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Theorem gsumpt 18184
 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 4281 . . . 4 (𝜑 → {𝑋} ⊆ 𝐴)
41, 3feqresmpt 6160 . . 3 (𝜑 → (𝐹 ↾ {𝑋}) = (𝑎 ∈ {𝑋} ↦ (𝐹𝑎)))
54oveq2d 6565 . 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 2610 . . 3 (Cntz‘𝐺) = (Cntz‘𝐺)
9 gsumpt.g . . 3 (𝜑𝐺 ∈ Mnd)
10 gsumpt.a . . 3 (𝜑𝐴𝑉)
111, 2ffvelrnd 6268 . . . . . . . 8 (𝜑 → (𝐹𝑋) ∈ 𝐵)
12 eqidd 2611 . . . . . . . 8 (𝜑 → ((𝐹𝑋)(+g𝐺)(𝐹𝑋)) = ((𝐹𝑋)(+g𝐺)(𝐹𝑋)))
13 eqid 2610 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
146, 13, 8elcntzsn 17581 . . . . . . . . 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 959 . . . . . . 7 (𝜑 → (𝐹𝑋) ∈ ((Cntz‘𝐺)‘{(𝐹𝑋)}))
1716snssd 4281 . . . . . 6 (𝜑 → {(𝐹𝑋)} ⊆ ((Cntz‘𝐺)‘{(𝐹𝑋)}))
18 eqid 2610 . . . . . . 7 (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
19 eqid 2610 . . . . . . 7 (𝐺s ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})) = (𝐺s ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
208, 18, 19cntzspan 18070 . . . . . 6 ((𝐺 ∈ Mnd ∧ {(𝐹𝑋)} ⊆ ((Cntz‘𝐺)‘{(𝐹𝑋)})) → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})) ∈ CMnd)
219, 17, 20syl2anc 691 . . . . 5 (𝜑 → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})) ∈ CMnd)
226submacs 17188 . . . . . . . 8 (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
23 acsmre 16136 . . . . . . . 8 ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
249, 22, 233syl 18 . . . . . . 7 (𝜑 → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
2511snssd 4281 . . . . . . 7 (𝜑 → {(𝐹𝑋)} ⊆ 𝐵)
2618mrccl 16094 . . . . . . 7 (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ {(𝐹𝑋)} ⊆ 𝐵) → ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ∈ (SubMnd‘𝐺))
2724, 25, 26syl2anc 691 . . . . . 6 (𝜑 → ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ∈ (SubMnd‘𝐺))
2819, 8submcmn2 18067 . . . . . 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 221 . . . 4 (𝜑 → ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ⊆ ((Cntz‘𝐺)‘((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})))
31 ffn 5958 . . . . . . 7 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
321, 31syl 17 . . . . . 6 (𝜑𝐹 Fn 𝐴)
33 simpr 476 . . . . . . . . . 10 (((𝜑𝑎𝐴) ∧ 𝑎 = 𝑋) → 𝑎 = 𝑋)
3433fveq2d 6107 . . . . . . . . 9 (((𝜑𝑎𝐴) ∧ 𝑎 = 𝑋) → (𝐹𝑎) = (𝐹𝑋))
3524, 18, 25mrcssidd 16108 . . . . . . . . . . 11 (𝜑 → {(𝐹𝑋)} ⊆ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
36 fvex 6113 . . . . . . . . . . . 12 (𝐹𝑋) ∈ V
3736snss 4259 . . . . . . . . . . 11 ((𝐹𝑋) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ↔ {(𝐹𝑋)} ⊆ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
3835, 37sylibr 223 . . . . . . . . . 10 (𝜑 → (𝐹𝑋) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
3938ad2antrr 758 . . . . . . . . 9 (((𝜑𝑎𝐴) ∧ 𝑎 = 𝑋) → (𝐹𝑋) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
4034, 39eqeltrd 2688 . . . . . . . 8 (((𝜑𝑎𝐴) ∧ 𝑎 = 𝑋) → (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
41 eldifsn 4260 . . . . . . . . . . 11 (𝑎 ∈ (𝐴 ∖ {𝑋}) ↔ (𝑎𝐴𝑎𝑋))
42 gsumpt.s . . . . . . . . . . . 12 (𝜑 → (𝐹 supp 0 ) ⊆ {𝑋})
43 fvex 6113 . . . . . . . . . . . . . 14 (0g𝐺) ∈ V
447, 43eqeltri 2684 . . . . . . . . . . . . 13 0 ∈ V
4544a1i 11 . . . . . . . . . . . 12 (𝜑0 ∈ V)
461, 42, 10, 45suppssr 7213 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (𝐴 ∖ {𝑋})) → (𝐹𝑎) = 0 )
4741, 46sylan2br 492 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑎𝑋)) → (𝐹𝑎) = 0 )
487subm0cl 17175 . . . . . . . . . . . 12 (((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ∈ (SubMnd‘𝐺) → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
4927, 48syl 17 . . . . . . . . . . 11 (𝜑0 ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
5049adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑎𝑋)) → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
5147, 50eqeltrd 2688 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝑎𝑋)) → (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
5251anassrs 678 . . . . . . . 8 (((𝜑𝑎𝐴) ∧ 𝑎𝑋) → (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
5340, 52pm2.61dane 2869 . . . . . . 7 ((𝜑𝑎𝐴) → (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
5453ralrimiva 2949 . . . . . 6 (𝜑 → ∀𝑎𝐴 (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
55 ffnfv 6295 . . . . . 6 (𝐹:𝐴⟶((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑎𝐴 (𝐹𝑎) ∈ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})))
5632, 54, 55sylanbrc 695 . . . . 5 (𝜑𝐹:𝐴⟶((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
57 frn 5966 . . . . 5 (𝐹:𝐴⟶((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
5856, 57syl 17 . . . 4 (𝜑 → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}))
598cntzidss 17593 . . . 4 ((((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)}) ⊆ ((Cntz‘𝐺)‘((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})) ∧ ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘{(𝐹𝑋)})) → ran 𝐹 ⊆ ((Cntz‘𝐺)‘ran 𝐹))
6030, 58, 59syl2anc 691 . . 3 (𝜑 → ran 𝐹 ⊆ ((Cntz‘𝐺)‘ran 𝐹))
61 ffun 5961 . . . . 5 (𝐹:𝐴𝐵 → Fun 𝐹)
621, 61syl 17 . . . 4 (𝜑 → Fun 𝐹)
63 snfi 7923 . . . . 5 {𝑋} ∈ Fin
64 ssfi 8065 . . . . 5 (({𝑋} ∈ Fin ∧ (𝐹 supp 0 ) ⊆ {𝑋}) → (𝐹 supp 0 ) ∈ Fin)
6563, 42, 64sylancr 694 . . . 4 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
66 fex 6394 . . . . . 6 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
671, 10, 66syl2anc 691 . . . . 5 (𝜑𝐹 ∈ V)
68 isfsupp 8162 . . . . 5 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 finSupp 0 ↔ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin)))
6967, 45, 68syl2anc 691 . . . 4 (𝜑 → (𝐹 finSupp 0 ↔ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin)))
7062, 65, 69mpbir2and 959 . . 3 (𝜑𝐹 finSupp 0 )
716, 7, 8, 9, 10, 1, 60, 42, 70gsumzres 18133 . 2 (𝜑 → (𝐺 Σg (𝐹 ↾ {𝑋})) = (𝐺 Σg 𝐹))
72 fveq2 6103 . . . 4 (𝑎 = 𝑋 → (𝐹𝑎) = (𝐹𝑋))
736, 72gsumsn 18177 . . 3 ((𝐺 ∈ Mnd ∧ 𝑋𝐴 ∧ (𝐹𝑋) ∈ 𝐵) → (𝐺 Σg (𝑎 ∈ {𝑋} ↦ (𝐹𝑎))) = (𝐹𝑋))
749, 2, 11, 73syl3anc 1318 . 2 (𝜑 → (𝐺 Σg (𝑎 ∈ {𝑋} ↦ (𝐹𝑎))) = (𝐹𝑋))
755, 71, 743eqtr3d 2652 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐹𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039   ↾ cres 5040  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   supp csupp 7182  Fincfn 7841   finSupp cfsupp 8158  Basecbs 15695   ↾s cress 15696  +gcplusg 15768  0gc0g 15923   Σg cgsu 15924  Moorecmre 16065  mrClscmrc 16066  ACScacs 16068  Mndcmnd 17117  SubMndcsubmnd 17157  Cntzccntz 17571  CMndccmn 18016 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018 This theorem is referenced by:  gsummpt1n0  18187  dprdfid  18239  evlslem3  19335  evlslem1  19336  coe1tmmul2  19467  coe1tmmul  19468  uvcresum  19951  frlmup2  19957  mamulid  20066  mamurid  20067  coe1mul3  23663  tayl0  23920  jensen  24515  linc1  42008
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