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Theorem gsumpt 19076
Description: Sum of a family that is nonzero at at most one point. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.)
Hypotheses
Ref Expression
gsumpt.b 𝐵 = (Base‘𝐺)
gsumpt.z 0 = (0g𝐺)
gsumpt.g (𝜑𝐺 ∈ Mnd)
gsumpt.a (𝜑𝐴𝑉)
gsumpt.x (𝜑𝑋𝐴)
gsumpt.f (𝜑𝐹:𝐴𝐵)
gsumpt.s (𝜑 → (𝐹 supp 0 ) ⊆ {𝑋})
Assertion
Ref Expression
gsumpt (𝜑 → (𝐺 Σg 𝐹) = (𝐹𝑋))

Proof of Theorem gsumpt
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 gsumpt.f . . . 4 (𝜑𝐹:𝐴𝐵)
2 gsumpt.x . . . . 5 (𝜑𝑋𝐴)
32snssd 4736 . . . 4 (𝜑 → {𝑋} ⊆ 𝐴)
41, 3feqresmpt 6729 . . 3 (𝜑 → (𝐹 ↾ {𝑋}) = (𝑎 ∈ {𝑋} ↦ (𝐹𝑎)))
54oveq2d 7166 . 2 (𝜑 → (𝐺 Σg (𝐹 ↾ {𝑋})) = (𝐺 Σg (𝑎 ∈ {𝑋} ↦ (𝐹𝑎))))
6 gsumpt.b . . 3 𝐵 = (Base‘𝐺)
7 gsumpt.z . . 3 0 = (0g𝐺)
8 eqid 2821 . . 3 (Cntz‘𝐺) = (Cntz‘𝐺)
9 gsumpt.g . . 3 (𝜑𝐺 ∈ Mnd)
10 gsumpt.a . . 3 (𝜑𝐴𝑉)
111, 2ffvelrnd 6847 . . . . . . . 8 (𝜑 → (𝐹𝑋) ∈ 𝐵)
12 eqidd 2822 . . . . . . . 8 (𝜑 → ((𝐹𝑋)(+g𝐺)(𝐹𝑋)) = ((𝐹𝑋)(+g𝐺)(𝐹𝑋)))
13 eqid 2821 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
146, 13, 8elcntzsn 18449 . . . . . . . . 9 ((𝐹𝑋) ∈ 𝐵 → ((𝐹𝑋) ∈ ((Cntz‘𝐺)‘{(𝐹𝑋)}) ↔ ((𝐹𝑋) ∈ 𝐵 ∧ ((𝐹𝑋)(+g𝐺)(𝐹𝑋)) = ((𝐹𝑋)(+g𝐺)(𝐹𝑋)))))
1511, 14syl 17 . . . . . . . 8 (𝜑 → ((𝐹𝑋) ∈ ((Cntz‘𝐺)‘{(𝐹𝑋)}) ↔ ((𝐹𝑋) ∈ 𝐵 ∧ ((𝐹𝑋)(+g𝐺)(𝐹𝑋)) = ((𝐹𝑋)(+g𝐺)(𝐹𝑋)))))
1611, 12, 15mpbir2and 711 . . . . . . 7 (𝜑 → (𝐹𝑋) ∈ ((Cntz‘𝐺)‘{(𝐹𝑋)}))
1716snssd 4736 . . . . . 6 (𝜑 → {(𝐹𝑋)} ⊆ ((Cntz‘𝐺)‘{(𝐹𝑋)}))
18 eqid 2821 . . . . . . 7 (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
19 eqid 2821 . . . . . . 7 (𝐺s ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})) = (𝐺s ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
208, 18, 19cntzspan 18958 . . . . . 6 ((𝐺 ∈ Mnd ∧ {(𝐹𝑋)} ⊆ ((Cntz‘𝐺)‘{(𝐹𝑋)})) → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})) ∈ CMnd)
219, 17, 20syl2anc 586 . . . . 5 (𝜑 → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})) ∈ CMnd)
226submacs 17985 . . . . . . . 8 (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
23 acsmre 16917 . . . . . . . 8 ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
249, 22, 233syl 18 . . . . . . 7 (𝜑 → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
2511snssd 4736 . . . . . . 7 (𝜑 → {(𝐹𝑋)} ⊆ 𝐵)
2618mrccl 16876 . . . . . . 7 (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ {(𝐹𝑋)} ⊆ 𝐵) → ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ∈ (SubMnd‘𝐺))
2724, 25, 26syl2anc 586 . . . . . 6 (𝜑 → ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ∈ (SubMnd‘𝐺))
2819, 8submcmn2 18953 . . . . . 6 (((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ∈ (SubMnd‘𝐺) → ((𝐺s ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})) ∈ CMnd ↔ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ⊆ ((Cntz‘𝐺)‘((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))))
2927, 28syl 17 . . . . 5 (𝜑 → ((𝐺s ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})) ∈ CMnd ↔ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ⊆ ((Cntz‘𝐺)‘((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))))
3021, 29mpbid 234 . . . 4 (𝜑 → ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ⊆ ((Cntz‘𝐺)‘((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})))
311ffnd 6510 . . . . . 6 (𝜑𝐹 Fn 𝐴)
32 simpr 487 . . . . . . . . . 10 (((𝜑𝑎𝐴) ∧ 𝑎 = 𝑋) → 𝑎 = 𝑋)
3332fveq2d 6669 . . . . . . . . 9 (((𝜑𝑎𝐴) ∧ 𝑎 = 𝑋) → (𝐹𝑎) = (𝐹𝑋))
3424, 18, 25mrcssidd 16890 . . . . . . . . . . 11 (𝜑 → {(𝐹𝑋)} ⊆ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
35 fvex 6678 . . . . . . . . . . . 12 (𝐹𝑋) ∈ V
3635snss 4712 . . . . . . . . . . 11 ((𝐹𝑋) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ↔ {(𝐹𝑋)} ⊆ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
3734, 36sylibr 236 . . . . . . . . . 10 (𝜑 → (𝐹𝑋) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
3837ad2antrr 724 . . . . . . . . 9 (((𝜑𝑎𝐴) ∧ 𝑎 = 𝑋) → (𝐹𝑋) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
3933, 38eqeltrd 2913 . . . . . . . 8 (((𝜑𝑎𝐴) ∧ 𝑎 = 𝑋) → (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
40 eldifsn 4713 . . . . . . . . . . 11 (𝑎 ∈ (𝐴 ∖ {𝑋}) ↔ (𝑎𝐴𝑎𝑋))
41 gsumpt.s . . . . . . . . . . . 12 (𝜑 → (𝐹 supp 0 ) ⊆ {𝑋})
427fvexi 6679 . . . . . . . . . . . . 13 0 ∈ V
4342a1i 11 . . . . . . . . . . . 12 (𝜑0 ∈ V)
441, 41, 10, 43suppssr 7855 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (𝐴 ∖ {𝑋})) → (𝐹𝑎) = 0 )
4540, 44sylan2br 596 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑎𝑋)) → (𝐹𝑎) = 0 )
467subm0cl 17970 . . . . . . . . . . . 12 (((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ∈ (SubMnd‘𝐺) → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
4727, 46syl 17 . . . . . . . . . . 11 (𝜑0 ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
4847adantr 483 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑎𝑋)) → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
4945, 48eqeltrd 2913 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝑎𝑋)) → (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
5049anassrs 470 . . . . . . . 8 (((𝜑𝑎𝐴) ∧ 𝑎𝑋) → (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
5139, 50pm2.61dane 3104 . . . . . . 7 ((𝜑𝑎𝐴) → (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
5251ralrimiva 3182 . . . . . 6 (𝜑 → ∀𝑎𝐴 (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
53 ffnfv 6877 . . . . . 6 (𝐹:𝐴⟶((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑎𝐴 (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})))
5431, 52, 53sylanbrc 585 . . . . 5 (𝜑𝐹:𝐴⟶((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
5554frnd 6516 . . . 4 (𝜑 → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
568cntzidss 18462 . . . 4 ((((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ⊆ ((Cntz‘𝐺)‘((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})) ∧ ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})) → ran 𝐹 ⊆ ((Cntz‘𝐺)‘ran 𝐹))
5730, 55, 56syl2anc 586 . . 3 (𝜑 → ran 𝐹 ⊆ ((Cntz‘𝐺)‘ran 𝐹))
581ffund 6513 . . . 4 (𝜑 → Fun 𝐹)
59 snfi 8588 . . . . 5 {𝑋} ∈ Fin
60 ssfi 8732 . . . . 5 (({𝑋} ∈ Fin ∧ (𝐹 supp 0 ) ⊆ {𝑋}) → (𝐹 supp 0 ) ∈ Fin)
6159, 41, 60sylancr 589 . . . 4 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
62 fex 6983 . . . . . 6 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
631, 10, 62syl2anc 586 . . . . 5 (𝜑𝐹 ∈ V)
64 isfsupp 8831 . . . . 5 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 finSupp 0 ↔ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin)))
6563, 43, 64syl2anc 586 . . . 4 (𝜑 → (𝐹 finSupp 0 ↔ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin)))
6658, 61, 65mpbir2and 711 . . 3 (𝜑𝐹 finSupp 0 )
676, 7, 8, 9, 10, 1, 57, 41, 66gsumzres 19023 . 2 (𝜑 → (𝐺 Σg (𝐹 ↾ {𝑋})) = (𝐺 Σg 𝐹))
68 fveq2 6665 . . . 4 (𝑎 = 𝑋 → (𝐹𝑎) = (𝐹𝑋))
696, 68gsumsn 19068 . . 3 ((𝐺 ∈ Mnd ∧ 𝑋𝐴 ∧ (𝐹𝑋) ∈ 𝐵) → (𝐺 Σg (𝑎 ∈ {𝑋} ↦ (𝐹𝑎))) = (𝐹𝑋))
709, 2, 11, 69syl3anc 1367 . 2 (𝜑 → (𝐺 Σg (𝑎 ∈ {𝑋} ↦ (𝐹𝑎))) = (𝐹𝑋))
715, 67, 703eqtr3d 2864 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  wne 3016  wral 3138  Vcvv 3495  cdif 3933  wss 3936  {csn 4561   class class class wbr 5059  cmpt 5139  ran crn 5551  cres 5552  Fun wfun 6344   Fn wfn 6345  wf 6346  cfv 6350  (class class class)co 7150   supp csupp 7824  Fincfn 8503   finSupp cfsupp 8827  Basecbs 16477  s cress 16478  +gcplusg 16559  0gc0g 16707   Σg cgsu 16708  Moorecmre 16847  mrClscmrc 16848  ACScacs 16850  Mndcmnd 17905  SubMndcsubmnd 17949  Cntzccntz 18439  CMndccmn 18900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-se 5510  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-isom 6359  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fsupp 8828  df-oi 8968  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12887  df-fzo 13028  df-seq 13364  df-hash 13685  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-0g 16709  df-gsum 16710  df-mre 16851  df-mrc 16852  df-acs 16854  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-submnd 17951  df-mulg 18219  df-cntz 18441  df-cmn 18902
This theorem is referenced by:  gsummpt1n0  19079  dprdfid  19133  evlslem3  20287  evlslem1  20289  coe1tmmul2  20438  coe1tmmul  20439  uvcresum  20931  frlmup2  20937  mamulid  21044  mamurid  21045  coe1mul3  24687  tayl0  24944  jensen  25560  linc1  44473
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