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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 3972 | . . . 4 ⊢ 𝑂 ⊆ 𝐵 |
10 | fss 6522 | . . . 4 ⊢ ((𝐹:𝐴⟶𝑂 ∧ 𝑂 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵) | |
11 | 8, 9, 10 | sylancl 589 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
12 | 1, 2, 3, 4, 5, 6, 7, 11 | gsumval 18004 | . 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 6512 | . . 3 ⊢ (𝐹:𝐴⟶𝑂 → ran 𝐹 ⊆ 𝑂) | |
14 | iftrue 4421 | . . 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 2773 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∃wex 1786 ∈ wcel 2113 ∀wral 3053 ∃wrex 3054 {crab 3057 Vcvv 3398 ∖ cdif 3841 ⊆ wss 3844 ifcif 4415 ◡ccnv 5525 ran crn 5527 “ cima 5529 ∘ ccom 5530 ℩cio 6296 ⟶wf 6336 –1-1-onto→wf1o 6339 ‘cfv 6340 (class class class)co 7171 1c1 10617 ℤ≥cuz 12325 ...cfz 12982 seqcseq 13461 ♯chash 13783 Basecbs 16587 +gcplusg 16669 0gc0g 16817 Σg cgsu 16818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-ov 7174 df-oprab 7175 df-mpo 7176 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-seq 13462 df-gsum 16820 |
This theorem is referenced by: gsum0 18011 gsumval2 18013 gsumz 18117 |
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