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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 2736 | . . 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 4092 | . . . 4 ⊢ 𝑂 ⊆ 𝐵 |
10 | fss 6753 | . . . 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 18703 | . 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 6744 | . . 3 ⊢ (𝐹:𝐴⟶𝑂 → ran 𝐹 ⊆ 𝑂) | |
14 | iftrue 4537 | . . 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 2775 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 {crab 3433 Vcvv 3478 ∖ cdif 3960 ⊆ wss 3963 ifcif 4531 ◡ccnv 5688 ran crn 5690 “ cima 5692 ∘ ccom 5693 ℩cio 6514 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 1c1 11154 ℤ≥cuz 12876 ...cfz 13544 seqcseq 14039 ♯chash 14366 Basecbs 17245 +gcplusg 17298 0gc0g 17486 Σg cgsu 17487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-seq 14040 df-gsum 17489 |
This theorem is referenced by: gsum0 18710 gsumval2 18712 gsumz 18862 |
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