MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumval3a Structured version   Visualization version   GIF version

Theorem gsumval3a 19813
Description: Value of the group sum operation over an index set with finite support. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by AV, 29-May-2019.)
Hypotheses
Ref Expression
gsumval3.b 𝐡 = (Baseβ€˜πΊ)
gsumval3.0 0 = (0gβ€˜πΊ)
gsumval3.p + = (+gβ€˜πΊ)
gsumval3.z 𝑍 = (Cntzβ€˜πΊ)
gsumval3.g (πœ‘ β†’ 𝐺 ∈ Mnd)
gsumval3.a (πœ‘ β†’ 𝐴 ∈ 𝑉)
gsumval3.f (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
gsumval3.c (πœ‘ β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
gsumval3a.t (πœ‘ β†’ π‘Š ∈ Fin)
gsumval3a.n (πœ‘ β†’ π‘Š β‰  βˆ…)
gsumval3a.w π‘Š = (𝐹 supp 0 )
gsumval3a.i (πœ‘ β†’ Β¬ 𝐴 ∈ ran ...)
Assertion
Ref Expression
gsumval3a (πœ‘ β†’ (𝐺 Ξ£g 𝐹) = (β„©π‘₯βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)))))
Distinct variable groups:   π‘₯,𝑓, +   𝐴,𝑓,π‘₯   πœ‘,𝑓,π‘₯   π‘₯, 0   𝑓,𝐺,π‘₯   π‘₯,𝑉   𝐡,𝑓,π‘₯   𝑓,𝐹,π‘₯   𝑓,π‘Š,π‘₯
Allowed substitution hints:   𝑉(𝑓)   0 (𝑓)   𝑍(π‘₯,𝑓)

Proof of Theorem gsumval3a
Dummy variables π‘š 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumval3.b . . 3 𝐡 = (Baseβ€˜πΊ)
2 gsumval3.0 . . 3 0 = (0gβ€˜πΊ)
3 gsumval3.p . . 3 + = (+gβ€˜πΊ)
4 eqid 2731 . . 3 {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}
5 gsumval3a.w . . . . 5 π‘Š = (𝐹 supp 0 )
65a1i 11 . . . 4 (πœ‘ β†’ π‘Š = (𝐹 supp 0 ))
7 gsumval3.f . . . . . 6 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
8 gsumval3.a . . . . . 6 (πœ‘ β†’ 𝐴 ∈ 𝑉)
97, 8fexd 7232 . . . . 5 (πœ‘ β†’ 𝐹 ∈ V)
102fvexi 6906 . . . . 5 0 ∈ V
11 suppimacnv 8162 . . . . 5 ((𝐹 ∈ V ∧ 0 ∈ V) β†’ (𝐹 supp 0 ) = (◑𝐹 β€œ (V βˆ– { 0 })))
129, 10, 11sylancl 585 . . . 4 (πœ‘ β†’ (𝐹 supp 0 ) = (◑𝐹 β€œ (V βˆ– { 0 })))
13 gsumval3.g . . . . . . . 8 (πœ‘ β†’ 𝐺 ∈ Mnd)
141, 2, 3, 4gsumvallem2 18752 . . . . . . . 8 (𝐺 ∈ Mnd β†’ {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 })
1513, 14syl 17 . . . . . . 7 (πœ‘ β†’ {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 })
1615eqcomd 2737 . . . . . 6 (πœ‘ β†’ { 0 } = {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
1716difeq2d 4123 . . . . 5 (πœ‘ β†’ (V βˆ– { 0 }) = (V βˆ– {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
1817imaeq2d 6060 . . . 4 (πœ‘ β†’ (◑𝐹 β€œ (V βˆ– { 0 })) = (◑𝐹 β€œ (V βˆ– {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
196, 12, 183eqtrd 2775 . . 3 (πœ‘ β†’ π‘Š = (◑𝐹 β€œ (V βˆ– {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
201, 2, 3, 4, 19, 13, 8, 7gsumval 18603 . 2 (πœ‘ β†’ (𝐺 Ξ£g 𝐹) = if(ran 𝐹 βŠ† {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}, 0 , if(𝐴 ∈ ran ..., (β„©π‘₯βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)(𝐴 = (π‘š...𝑛) ∧ π‘₯ = (seqπ‘š( + , 𝐹)β€˜π‘›))), (β„©π‘₯βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)))))))
21 gsumval3a.n . . . 4 (πœ‘ β†’ π‘Š β‰  βˆ…)
2215sseq2d 4015 . . . . . 6 (πœ‘ β†’ (ran 𝐹 βŠ† {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} ↔ ran 𝐹 βŠ† { 0 }))
235a1i 11 . . . . . . . 8 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ π‘Š = (𝐹 supp 0 ))
247, 8jca 511 . . . . . . . . . . 11 (πœ‘ β†’ (𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ 𝑉))
2524adantr 480 . . . . . . . . . 10 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ (𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ 𝑉))
26 fex 7231 . . . . . . . . . 10 ((𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ 𝑉) β†’ 𝐹 ∈ V)
2725, 26syl 17 . . . . . . . . 9 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ 𝐹 ∈ V)
2827, 10, 11sylancl 585 . . . . . . . 8 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ (𝐹 supp 0 ) = (◑𝐹 β€œ (V βˆ– { 0 })))
297ffnd 6719 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹 Fn 𝐴)
3029adantr 480 . . . . . . . . . 10 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ 𝐹 Fn 𝐴)
31 simpr 484 . . . . . . . . . 10 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ ran 𝐹 βŠ† { 0 })
32 df-f 6548 . . . . . . . . . 10 (𝐹:𝐴⟢{ 0 } ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† { 0 }))
3330, 31, 32sylanbrc 582 . . . . . . . . 9 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ 𝐹:𝐴⟢{ 0 })
34 disjdif 4472 . . . . . . . . 9 ({ 0 } ∩ (V βˆ– { 0 })) = βˆ…
35 fimacnvdisj 6770 . . . . . . . . 9 ((𝐹:𝐴⟢{ 0 } ∧ ({ 0 } ∩ (V βˆ– { 0 })) = βˆ…) β†’ (◑𝐹 β€œ (V βˆ– { 0 })) = βˆ…)
3633, 34, 35sylancl 585 . . . . . . . 8 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ (◑𝐹 β€œ (V βˆ– { 0 })) = βˆ…)
3723, 28, 363eqtrd 2775 . . . . . . 7 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ π‘Š = βˆ…)
3837ex 412 . . . . . 6 (πœ‘ β†’ (ran 𝐹 βŠ† { 0 } β†’ π‘Š = βˆ…))
3922, 38sylbid 239 . . . . 5 (πœ‘ β†’ (ran 𝐹 βŠ† {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} β†’ π‘Š = βˆ…))
4039necon3ad 2952 . . . 4 (πœ‘ β†’ (π‘Š β‰  βˆ… β†’ Β¬ ran 𝐹 βŠ† {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
4121, 40mpd 15 . . 3 (πœ‘ β†’ Β¬ ran 𝐹 βŠ† {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
4241iffalsed 4540 . 2 (πœ‘ β†’ if(ran 𝐹 βŠ† {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}, 0 , if(𝐴 ∈ ran ..., (β„©π‘₯βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)(𝐴 = (π‘š...𝑛) ∧ π‘₯ = (seqπ‘š( + , 𝐹)β€˜π‘›))), (β„©π‘₯βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)))))) = if(𝐴 ∈ ran ..., (β„©π‘₯βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)(𝐴 = (π‘š...𝑛) ∧ π‘₯ = (seqπ‘š( + , 𝐹)β€˜π‘›))), (β„©π‘₯βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))))))
43 gsumval3a.i . . 3 (πœ‘ β†’ Β¬ 𝐴 ∈ ran ...)
4443iffalsed 4540 . 2 (πœ‘ β†’ if(𝐴 ∈ ran ..., (β„©π‘₯βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)(𝐴 = (π‘š...𝑛) ∧ π‘₯ = (seqπ‘š( + , 𝐹)β€˜π‘›))), (β„©π‘₯βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))))) = (β„©π‘₯βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)))))
4520, 42, 443eqtrd 2775 1 (πœ‘ β†’ (𝐺 Ξ£g 𝐹) = (β„©π‘₯βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1540  βˆƒwex 1780   ∈ wcel 2105   β‰  wne 2939  βˆ€wral 3060  βˆƒwrex 3069  {crab 3431  Vcvv 3473   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  ifcif 4529  {csn 4629  β—‘ccnv 5676  ran crn 5678   β€œ cima 5680   ∘ ccom 5681  β„©cio 6494   Fn wfn 6539  βŸΆwf 6540  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  (class class class)co 7412   supp csupp 8149  Fincfn 8942  1c1 11114  β„€β‰₯cuz 12827  ...cfz 13489  seqcseq 13971  β™―chash 14295  Basecbs 17149  +gcplusg 17202  0gc0g 17390   Ξ£g cgsu 17391  Mndcmnd 18660  Cntzccntz 19221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-supp 8150  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-seq 13972  df-0g 17392  df-gsum 17393  df-mgm 18566  df-sgrp 18645  df-mnd 18661
This theorem is referenced by:  gsumval3lem2  19816
  Copyright terms: Public domain W3C validator