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Theorem gsumzcl2 19867
Description: Closure of a finite group sum. This theorem has a weaker hypothesis than gsumzcl 19868, because it is not required that 𝐹 is a function (actually, the hypothesis always holds for any proper class 𝐹). (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 1-Jun-2019.)
Hypotheses
Ref Expression
gsumzcl.b 𝐡 = (Baseβ€˜πΊ)
gsumzcl.0 0 = (0gβ€˜πΊ)
gsumzcl.z 𝑍 = (Cntzβ€˜πΊ)
gsumzcl.g (πœ‘ β†’ 𝐺 ∈ Mnd)
gsumzcl.a (πœ‘ β†’ 𝐴 ∈ 𝑉)
gsumzcl.f (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
gsumzcl.c (πœ‘ β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
gsumzcl2.w (πœ‘ β†’ (𝐹 supp 0 ) ∈ Fin)
Assertion
Ref Expression
gsumzcl2 (πœ‘ β†’ (𝐺 Ξ£g 𝐹) ∈ 𝐡)
Proof of Theorem gsumzcl2
Dummy variables 𝑓 π‘˜ π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzcl.f . . . . . . 7 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
2 gsumzcl.a . . . . . . 7 (πœ‘ β†’ 𝐴 ∈ 𝑉)
3 gsumzcl.0 . . . . . . . . 9 0 = (0gβ€˜πΊ)
43fvexi 6904 . . . . . . . 8 0 ∈ V
54a1i 11 . . . . . . 7 (πœ‘ β†’ 0 ∈ V)
6 ssidd 3995 . . . . . . 7 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† (𝐹 supp 0 ))
71, 2, 5, 6gsumcllem 19865 . . . . . 6 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ 𝐹 = (π‘˜ ∈ 𝐴 ↦ 0 ))
87oveq2d 7430 . . . . 5 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )))
9 gsumzcl.g . . . . . . 7 (πœ‘ β†’ 𝐺 ∈ Mnd)
103gsumz 18790 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )) = 0 )
119, 2, 10syl2anc 582 . . . . . 6 (πœ‘ β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )) = 0 )
1211adantr 479 . . . . 5 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )) = 0 )
138, 12eqtrd 2765 . . . 4 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g 𝐹) = 0 )
14 gsumzcl.b . . . . . . 7 𝐡 = (Baseβ€˜πΊ)
1514, 3mndidcl 18706 . . . . . 6 (𝐺 ∈ Mnd β†’ 0 ∈ 𝐡)
169, 15syl 17 . . . . 5 (πœ‘ β†’ 0 ∈ 𝐡)
1716adantr 479 . . . 4 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ 0 ∈ 𝐡)
1813, 17eqeltrd 2825 . . 3 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g 𝐹) ∈ 𝐡)
1918ex 411 . 2 (πœ‘ β†’ ((𝐹 supp 0 ) = βˆ… β†’ (𝐺 Ξ£g 𝐹) ∈ 𝐡))
20 eqid 2725 . . . . . . 7 (+gβ€˜πΊ) = (+gβ€˜πΊ)
21 gsumzcl.z . . . . . . 7 𝑍 = (Cntzβ€˜πΊ)
229adantr 479 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝐺 ∈ Mnd)
232adantr 479 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝐴 ∈ 𝑉)
241adantr 479 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝐹:𝐴⟢𝐡)
25 gsumzcl.c . . . . . . . 8 (πœ‘ β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
2625adantr 479 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
27 simprl 769 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (β™―β€˜(𝐹 supp 0 )) ∈ β„•)
28 f1of1 6831 . . . . . . . . 9 (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1β†’(𝐹 supp 0 ))
2928ad2antll 727 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1β†’(𝐹 supp 0 ))
30 suppssdm 8178 . . . . . . . . . 10 (𝐹 supp 0 ) βŠ† dom 𝐹
3130, 1fssdm 6735 . . . . . . . . 9 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† 𝐴)
3231adantr 479 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐹 supp 0 ) βŠ† 𝐴)
33 f1ss 6792 . . . . . . . 8 ((𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1β†’(𝐹 supp 0 ) ∧ (𝐹 supp 0 ) βŠ† 𝐴) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1→𝐴)
3429, 32, 33syl2anc 582 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1→𝐴)
35 ssid 3994 . . . . . . . 8 (𝐹 supp 0 ) βŠ† (𝐹 supp 0 )
36 f1ofo 6839 . . . . . . . . . 10 (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–ontoβ†’(𝐹 supp 0 ))
37 forn 6807 . . . . . . . . . 10 (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–ontoβ†’(𝐹 supp 0 ) β†’ ran 𝑓 = (𝐹 supp 0 ))
3836, 37syl 17 . . . . . . . . 9 (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) β†’ ran 𝑓 = (𝐹 supp 0 ))
3938ad2antll 727 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ ran 𝑓 = (𝐹 supp 0 ))
4035, 39sseqtrrid 4025 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐹 supp 0 ) βŠ† ran 𝑓)
41 eqid 2725 . . . . . . 7 ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
4214, 3, 20, 21, 22, 23, 24, 26, 27, 34, 40, 41gsumval3 19864 . . . . . 6 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐺 Ξ£g 𝐹) = (seq1((+gβ€˜πΊ), (𝐹 ∘ 𝑓))β€˜(β™―β€˜(𝐹 supp 0 ))))
43 nnuz 12893 . . . . . . . 8 β„• = (β„€β‰₯β€˜1)
4427, 43eleqtrdi 2835 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (β™―β€˜(𝐹 supp 0 )) ∈ (β„€β‰₯β€˜1))
45 f1f 6786 . . . . . . . . . 10 (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1→𝐴 β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))⟢𝐴)
4634, 45syl 17 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))⟢𝐴)
47 fco 6740 . . . . . . . . 9 ((𝐹:𝐴⟢𝐡 ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))⟢𝐴) β†’ (𝐹 ∘ 𝑓):(1...(β™―β€˜(𝐹 supp 0 )))⟢𝐡)
4824, 46, 47syl2anc 582 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐹 ∘ 𝑓):(1...(β™―β€˜(𝐹 supp 0 )))⟢𝐡)
4948ffvelcdmda 7087 . . . . . . 7 (((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) ∧ π‘˜ ∈ (1...(β™―β€˜(𝐹 supp 0 )))) β†’ ((𝐹 ∘ 𝑓)β€˜π‘˜) ∈ 𝐡)
5014, 20mndcl 18699 . . . . . . . . 9 ((𝐺 ∈ Mnd ∧ π‘˜ ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) β†’ (π‘˜(+gβ€˜πΊ)π‘₯) ∈ 𝐡)
51503expb 1117 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ (π‘˜ ∈ 𝐡 ∧ π‘₯ ∈ 𝐡)) β†’ (π‘˜(+gβ€˜πΊ)π‘₯) ∈ 𝐡)
5222, 51sylan 578 . . . . . . 7 (((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) ∧ (π‘˜ ∈ 𝐡 ∧ π‘₯ ∈ 𝐡)) β†’ (π‘˜(+gβ€˜πΊ)π‘₯) ∈ 𝐡)
5344, 49, 52seqcl 14017 . . . . . 6 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (seq1((+gβ€˜πΊ), (𝐹 ∘ 𝑓))β€˜(β™―β€˜(𝐹 supp 0 ))) ∈ 𝐡)
5442, 53eqeltrd 2825 . . . . 5 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐺 Ξ£g 𝐹) ∈ 𝐡)
5554expr 455 . . . 4 ((πœ‘ ∧ (β™―β€˜(𝐹 supp 0 )) ∈ β„•) β†’ (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) β†’ (𝐺 Ξ£g 𝐹) ∈ 𝐡))
5655exlimdv 1928 . . 3 ((πœ‘ ∧ (β™―β€˜(𝐹 supp 0 )) ∈ β„•) β†’ (βˆƒπ‘“ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) β†’ (𝐺 Ξ£g 𝐹) ∈ 𝐡))
5756expimpd 452 . 2 (πœ‘ β†’ (((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 )) β†’ (𝐺 Ξ£g 𝐹) ∈ 𝐡))
58 gsumzcl2.w . . 3 (πœ‘ β†’ (𝐹 supp 0 ) ∈ Fin)
59 fz1f1o 15686 . . 3 ((𝐹 supp 0 ) ∈ Fin β†’ ((𝐹 supp 0 ) = βˆ… ∨ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))))
6058, 59syl 17 . 2 (πœ‘ β†’ ((𝐹 supp 0 ) = βˆ… ∨ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))))
6119, 57, 60mpjaod 858 1 (πœ‘ β†’ (𝐺 Ξ£g 𝐹) ∈ 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∨ wo 845   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  Vcvv 3463   βŠ† wss 3939  βˆ…c0 4316   ↦ cmpt 5224  ran crn 5671   ∘ ccom 5674  βŸΆwf 6537  β€“1-1β†’wf1 6538  β€“ontoβ†’wfo 6539  β€“1-1-ontoβ†’wf1o 6540  β€˜cfv 6541  (class class class)co 7414   supp csupp 8161  Fincfn 8960  1c1 11137  β„•cn 12240  β„€β‰₯cuz 12850  ...cfz 13514  seqcseq 13996  β™―chash 14319  Basecbs 17177  +gcplusg 17230  0gc0g 17418   Ξ£g cgsu 17419  Mndcmnd 18691  Cntzccntz 19268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5357  ax-pr 5421  ax-un 7736  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3958  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4943  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7867  df-1st 7989  df-2nd 7990  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-oi 9531  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-n0 12501  df-z 12587  df-uz 12851  df-fz 13515  df-fzo 13658  df-seq 13997  df-hash 14320  df-0g 17420  df-gsum 17421  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-cntz 19270
This theorem is referenced by:  gsumzcl  19868  gsumcl2  19871
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