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Theorem gsum2d2lem 19205
Description: Lemma for gsum2d2 19206: show the function is finitely supported. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 9-Jun-2019.)
Hypotheses
Ref Expression
gsum2d2.b 𝐵 = (Base‘𝐺)
gsum2d2.z 0 = (0g𝐺)
gsum2d2.g (𝜑𝐺 ∈ CMnd)
gsum2d2.a (𝜑𝐴𝑉)
gsum2d2.r ((𝜑𝑗𝐴) → 𝐶𝑊)
gsum2d2.f ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)
gsum2d2.u (𝜑𝑈 ∈ Fin)
gsum2d2.n ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )
Assertion
Ref Expression
gsum2d2lem (𝜑 → (𝑗𝐴, 𝑘𝐶𝑋) finSupp 0 )
Distinct variable groups:   𝑗,𝑘,𝐵   𝜑,𝑗,𝑘   𝐴,𝑗,𝑘   𝑗,𝐺,𝑘   𝑈,𝑗,𝑘   𝐶,𝑘   𝑗,𝑉   0 ,𝑗,𝑘
Allowed substitution hints:   𝐶(𝑗)   𝑉(𝑘)   𝑊(𝑗,𝑘)   𝑋(𝑗,𝑘)

Proof of Theorem gsum2d2lem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqid 2738 . . . 4 (𝑗𝐴, 𝑘𝐶𝑋) = (𝑗𝐴, 𝑘𝐶𝑋)
21mpofun 7284 . . 3 Fun (𝑗𝐴, 𝑘𝐶𝑋)
32a1i 11 . 2 (𝜑 → Fun (𝑗𝐴, 𝑘𝐶𝑋))
4 gsum2d2.u . . 3 (𝜑𝑈 ∈ Fin)
5 gsum2d2.f . . . . . 6 ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)
65ralrimivva 3103 . . . . 5 (𝜑 → ∀𝑗𝐴𝑘𝐶 𝑋𝐵)
71fmpox 7783 . . . . 5 (∀𝑗𝐴𝑘𝐶 𝑋𝐵 ↔ (𝑗𝐴, 𝑘𝐶𝑋): 𝑗𝐴 ({𝑗} × 𝐶)⟶𝐵)
86, 7sylib 221 . . . 4 (𝜑 → (𝑗𝐴, 𝑘𝐶𝑋): 𝑗𝐴 ({𝑗} × 𝐶)⟶𝐵)
9 nfv 1920 . . . . . 6 𝑗𝜑
10 nfiu1 4912 . . . . . . . 8 𝑗 𝑗𝐴 ({𝑗} × 𝐶)
11 nfcv 2899 . . . . . . . 8 𝑗𝑈
1210, 11nfdif 4014 . . . . . . 7 𝑗( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)
1312nfcri 2886 . . . . . 6 𝑗 𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)
149, 13nfan 1905 . . . . 5 𝑗(𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
15 nfmpo1 7242 . . . . . . 7 𝑗(𝑗𝐴, 𝑘𝐶𝑋)
16 nfcv 2899 . . . . . . 7 𝑗𝑧
1715, 16nffv 6678 . . . . . 6 𝑗((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧)
1817nfeq1 2914 . . . . 5 𝑗((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0
19 relxp 5537 . . . . . . . 8 Rel ({𝑗} × 𝐶)
2019rgenw 3065 . . . . . . 7 𝑗𝐴 Rel ({𝑗} × 𝐶)
21 reliun 5654 . . . . . . 7 (Rel 𝑗𝐴 ({𝑗} × 𝐶) ↔ ∀𝑗𝐴 Rel ({𝑗} × 𝐶))
2220, 21mpbir 234 . . . . . 6 Rel 𝑗𝐴 ({𝑗} × 𝐶)
23 eldifi 4015 . . . . . . 7 (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) → 𝑧 𝑗𝐴 ({𝑗} × 𝐶))
2423adantl 485 . . . . . 6 ((𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → 𝑧 𝑗𝐴 ({𝑗} × 𝐶))
25 elrel 5636 . . . . . 6 ((Rel 𝑗𝐴 ({𝑗} × 𝐶) ∧ 𝑧 𝑗𝐴 ({𝑗} × 𝐶)) → ∃𝑗𝑘 𝑧 = ⟨𝑗, 𝑘⟩)
2622, 24, 25sylancr 590 . . . . 5 ((𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → ∃𝑗𝑘 𝑧 = ⟨𝑗, 𝑘⟩)
27 nfv 1920 . . . . . 6 𝑘(𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
28 nfmpo2 7243 . . . . . . . 8 𝑘(𝑗𝐴, 𝑘𝐶𝑋)
29 nfcv 2899 . . . . . . . 8 𝑘𝑧
3028, 29nffv 6678 . . . . . . 7 𝑘((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧)
3130nfeq1 2914 . . . . . 6 𝑘((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0
32 simprr 773 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑧 = ⟨𝑗, 𝑘⟩)
3332fveq2d 6672 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = ((𝑗𝐴, 𝑘𝐶𝑋)‘⟨𝑗, 𝑘⟩))
34 df-ov 7167 . . . . . . . . 9 (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘) = ((𝑗𝐴, 𝑘𝐶𝑋)‘⟨𝑗, 𝑘⟩)
35 simprl 771 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
3632, 35eqeltrrd 2834 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ⟨𝑗, 𝑘⟩ ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
3736eldifad 3853 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ⟨𝑗, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶))
38 opeliunxp 5584 . . . . . . . . . . . 12 (⟨𝑗, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ (𝑗𝐴𝑘𝐶))
3937, 38sylib 221 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑗𝐴𝑘𝐶))
4039simpld 498 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑗𝐴)
4139simprd 499 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑘𝐶)
4239, 5syldan 594 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑋𝐵)
431ovmpt4g 7306 . . . . . . . . . 10 ((𝑗𝐴𝑘𝐶𝑋𝐵) → (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘) = 𝑋)
4440, 41, 42, 43syl3anc 1372 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘) = 𝑋)
4534, 44eqtr3id 2787 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗𝐴, 𝑘𝐶𝑋)‘⟨𝑗, 𝑘⟩) = 𝑋)
46 eldifn 4016 . . . . . . . . . . . 12 (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) → ¬ 𝑧𝑈)
4746ad2antrl 728 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ¬ 𝑧𝑈)
4832eleq1d 2817 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑧𝑈 ↔ ⟨𝑗, 𝑘⟩ ∈ 𝑈))
49 df-br 5028 . . . . . . . . . . . 12 (𝑗𝑈𝑘 ↔ ⟨𝑗, 𝑘⟩ ∈ 𝑈)
5048, 49bitr4di 292 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑧𝑈𝑗𝑈𝑘))
5147, 50mtbid 327 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ¬ 𝑗𝑈𝑘)
5239, 51jca 515 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘))
53 gsum2d2.n . . . . . . . . 9 ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )
5452, 53syldan 594 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑋 = 0 )
5533, 45, 543eqtrd 2777 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0 )
5655expr 460 . . . . . 6 ((𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → (𝑧 = ⟨𝑗, 𝑘⟩ → ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0 ))
5727, 31, 56exlimd 2219 . . . . 5 ((𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → (∃𝑘 𝑧 = ⟨𝑗, 𝑘⟩ → ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0 ))
5814, 18, 26, 57exlimimdd 2220 . . . 4 ((𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0 )
598, 58suppss 7882 . . 3 (𝜑 → ((𝑗𝐴, 𝑘𝐶𝑋) supp 0 ) ⊆ 𝑈)
604, 59ssfid 8812 . 2 (𝜑 → ((𝑗𝐴, 𝑘𝐶𝑋) supp 0 ) ∈ Fin)
61 gsum2d2.a . . . 4 (𝜑𝐴𝑉)
62 gsum2d2.r . . . . 5 ((𝜑𝑗𝐴) → 𝐶𝑊)
6362ralrimiva 3096 . . . 4 (𝜑 → ∀𝑗𝐴 𝐶𝑊)
641mpoexxg 7792 . . . 4 ((𝐴𝑉 ∧ ∀𝑗𝐴 𝐶𝑊) → (𝑗𝐴, 𝑘𝐶𝑋) ∈ V)
6561, 63, 64syl2anc 587 . . 3 (𝜑 → (𝑗𝐴, 𝑘𝐶𝑋) ∈ V)
66 gsum2d2.z . . . . 5 0 = (0g𝐺)
6766fvexi 6682 . . . 4 0 ∈ V
6867a1i 11 . . 3 (𝜑0 ∈ V)
69 isfsupp 8903 . . 3 (((𝑗𝐴, 𝑘𝐶𝑋) ∈ V ∧ 0 ∈ V) → ((𝑗𝐴, 𝑘𝐶𝑋) finSupp 0 ↔ (Fun (𝑗𝐴, 𝑘𝐶𝑋) ∧ ((𝑗𝐴, 𝑘𝐶𝑋) supp 0 ) ∈ Fin)))
7065, 68, 69syl2anc 587 . 2 (𝜑 → ((𝑗𝐴, 𝑘𝐶𝑋) finSupp 0 ↔ (Fun (𝑗𝐴, 𝑘𝐶𝑋) ∧ ((𝑗𝐴, 𝑘𝐶𝑋) supp 0 ) ∈ Fin)))
713, 60, 70mpbir2and 713 1 (𝜑 → (𝑗𝐴, 𝑘𝐶𝑋) finSupp 0 )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1542  wex 1786  wcel 2113  wral 3053  Vcvv 3397  cdif 3838  {csn 4513  cop 4519   ciun 4878   class class class wbr 5027   × cxp 5517  Rel wrel 5524  Fun wfun 6327  wf 6329  cfv 6333  (class class class)co 7164  cmpo 7166   supp csupp 7849  Fincfn 8548   finSupp cfsupp 8899  Basecbs 16579  0gc0g 16809  CMndccmn 19017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-supp 7850  df-1o 8124  df-en 8549  df-fin 8552  df-fsupp 8900
This theorem is referenced by:  gsum2d2  19206  gsumcom2  19207
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