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Theorem gsumpt 19747
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 𝐡 = (Baseβ€˜πΊ)
gsumpt.z 0 = (0gβ€˜πΊ)
gsumpt.g (πœ‘ β†’ 𝐺 ∈ Mnd)
gsumpt.a (πœ‘ β†’ 𝐴 ∈ 𝑉)
gsumpt.x (πœ‘ β†’ 𝑋 ∈ 𝐴)
gsumpt.f (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
gsumpt.s (πœ‘ β†’ (𝐹 supp 0 ) βŠ† {𝑋})
Assertion
Ref Expression
gsumpt (πœ‘ β†’ (𝐺 Ξ£g 𝐹) = (πΉβ€˜π‘‹))

Proof of Theorem gsumpt
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 gsumpt.f . . . 4 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
2 gsumpt.x . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝐴)
32snssd 4773 . . . 4 (πœ‘ β†’ {𝑋} βŠ† 𝐴)
41, 3feqresmpt 6915 . . 3 (πœ‘ β†’ (𝐹 β†Ύ {𝑋}) = (π‘Ž ∈ {𝑋} ↦ (πΉβ€˜π‘Ž)))
54oveq2d 7377 . 2 (πœ‘ β†’ (𝐺 Ξ£g (𝐹 β†Ύ {𝑋})) = (𝐺 Ξ£g (π‘Ž ∈ {𝑋} ↦ (πΉβ€˜π‘Ž))))
6 gsumpt.b . . 3 𝐡 = (Baseβ€˜πΊ)
7 gsumpt.z . . 3 0 = (0gβ€˜πΊ)
8 eqid 2733 . . 3 (Cntzβ€˜πΊ) = (Cntzβ€˜πΊ)
9 gsumpt.g . . 3 (πœ‘ β†’ 𝐺 ∈ Mnd)
10 gsumpt.a . . 3 (πœ‘ β†’ 𝐴 ∈ 𝑉)
111, 2ffvelcdmd 7040 . . . . . . . 8 (πœ‘ β†’ (πΉβ€˜π‘‹) ∈ 𝐡)
12 eqidd 2734 . . . . . . . 8 (πœ‘ β†’ ((πΉβ€˜π‘‹)(+gβ€˜πΊ)(πΉβ€˜π‘‹)) = ((πΉβ€˜π‘‹)(+gβ€˜πΊ)(πΉβ€˜π‘‹)))
13 eqid 2733 . . . . . . . . . 10 (+gβ€˜πΊ) = (+gβ€˜πΊ)
146, 13, 8elcntzsn 19113 . . . . . . . . 9 ((πΉβ€˜π‘‹) ∈ 𝐡 β†’ ((πΉβ€˜π‘‹) ∈ ((Cntzβ€˜πΊ)β€˜{(πΉβ€˜π‘‹)}) ↔ ((πΉβ€˜π‘‹) ∈ 𝐡 ∧ ((πΉβ€˜π‘‹)(+gβ€˜πΊ)(πΉβ€˜π‘‹)) = ((πΉβ€˜π‘‹)(+gβ€˜πΊ)(πΉβ€˜π‘‹)))))
1511, 14syl 17 . . . . . . . 8 (πœ‘ β†’ ((πΉβ€˜π‘‹) ∈ ((Cntzβ€˜πΊ)β€˜{(πΉβ€˜π‘‹)}) ↔ ((πΉβ€˜π‘‹) ∈ 𝐡 ∧ ((πΉβ€˜π‘‹)(+gβ€˜πΊ)(πΉβ€˜π‘‹)) = ((πΉβ€˜π‘‹)(+gβ€˜πΊ)(πΉβ€˜π‘‹)))))
1611, 12, 15mpbir2and 712 . . . . . . 7 (πœ‘ β†’ (πΉβ€˜π‘‹) ∈ ((Cntzβ€˜πΊ)β€˜{(πΉβ€˜π‘‹)}))
1716snssd 4773 . . . . . 6 (πœ‘ β†’ {(πΉβ€˜π‘‹)} βŠ† ((Cntzβ€˜πΊ)β€˜{(πΉβ€˜π‘‹)}))
18 eqid 2733 . . . . . . 7 (mrClsβ€˜(SubMndβ€˜πΊ)) = (mrClsβ€˜(SubMndβ€˜πΊ))
19 eqid 2733 . . . . . . 7 (𝐺 β†Ύs ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)})) = (𝐺 β†Ύs ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}))
208, 18, 19cntzspan 19630 . . . . . 6 ((𝐺 ∈ Mnd ∧ {(πΉβ€˜π‘‹)} βŠ† ((Cntzβ€˜πΊ)β€˜{(πΉβ€˜π‘‹)})) β†’ (𝐺 β†Ύs ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)})) ∈ CMnd)
219, 17, 20syl2anc 585 . . . . 5 (πœ‘ β†’ (𝐺 β†Ύs ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)})) ∈ CMnd)
226submacs 18645 . . . . . . . 8 (𝐺 ∈ Mnd β†’ (SubMndβ€˜πΊ) ∈ (ACSβ€˜π΅))
23 acsmre 17540 . . . . . . . 8 ((SubMndβ€˜πΊ) ∈ (ACSβ€˜π΅) β†’ (SubMndβ€˜πΊ) ∈ (Mooreβ€˜π΅))
249, 22, 233syl 18 . . . . . . 7 (πœ‘ β†’ (SubMndβ€˜πΊ) ∈ (Mooreβ€˜π΅))
2511snssd 4773 . . . . . . 7 (πœ‘ β†’ {(πΉβ€˜π‘‹)} βŠ† 𝐡)
2618mrccl 17499 . . . . . . 7 (((SubMndβ€˜πΊ) ∈ (Mooreβ€˜π΅) ∧ {(πΉβ€˜π‘‹)} βŠ† 𝐡) β†’ ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}) ∈ (SubMndβ€˜πΊ))
2724, 25, 26syl2anc 585 . . . . . 6 (πœ‘ β†’ ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}) ∈ (SubMndβ€˜πΊ))
2819, 8submcmn2 19625 . . . . . 6 (((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}) ∈ (SubMndβ€˜πΊ) β†’ ((𝐺 β†Ύs ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)})) ∈ CMnd ↔ ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}) βŠ† ((Cntzβ€˜πΊ)β€˜((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}))))
2927, 28syl 17 . . . . 5 (πœ‘ β†’ ((𝐺 β†Ύs ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)})) ∈ CMnd ↔ ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}) βŠ† ((Cntzβ€˜πΊ)β€˜((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}))))
3021, 29mpbid 231 . . . 4 (πœ‘ β†’ ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}) βŠ† ((Cntzβ€˜πΊ)β€˜((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)})))
311ffnd 6673 . . . . . 6 (πœ‘ β†’ 𝐹 Fn 𝐴)
32 simpr 486 . . . . . . . . . 10 (((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘Ž = 𝑋) β†’ π‘Ž = 𝑋)
3332fveq2d 6850 . . . . . . . . 9 (((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘Ž = 𝑋) β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π‘‹))
3424, 18, 25mrcssidd 17513 . . . . . . . . . . 11 (πœ‘ β†’ {(πΉβ€˜π‘‹)} βŠ† ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}))
35 fvex 6859 . . . . . . . . . . . 12 (πΉβ€˜π‘‹) ∈ V
3635snss 4750 . . . . . . . . . . 11 ((πΉβ€˜π‘‹) ∈ ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}) ↔ {(πΉβ€˜π‘‹)} βŠ† ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}))
3734, 36sylibr 233 . . . . . . . . . 10 (πœ‘ β†’ (πΉβ€˜π‘‹) ∈ ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}))
3837ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘Ž = 𝑋) β†’ (πΉβ€˜π‘‹) ∈ ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}))
3933, 38eqeltrd 2834 . . . . . . . 8 (((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘Ž = 𝑋) β†’ (πΉβ€˜π‘Ž) ∈ ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}))
40 eldifsn 4751 . . . . . . . . . . 11 (π‘Ž ∈ (𝐴 βˆ– {𝑋}) ↔ (π‘Ž ∈ 𝐴 ∧ π‘Ž β‰  𝑋))
41 gsumpt.s . . . . . . . . . . . 12 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† {𝑋})
427fvexi 6860 . . . . . . . . . . . . 13 0 ∈ V
4342a1i 11 . . . . . . . . . . . 12 (πœ‘ β†’ 0 ∈ V)
441, 41, 10, 43suppssr 8131 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ (𝐴 βˆ– {𝑋})) β†’ (πΉβ€˜π‘Ž) = 0 )
4540, 44sylan2br 596 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐴 ∧ π‘Ž β‰  𝑋)) β†’ (πΉβ€˜π‘Ž) = 0 )
467subm0cl 18630 . . . . . . . . . . . 12 (((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}) ∈ (SubMndβ€˜πΊ) β†’ 0 ∈ ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}))
4727, 46syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 0 ∈ ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}))
4847adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐴 ∧ π‘Ž β‰  𝑋)) β†’ 0 ∈ ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}))
4945, 48eqeltrd 2834 . . . . . . . . 9 ((πœ‘ ∧ (π‘Ž ∈ 𝐴 ∧ π‘Ž β‰  𝑋)) β†’ (πΉβ€˜π‘Ž) ∈ ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}))
5049anassrs 469 . . . . . . . 8 (((πœ‘ ∧ π‘Ž ∈ 𝐴) ∧ π‘Ž β‰  𝑋) β†’ (πΉβ€˜π‘Ž) ∈ ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}))
5139, 50pm2.61dane 3029 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ (πΉβ€˜π‘Ž) ∈ ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}))
5251ralrimiva 3140 . . . . . 6 (πœ‘ β†’ βˆ€π‘Ž ∈ 𝐴 (πΉβ€˜π‘Ž) ∈ ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}))
53 ffnfv 7070 . . . . . 6 (𝐹:𝐴⟢((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}) ↔ (𝐹 Fn 𝐴 ∧ βˆ€π‘Ž ∈ 𝐴 (πΉβ€˜π‘Ž) ∈ ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)})))
5431, 52, 53sylanbrc 584 . . . . 5 (πœ‘ β†’ 𝐹:𝐴⟢((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}))
5554frnd 6680 . . . 4 (πœ‘ β†’ ran 𝐹 βŠ† ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}))
568cntzidss 19126 . . . 4 ((((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)}) βŠ† ((Cntzβ€˜πΊ)β€˜((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)})) ∧ ran 𝐹 βŠ† ((mrClsβ€˜(SubMndβ€˜πΊ))β€˜{(πΉβ€˜π‘‹)})) β†’ ran 𝐹 βŠ† ((Cntzβ€˜πΊ)β€˜ran 𝐹))
5730, 55, 56syl2anc 585 . . 3 (πœ‘ β†’ ran 𝐹 βŠ† ((Cntzβ€˜πΊ)β€˜ran 𝐹))
581ffund 6676 . . . 4 (πœ‘ β†’ Fun 𝐹)
59 snfi 8994 . . . . 5 {𝑋} ∈ Fin
60 ssfi 9123 . . . . 5 (({𝑋} ∈ Fin ∧ (𝐹 supp 0 ) βŠ† {𝑋}) β†’ (𝐹 supp 0 ) ∈ Fin)
6159, 41, 60sylancr 588 . . . 4 (πœ‘ β†’ (𝐹 supp 0 ) ∈ Fin)
621, 10fexd 7181 . . . . 5 (πœ‘ β†’ 𝐹 ∈ V)
63 isfsupp 9315 . . . . 5 ((𝐹 ∈ V ∧ 0 ∈ V) β†’ (𝐹 finSupp 0 ↔ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin)))
6462, 43, 63syl2anc 585 . . . 4 (πœ‘ β†’ (𝐹 finSupp 0 ↔ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin)))
6558, 61, 64mpbir2and 712 . . 3 (πœ‘ β†’ 𝐹 finSupp 0 )
666, 7, 8, 9, 10, 1, 57, 41, 65gsumzres 19694 . 2 (πœ‘ β†’ (𝐺 Ξ£g (𝐹 β†Ύ {𝑋})) = (𝐺 Ξ£g 𝐹))
67 fveq2 6846 . . . 4 (π‘Ž = 𝑋 β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π‘‹))
686, 67gsumsn 19739 . . 3 ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐴 ∧ (πΉβ€˜π‘‹) ∈ 𝐡) β†’ (𝐺 Ξ£g (π‘Ž ∈ {𝑋} ↦ (πΉβ€˜π‘Ž))) = (πΉβ€˜π‘‹))
699, 2, 11, 68syl3anc 1372 . 2 (πœ‘ β†’ (𝐺 Ξ£g (π‘Ž ∈ {𝑋} ↦ (πΉβ€˜π‘Ž))) = (πΉβ€˜π‘‹))
705, 66, 693eqtr3d 2781 1 (πœ‘ β†’ (𝐺 Ξ£g 𝐹) = (πΉβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  Vcvv 3447   βˆ– cdif 3911   βŠ† wss 3914  {csn 4590   class class class wbr 5109   ↦ cmpt 5192  ran crn 5638   β†Ύ cres 5639  Fun wfun 6494   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   supp csupp 8096  Fincfn 8889   finSupp cfsupp 9311  Basecbs 17091   β†Ύs cress 17120  +gcplusg 17141  0gc0g 17329   Ξ£g cgsu 17330  Moorecmre 17470  mrClscmrc 17471  ACScacs 17473  Mndcmnd 18564  SubMndcsubmnd 18608  Cntzccntz 19103  CMndccmn 19570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-supp 8097  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-fsupp 9312  df-oi 9454  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-fzo 13577  df-seq 13916  df-hash 14240  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-0g 17331  df-gsum 17332  df-mre 17474  df-mrc 17475  df-acs 17477  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-submnd 18610  df-mulg 18881  df-cntz 19105  df-cmn 19572
This theorem is referenced by:  gsummpt1n0  19750  dprdfid  19804  uvcresum  21222  frlmup2  21228  evlslem3  21513  evlslem1  21515  coe1tmmul2  21670  coe1tmmul  21671  mamulid  21813  mamurid  21814  coe1mul3  25487  tayl0  25744  jensen  26361  linc1  46596
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