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Theorem gsumval3a 19770
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 2732 . . 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 7228 . . . . 5 (πœ‘ β†’ 𝐹 ∈ V)
102fvexi 6905 . . . . 5 0 ∈ V
11 suppimacnv 8158 . . . . 5 ((𝐹 ∈ V ∧ 0 ∈ V) β†’ (𝐹 supp 0 ) = (◑𝐹 β€œ (V βˆ– { 0 })))
129, 10, 11sylancl 586 . . . 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 18714 . . . . . . . 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 2738 . . . . . 6 (πœ‘ β†’ { 0 } = {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
1716difeq2d 4122 . . . . 5 (πœ‘ β†’ (V βˆ– { 0 }) = (V βˆ– {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
1817imaeq2d 6059 . . . 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 2776 . . 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 18595 . 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 4014 . . . . . 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 512 . . . . . . . . . . 11 (πœ‘ β†’ (𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ 𝑉))
2524adantr 481 . . . . . . . . . 10 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ (𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ 𝑉))
26 fex 7227 . . . . . . . . . 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 586 . . . . . . . 8 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ (𝐹 supp 0 ) = (◑𝐹 β€œ (V βˆ– { 0 })))
297ffnd 6718 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹 Fn 𝐴)
3029adantr 481 . . . . . . . . . 10 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ 𝐹 Fn 𝐴)
31 simpr 485 . . . . . . . . . 10 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ ran 𝐹 βŠ† { 0 })
32 df-f 6547 . . . . . . . . . 10 (𝐹:𝐴⟢{ 0 } ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† { 0 }))
3330, 31, 32sylanbrc 583 . . . . . . . . 9 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ 𝐹:𝐴⟢{ 0 })
34 disjdif 4471 . . . . . . . . 9 ({ 0 } ∩ (V βˆ– { 0 })) = βˆ…
35 fimacnvdisj 6769 . . . . . . . . 9 ((𝐹:𝐴⟢{ 0 } ∧ ({ 0 } ∩ (V βˆ– { 0 })) = βˆ…) β†’ (◑𝐹 β€œ (V βˆ– { 0 })) = βˆ…)
3633, 34, 35sylancl 586 . . . . . . . 8 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ (◑𝐹 β€œ (V βˆ– { 0 })) = βˆ…)
3723, 28, 363eqtrd 2776 . . . . . . 7 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ π‘Š = βˆ…)
3837ex 413 . . . . . 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 2953 . . . 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 4539 . 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 4539 . 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 2776 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 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βˆ– cdif 3945   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  ifcif 4528  {csn 4628  β—‘ccnv 5675  ran crn 5677   β€œ cima 5679   ∘ ccom 5680  β„©cio 6493   Fn wfn 6538  βŸΆwf 6539  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7408   supp csupp 8145  Fincfn 8938  1c1 11110  β„€β‰₯cuz 12821  ...cfz 13483  seqcseq 13965  β™―chash 14289  Basecbs 17143  +gcplusg 17196  0gc0g 17384   Ξ£g cgsu 17385  Mndcmnd 18624  Cntzccntz 19178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-supp 8146  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-seq 13966  df-0g 17386  df-gsum 17387  df-mgm 18560  df-sgrp 18609  df-mnd 18625
This theorem is referenced by:  gsumval3lem2  19773
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