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Theorem gsumval3a 19812
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 2730 . . 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 7230 . . . . 5 (πœ‘ β†’ 𝐹 ∈ V)
102fvexi 6904 . . . . 5 0 ∈ V
11 suppimacnv 8161 . . . . 5 ((𝐹 ∈ V ∧ 0 ∈ V) β†’ (𝐹 supp 0 ) = (◑𝐹 β€œ (V βˆ– { 0 })))
129, 10, 11sylancl 584 . . . 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 18751 . . . . . . . 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 2736 . . . . . 6 (πœ‘ β†’ { 0 } = {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
1716difeq2d 4121 . . . . 5 (πœ‘ β†’ (V βˆ– { 0 }) = (V βˆ– {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
1817imaeq2d 6058 . . . 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 2774 . . 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 18602 . 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 4013 . . . . . 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 510 . . . . . . . . . . 11 (πœ‘ β†’ (𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ 𝑉))
2524adantr 479 . . . . . . . . . 10 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ (𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ 𝑉))
26 fex 7229 . . . . . . . . . 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 584 . . . . . . . 8 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ (𝐹 supp 0 ) = (◑𝐹 β€œ (V βˆ– { 0 })))
297ffnd 6717 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹 Fn 𝐴)
3029adantr 479 . . . . . . . . . 10 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ 𝐹 Fn 𝐴)
31 simpr 483 . . . . . . . . . 10 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ ran 𝐹 βŠ† { 0 })
32 df-f 6546 . . . . . . . . . 10 (𝐹:𝐴⟢{ 0 } ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† { 0 }))
3330, 31, 32sylanbrc 581 . . . . . . . . 9 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ 𝐹:𝐴⟢{ 0 })
34 disjdif 4470 . . . . . . . . 9 ({ 0 } ∩ (V βˆ– { 0 })) = βˆ…
35 fimacnvdisj 6768 . . . . . . . . 9 ((𝐹:𝐴⟢{ 0 } ∧ ({ 0 } ∩ (V βˆ– { 0 })) = βˆ…) β†’ (◑𝐹 β€œ (V βˆ– { 0 })) = βˆ…)
3633, 34, 35sylancl 584 . . . . . . . 8 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ (◑𝐹 β€œ (V βˆ– { 0 })) = βˆ…)
3723, 28, 363eqtrd 2774 . . . . . . 7 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ π‘Š = βˆ…)
3837ex 411 . . . . . 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 2951 . . . 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 4538 . 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 4538 . 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 2774 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 394   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068  {crab 3430  Vcvv 3472   βˆ– cdif 3944   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  ifcif 4527  {csn 4627  β—‘ccnv 5674  ran crn 5676   β€œ cima 5678   ∘ ccom 5679  β„©cio 6492   Fn wfn 6537  βŸΆwf 6538  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7411   supp csupp 8148  Fincfn 8941  1c1 11113  β„€β‰₯cuz 12826  ...cfz 13488  seqcseq 13970  β™―chash 14294  Basecbs 17148  +gcplusg 17201  0gc0g 17389   Ξ£g cgsu 17390  Mndcmnd 18659  Cntzccntz 19220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-seq 13971  df-0g 17391  df-gsum 17392  df-mgm 18565  df-sgrp 18644  df-mnd 18660
This theorem is referenced by:  gsumval3lem2  19815
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