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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 2737 | . . 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 4081 | . . . 4 ⊢ 𝑂 ⊆ 𝐵 | 
| 10 | fss 6751 | . . . 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 18691 | . 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 6742 | . . 3 ⊢ (𝐹:𝐴⟶𝑂 → ran 𝐹 ⊆ 𝑂) | |
| 14 | iftrue 4530 | . . 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 2776 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = 0 ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 {crab 3435 Vcvv 3479 ∖ cdif 3947 ⊆ wss 3950 ifcif 4524 ◡ccnv 5683 ran crn 5685 “ cima 5687 ∘ ccom 5688 ℩cio 6511 ⟶wf 6556 –1-1-onto→wf1o 6559 ‘cfv 6560 (class class class)co 7432 1c1 11157 ℤ≥cuz 12879 ...cfz 13548 seqcseq 14043 ♯chash 14370 Basecbs 17248 +gcplusg 17298 0gc0g 17485 Σg cgsu 17486 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-seq 14044 df-gsum 17488 | 
| This theorem is referenced by: gsum0 18698 gsumval2 18700 gsumz 18850 | 
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