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Mirrors > Home > MPE Home > Th. List > gsumval1 | Structured version Visualization version GIF version |
Description: Value of the group sum operation when every element being summed is an identity of 𝐺. (Contributed by Mario Carneiro, 7-Dec-2014.) |
Ref | Expression |
---|---|
gsumval1.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumval1.z | ⊢ 0 = (0g‘𝐺) |
gsumval1.p | ⊢ + = (+g‘𝐺) |
gsumval1.o | ⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} |
gsumval1.g | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
gsumval1.a | ⊢ (𝜑 → 𝐴 ∈ 𝑊) |
gsumval1.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝑂) |
Ref | Expression |
---|---|
gsumval1 | ⊢ (𝜑 → (𝐺 Σg 𝐹) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumval1.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumval1.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | gsumval1.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | gsumval1.o | . . 3 ⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} | |
5 | eqidd 2739 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (V ∖ 𝑂)) = (◡𝐹 “ (V ∖ 𝑂))) | |
6 | gsumval1.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
7 | gsumval1.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊) | |
8 | gsumval1.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝑂) | |
9 | 4 | ssrab3 4015 | . . . 4 ⊢ 𝑂 ⊆ 𝐵 |
10 | fss 6617 | . . . 4 ⊢ ((𝐹:𝐴⟶𝑂 ∧ 𝑂 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵) | |
11 | 8, 9, 10 | sylancl 586 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
12 | 1, 2, 3, 4, 5, 6, 7, 11 | gsumval 18361 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ 𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ 𝑂))))–1-1-onto→(◡𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ 𝑂))))))))) |
13 | frn 6607 | . . 3 ⊢ (𝐹:𝐴⟶𝑂 → ran 𝐹 ⊆ 𝑂) | |
14 | iftrue 4465 | . . 3 ⊢ (ran 𝐹 ⊆ 𝑂 → if(ran 𝐹 ⊆ 𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ 𝑂))))–1-1-onto→(◡𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ 𝑂)))))))) = 0 ) | |
15 | 8, 13, 14 | 3syl 18 | . 2 ⊢ (𝜑 → if(ran 𝐹 ⊆ 𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ 𝑂))))–1-1-onto→(◡𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ 𝑂)))))))) = 0 ) |
16 | 12, 15 | eqtrd 2778 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 {crab 3068 Vcvv 3432 ∖ cdif 3884 ⊆ wss 3887 ifcif 4459 ◡ccnv 5588 ran crn 5590 “ cima 5592 ∘ ccom 5593 ℩cio 6389 ⟶wf 6429 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 1c1 10872 ℤ≥cuz 12582 ...cfz 13239 seqcseq 13721 ♯chash 14044 Basecbs 16912 +gcplusg 16962 0gc0g 17150 Σg cgsu 17151 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-seq 13722 df-gsum 17153 |
This theorem is referenced by: gsum0 18368 gsumval2 18370 gsumz 18474 |
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