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Theorem gsumzcl2 18227
Description: Closure of a finite group sum. This theorem has a weaker hypothesis than gsumzcl 18228, 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 𝐵 = (Base‘𝐺)
gsumzcl.0 0 = (0g𝐺)
gsumzcl.z 𝑍 = (Cntz‘𝐺)
gsumzcl.g (𝜑𝐺 ∈ Mnd)
gsumzcl.a (𝜑𝐴𝑉)
gsumzcl.f (𝜑𝐹:𝐴𝐵)
gsumzcl.c (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumzcl2.w (𝜑 → (𝐹 supp 0 ) ∈ Fin)
Assertion
Ref Expression
gsumzcl2 (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵)

Proof of Theorem gsumzcl2
Dummy variables 𝑓 𝑘 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzcl.f . . . . . . 7 (𝜑𝐹:𝐴𝐵)
2 gsumzcl.a . . . . . . 7 (𝜑𝐴𝑉)
3 gsumzcl.0 . . . . . . . . 9 0 = (0g𝐺)
4 fvex 6160 . . . . . . . . 9 (0g𝐺) ∈ V
53, 4eqeltri 2700 . . . . . . . 8 0 ∈ V
65a1i 11 . . . . . . 7 (𝜑0 ∈ V)
7 ssid 3608 . . . . . . . 8 (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )
87a1i 11 . . . . . . 7 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
91, 2, 6, 8gsumcllem 18225 . . . . . 6 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘𝐴0 ))
109oveq2d 6621 . . . . 5 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘𝐴0 )))
11 gsumzcl.g . . . . . . 7 (𝜑𝐺 ∈ Mnd)
123gsumz 17290 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
1311, 2, 12syl2anc 692 . . . . . 6 (𝜑 → (𝐺 Σg (𝑘𝐴0 )) = 0 )
1413adantr 481 . . . . 5 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
1510, 14eqtrd 2660 . . . 4 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg 𝐹) = 0 )
16 gsumzcl.b . . . . . . 7 𝐵 = (Base‘𝐺)
1716, 3mndidcl 17224 . . . . . 6 (𝐺 ∈ Mnd → 0𝐵)
1811, 17syl 17 . . . . 5 (𝜑0𝐵)
1918adantr 481 . . . 4 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 0𝐵)
2015, 19eqeltrd 2704 . . 3 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg 𝐹) ∈ 𝐵)
2120ex 450 . 2 (𝜑 → ((𝐹 supp 0 ) = ∅ → (𝐺 Σg 𝐹) ∈ 𝐵))
22 eqid 2626 . . . . . . 7 (+g𝐺) = (+g𝐺)
23 gsumzcl.z . . . . . . 7 𝑍 = (Cntz‘𝐺)
2411adantr 481 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐺 ∈ Mnd)
252adantr 481 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐴𝑉)
261adantr 481 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐹:𝐴𝐵)
27 gsumzcl.c . . . . . . . 8 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
2827adantr 481 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
29 simprl 793 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (#‘(𝐹 supp 0 )) ∈ ℕ)
30 f1of1 6095 . . . . . . . . 9 (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ))
3130ad2antll 764 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ))
32 suppssdm 7254 . . . . . . . . . 10 (𝐹 supp 0 ) ⊆ dom 𝐹
33 fdm 6010 . . . . . . . . . . 11 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
341, 33syl 17 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝐴)
3532, 34syl5sseq 3637 . . . . . . . . 9 (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
3635adantr 481 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝐴)
37 f1ss 6065 . . . . . . . 8 ((𝑓:(1...(#‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ) ∧ (𝐹 supp 0 ) ⊆ 𝐴) → 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1𝐴)
3831, 36, 37syl2anc 692 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1𝐴)
39 f1ofo 6103 . . . . . . . . . 10 (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(#‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ))
40 forn 6077 . . . . . . . . . 10 (𝑓:(1...(#‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 ))
4139, 40syl 17 . . . . . . . . 9 (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 ))
4241ad2antll 764 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝑓 = (𝐹 supp 0 ))
437, 42syl5sseqr 3638 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
44 eqid 2626 . . . . . . 7 ((𝐹𝑓) supp 0 ) = ((𝐹𝑓) supp 0 )
4516, 3, 22, 23, 24, 25, 26, 28, 29, 38, 43, 44gsumval3 18224 . . . . . 6 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg 𝐹) = (seq1((+g𝐺), (𝐹𝑓))‘(#‘(𝐹 supp 0 ))))
46 nnuz 11667 . . . . . . . 8 ℕ = (ℤ‘1)
4729, 46syl6eleq 2714 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (#‘(𝐹 supp 0 )) ∈ (ℤ‘1))
48 f1f 6060 . . . . . . . . . 10 (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1𝐴𝑓:(1...(#‘(𝐹 supp 0 )))⟶𝐴)
4938, 48syl 17 . . . . . . . . 9 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓:(1...(#‘(𝐹 supp 0 )))⟶𝐴)
50 fco 6017 . . . . . . . . 9 ((𝐹:𝐴𝐵𝑓:(1...(#‘(𝐹 supp 0 )))⟶𝐴) → (𝐹𝑓):(1...(#‘(𝐹 supp 0 )))⟶𝐵)
5126, 49, 50syl2anc 692 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹𝑓):(1...(#‘(𝐹 supp 0 )))⟶𝐵)
5251ffvelrnda 6316 . . . . . . 7 (((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) ∧ 𝑘 ∈ (1...(#‘(𝐹 supp 0 )))) → ((𝐹𝑓)‘𝑘) ∈ 𝐵)
5316, 22mndcl 17217 . . . . . . . . 9 ((𝐺 ∈ Mnd ∧ 𝑘𝐵𝑥𝐵) → (𝑘(+g𝐺)𝑥) ∈ 𝐵)
54533expb 1263 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ (𝑘𝐵𝑥𝐵)) → (𝑘(+g𝐺)𝑥) ∈ 𝐵)
5524, 54sylan 488 . . . . . . 7 (((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) ∧ (𝑘𝐵𝑥𝐵)) → (𝑘(+g𝐺)𝑥) ∈ 𝐵)
5647, 52, 55seqcl 12758 . . . . . 6 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (seq1((+g𝐺), (𝐹𝑓))‘(#‘(𝐹 supp 0 ))) ∈ 𝐵)
5745, 56eqeltrd 2704 . . . . 5 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg 𝐹) ∈ 𝐵)
5857expr 642 . . . 4 ((𝜑 ∧ (#‘(𝐹 supp 0 )) ∈ ℕ) → (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σg 𝐹) ∈ 𝐵))
5958exlimdv 1863 . . 3 ((𝜑 ∧ (#‘(𝐹 supp 0 )) ∈ ℕ) → (∃𝑓 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σg 𝐹) ∈ 𝐵))
6059expimpd 628 . 2 (𝜑 → (((#‘(𝐹 supp 0 )) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )) → (𝐺 Σg 𝐹) ∈ 𝐵))
61 gsumzcl2.w . . 3 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
62 fz1f1o 14369 . . 3 ((𝐹 supp 0 ) ∈ Fin → ((𝐹 supp 0 ) = ∅ ∨ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))))
6361, 62syl 17 . 2 (𝜑 → ((𝐹 supp 0 ) = ∅ ∨ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))))
6421, 60, 63mpjaod 396 1 (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1992  Vcvv 3191  wss 3560  c0 3896  cmpt 4678  dom cdm 5079  ran crn 5080  ccom 5083  wf 5846  1-1wf1 5847  ontowfo 5848  1-1-ontowf1o 5849  cfv 5850  (class class class)co 6605   supp csupp 7241  Fincfn 7900  1c1 9882  cn 10965  cuz 11631  ...cfz 12265  seqcseq 12738  #chash 13054  Basecbs 15776  +gcplusg 15857  0gc0g 16016   Σg cgsu 16017  Mndcmnd 17210  Cntzccntz 17664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-supp 7242  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-fzo 12404  df-seq 12739  df-hash 13055  df-0g 16018  df-gsum 16019  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-cntz 17666
This theorem is referenced by:  gsumzcl  18228  gsumcl2  18231
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