumsnf 33062 sitg0 33345 mrsub0 34507 matunitlindflem1 36484 lincval0 47096 |
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