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Theorem gsumpt 19980
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 4809 . . . 4 (𝜑 → {𝑋} ⊆ 𝐴)
41, 3feqresmpt 6978 . . 3 (𝜑 → (𝐹 ↾ {𝑋}) = (𝑎 ∈ {𝑋} ↦ (𝐹𝑎)))
54oveq2d 7447 . 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 2737 . . 3 (Cntz‘𝐺) = (Cntz‘𝐺)
9 gsumpt.g . . 3 (𝜑𝐺 ∈ Mnd)
10 gsumpt.a . . 3 (𝜑𝐴𝑉)
111, 2ffvelcdmd 7105 . . . . . . . 8 (𝜑 → (𝐹𝑋) ∈ 𝐵)
12 eqidd 2738 . . . . . . . 8 (𝜑 → ((𝐹𝑋)(+g𝐺)(𝐹𝑋)) = ((𝐹𝑋)(+g𝐺)(𝐹𝑋)))
13 eqid 2737 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
146, 13, 8elcntzsn 19343 . . . . . . . . 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 713 . . . . . . 7 (𝜑 → (𝐹𝑋) ∈ ((Cntz‘𝐺)‘{(𝐹𝑋)}))
1716snssd 4809 . . . . . 6 (𝜑 → {(𝐹𝑋)} ⊆ ((Cntz‘𝐺)‘{(𝐹𝑋)}))
18 eqid 2737 . . . . . . 7 (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
19 eqid 2737 . . . . . . 7 (𝐺s ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})) = (𝐺s ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
208, 18, 19cntzspan 19862 . . . . . 6 ((𝐺 ∈ Mnd ∧ {(𝐹𝑋)} ⊆ ((Cntz‘𝐺)‘{(𝐹𝑋)})) → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})) ∈ CMnd)
219, 17, 20syl2anc 584 . . . . 5 (𝜑 → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})) ∈ CMnd)
226submacs 18840 . . . . . . . 8 (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
23 acsmre 17695 . . . . . . . 8 ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
249, 22, 233syl 18 . . . . . . 7 (𝜑 → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
2511snssd 4809 . . . . . . 7 (𝜑 → {(𝐹𝑋)} ⊆ 𝐵)
2618mrccl 17654 . . . . . . 7 (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ {(𝐹𝑋)} ⊆ 𝐵) → ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ∈ (SubMnd‘𝐺))
2724, 25, 26syl2anc 584 . . . . . 6 (𝜑 → ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ∈ (SubMnd‘𝐺))
2819, 8submcmn2 19857 . . . . . 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 232 . . . 4 (𝜑 → ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ⊆ ((Cntz‘𝐺)‘((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})))
311ffnd 6737 . . . . . 6 (𝜑𝐹 Fn 𝐴)
32 simpr 484 . . . . . . . . . 10 (((𝜑𝑎𝐴) ∧ 𝑎 = 𝑋) → 𝑎 = 𝑋)
3332fveq2d 6910 . . . . . . . . 9 (((𝜑𝑎𝐴) ∧ 𝑎 = 𝑋) → (𝐹𝑎) = (𝐹𝑋))
3424, 18, 25mrcssidd 17668 . . . . . . . . . . 11 (𝜑 → {(𝐹𝑋)} ⊆ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
35 fvex 6919 . . . . . . . . . . . 12 (𝐹𝑋) ∈ V
3635snss 4785 . . . . . . . . . . 11 ((𝐹𝑋) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ↔ {(𝐹𝑋)} ⊆ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
3734, 36sylibr 234 . . . . . . . . . 10 (𝜑 → (𝐹𝑋) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
3837ad2antrr 726 . . . . . . . . 9 (((𝜑𝑎𝐴) ∧ 𝑎 = 𝑋) → (𝐹𝑋) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
3933, 38eqeltrd 2841 . . . . . . . 8 (((𝜑𝑎𝐴) ∧ 𝑎 = 𝑋) → (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
40 eldifsn 4786 . . . . . . . . . . 11 (𝑎 ∈ (𝐴 ∖ {𝑋}) ↔ (𝑎𝐴𝑎𝑋))
41 gsumpt.s . . . . . . . . . . . 12 (𝜑 → (𝐹 supp 0 ) ⊆ {𝑋})
427fvexi 6920 . . . . . . . . . . . . 13 0 ∈ V
4342a1i 11 . . . . . . . . . . . 12 (𝜑0 ∈ V)
441, 41, 10, 43suppssr 8220 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (𝐴 ∖ {𝑋})) → (𝐹𝑎) = 0 )
4540, 44sylan2br 595 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑎𝑋)) → (𝐹𝑎) = 0 )
467subm0cl 18824 . . . . . . . . . . . 12 (((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ∈ (SubMnd‘𝐺) → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
4727, 46syl 17 . . . . . . . . . . 11 (𝜑0 ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
4847adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑎𝑋)) → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
4945, 48eqeltrd 2841 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝑎𝑋)) → (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
5049anassrs 467 . . . . . . . 8 (((𝜑𝑎𝐴) ∧ 𝑎𝑋) → (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
5139, 50pm2.61dane 3029 . . . . . . 7 ((𝜑𝑎𝐴) → (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
5251ralrimiva 3146 . . . . . 6 (𝜑 → ∀𝑎𝐴 (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
53 ffnfv 7139 . . . . . 6 (𝐹:𝐴⟶((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑎𝐴 (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})))
5431, 52, 53sylanbrc 583 . . . . 5 (𝜑𝐹:𝐴⟶((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
5554frnd 6744 . . . 4 (𝜑 → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
568cntzidss 19358 . . . 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 584 . . 3 (𝜑 → ran 𝐹 ⊆ ((Cntz‘𝐺)‘ran 𝐹))
581ffund 6740 . . . 4 (𝜑 → Fun 𝐹)
59 snfi 9083 . . . . 5 {𝑋} ∈ Fin
60 ssfi 9213 . . . . 5 (({𝑋} ∈ Fin ∧ (𝐹 supp 0 ) ⊆ {𝑋}) → (𝐹 supp 0 ) ∈ Fin)
6159, 41, 60sylancr 587 . . . 4 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
621, 10fexd 7247 . . . . 5 (𝜑𝐹 ∈ V)
63 isfsupp 9405 . . . . 5 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 finSupp 0 ↔ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin)))
6462, 43, 63syl2anc 584 . . . 4 (𝜑 → (𝐹 finSupp 0 ↔ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin)))
6558, 61, 64mpbir2and 713 . . 3 (𝜑𝐹 finSupp 0 )
666, 7, 8, 9, 10, 1, 57, 41, 65gsumzres 19927 . 2 (𝜑 → (𝐺 Σg (𝐹 ↾ {𝑋})) = (𝐺 Σg 𝐹))
67 fveq2 6906 . . . 4 (𝑎 = 𝑋 → (𝐹𝑎) = (𝐹𝑋))
686, 67gsumsn 19972 . . 3 ((𝐺 ∈ Mnd ∧ 𝑋𝐴 ∧ (𝐹𝑋) ∈ 𝐵) → (𝐺 Σg (𝑎 ∈ {𝑋} ↦ (𝐹𝑎))) = (𝐹𝑋))
699, 2, 11, 68syl3anc 1373 . 2 (𝜑 → (𝐺 Σg (𝑎 ∈ {𝑋} ↦ (𝐹𝑎))) = (𝐹𝑋))
705, 66, 693eqtr3d 2785 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  Vcvv 3480  cdif 3948  wss 3951  {csn 4626   class class class wbr 5143  cmpt 5225  ran crn 5686  cres 5687  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431   supp csupp 8185  Fincfn 8985   finSupp cfsupp 9401  Basecbs 17247  s cress 17274  +gcplusg 17297  0gc0g 17484   Σg cgsu 17485  Moorecmre 17625  mrClscmrc 17626  ACScacs 17628  Mndcmnd 18747  SubMndcsubmnd 18795  Cntzccntz 19333  CMndccmn 19798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-gsum 17487  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800
This theorem is referenced by:  gsummpt1n0  19983  dprdfid  20037  uvcresum  21813  frlmup2  21819  evlslem3  22104  evlslem1  22106  coe1tmmul2  22279  coe1tmmul  22280  mamulid  22447  mamurid  22448  coe1mul3  26138  tayl0  26403  jensen  27032  linc1  48342
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