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Theorem gsumval3a 19688
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 2733 . . 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 7181 . . . . 5 (πœ‘ β†’ 𝐹 ∈ V)
102fvexi 6860 . . . . 5 0 ∈ V
11 suppimacnv 8109 . . . . 5 ((𝐹 ∈ V ∧ 0 ∈ V) β†’ (𝐹 supp 0 ) = (◑𝐹 β€œ (V βˆ– { 0 })))
129, 10, 11sylancl 587 . . . 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 18652 . . . . . . . 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 2739 . . . . . 6 (πœ‘ β†’ { 0 } = {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
1716difeq2d 4086 . . . . 5 (πœ‘ β†’ (V βˆ– { 0 }) = (V βˆ– {𝑧 ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
1817imaeq2d 6017 . . . 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 2777 . . 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 18540 . 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 3980 . . . . . 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 513 . . . . . . . . . . 11 (πœ‘ β†’ (𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ 𝑉))
2524adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ (𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ 𝑉))
26 fex 7180 . . . . . . . . . 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 587 . . . . . . . 8 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ (𝐹 supp 0 ) = (◑𝐹 β€œ (V βˆ– { 0 })))
297ffnd 6673 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹 Fn 𝐴)
3029adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ 𝐹 Fn 𝐴)
31 simpr 486 . . . . . . . . . 10 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ ran 𝐹 βŠ† { 0 })
32 df-f 6504 . . . . . . . . . 10 (𝐹:𝐴⟢{ 0 } ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† { 0 }))
3330, 31, 32sylanbrc 584 . . . . . . . . 9 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ 𝐹:𝐴⟢{ 0 })
34 disjdif 4435 . . . . . . . . 9 ({ 0 } ∩ (V βˆ– { 0 })) = βˆ…
35 fimacnvdisj 6724 . . . . . . . . 9 ((𝐹:𝐴⟢{ 0 } ∧ ({ 0 } ∩ (V βˆ– { 0 })) = βˆ…) β†’ (◑𝐹 β€œ (V βˆ– { 0 })) = βˆ…)
3633, 34, 35sylancl 587 . . . . . . . 8 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ (◑𝐹 β€œ (V βˆ– { 0 })) = βˆ…)
3723, 28, 363eqtrd 2777 . . . . . . 7 ((πœ‘ ∧ ran 𝐹 βŠ† { 0 }) β†’ π‘Š = βˆ…)
3837ex 414 . . . . . 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 4501 . 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 4501 . 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 2777 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 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3406  Vcvv 3447   βˆ– cdif 3911   ∩ cin 3913   βŠ† wss 3914  βˆ…c0 4286  ifcif 4490  {csn 4590  β—‘ccnv 5636  ran crn 5638   β€œ cima 5640   ∘ ccom 5641  β„©cio 6450   Fn wfn 6495  βŸΆwf 6496  β€“1-1-ontoβ†’wf1o 6499  β€˜cfv 6500  (class class class)co 7361   supp csupp 8096  Fincfn 8889  1c1 11060  β„€β‰₯cuz 12771  ...cfz 13433  seqcseq 13915  β™―chash 14239  Basecbs 17091  +gcplusg 17141  0gc0g 17329   Ξ£g cgsu 17330  Mndcmnd 18564  Cntzccntz 19103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-supp 8097  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-seq 13916  df-0g 17331  df-gsum 17332  df-mgm 18505  df-sgrp 18554  df-mnd 18565
This theorem is referenced by:  gsumval3lem2  19691
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