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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 2822 | . . 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 4056 | . . . 4 ⊢ 𝑂 ⊆ 𝐵 |
10 | fss 6526 | . . . 4 ⊢ ((𝐹:𝐴⟶𝑂 ∧ 𝑂 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵) | |
11 | 8, 9, 10 | sylancl 588 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
12 | 1, 2, 3, 4, 5, 6, 7, 11 | gsumval 17886 | . 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 6519 | . . 3 ⊢ (𝐹:𝐴⟶𝑂 → ran 𝐹 ⊆ 𝑂) | |
14 | iftrue 4472 | . . 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 2856 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 {crab 3142 Vcvv 3494 ∖ cdif 3932 ⊆ wss 3935 ifcif 4466 ◡ccnv 5553 ran crn 5555 “ cima 5557 ∘ ccom 5558 ℩cio 6311 ⟶wf 6350 –1-1-onto→wf1o 6353 ‘cfv 6354 (class class class)co 7155 1c1 10537 ℤ≥cuz 12242 ...cfz 12891 seqcseq 13368 ♯chash 13689 Basecbs 16482 +gcplusg 16564 0gc0g 16712 Σg cgsu 16713 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-seq 13369 df-gsum 16715 |
This theorem is referenced by: gsum0 17893 gsumval2 17895 gsumz 17999 |
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