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Theorem gsum2d2lem 19072
Description: Lemma for gsum2d2 19073: 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 2820 . . . 4 (𝑗𝐴, 𝑘𝐶𝑋) = (𝑗𝐴, 𝑘𝐶𝑋)
21mpofun 7253 . . 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 3178 . . . . 5 (𝜑 → ∀𝑗𝐴𝑘𝐶 𝑋𝐵)
71fmpox 7743 . . . . 5 (∀𝑗𝐴𝑘𝐶 𝑋𝐵 ↔ (𝑗𝐴, 𝑘𝐶𝑋): 𝑗𝐴 ({𝑗} × 𝐶)⟶𝐵)
86, 7sylib 220 . . . 4 (𝜑 → (𝑗𝐴, 𝑘𝐶𝑋): 𝑗𝐴 ({𝑗} × 𝐶)⟶𝐵)
9 nfv 1915 . . . . . 6 𝑗𝜑
10 nfiu1 4929 . . . . . . . 8 𝑗 𝑗𝐴 ({𝑗} × 𝐶)
11 nfcv 2973 . . . . . . . 8 𝑗𝑈
1210, 11nfdif 4081 . . . . . . 7 𝑗( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)
1312nfcri 2967 . . . . . 6 𝑗 𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)
149, 13nfan 1900 . . . . 5 𝑗(𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
15 nfmpo1 7211 . . . . . . 7 𝑗(𝑗𝐴, 𝑘𝐶𝑋)
16 nfcv 2973 . . . . . . 7 𝑗𝑧
1715, 16nffv 6656 . . . . . 6 𝑗((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧)
1817nfeq1 2988 . . . . 5 𝑗((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0
19 relxp 5549 . . . . . . . 8 Rel ({𝑗} × 𝐶)
2019rgenw 3137 . . . . . . 7 𝑗𝐴 Rel ({𝑗} × 𝐶)
21 reliun 5665 . . . . . . 7 (Rel 𝑗𝐴 ({𝑗} × 𝐶) ↔ ∀𝑗𝐴 Rel ({𝑗} × 𝐶))
2220, 21mpbir 233 . . . . . 6 Rel 𝑗𝐴 ({𝑗} × 𝐶)
23 eldifi 4082 . . . . . . 7 (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) → 𝑧 𝑗𝐴 ({𝑗} × 𝐶))
2423adantl 484 . . . . . 6 ((𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → 𝑧 𝑗𝐴 ({𝑗} × 𝐶))
25 elrel 5647 . . . . . 6 ((Rel 𝑗𝐴 ({𝑗} × 𝐶) ∧ 𝑧 𝑗𝐴 ({𝑗} × 𝐶)) → ∃𝑗𝑘 𝑧 = ⟨𝑗, 𝑘⟩)
2622, 24, 25sylancr 589 . . . . 5 ((𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → ∃𝑗𝑘 𝑧 = ⟨𝑗, 𝑘⟩)
27 nfv 1915 . . . . . 6 𝑘(𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
28 nfmpo2 7212 . . . . . . . 8 𝑘(𝑗𝐴, 𝑘𝐶𝑋)
29 nfcv 2973 . . . . . . . 8 𝑘𝑧
3028, 29nffv 6656 . . . . . . 7 𝑘((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧)
3130nfeq1 2988 . . . . . 6 𝑘((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0
32 simprr 771 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑧 = ⟨𝑗, 𝑘⟩)
3332fveq2d 6650 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = ((𝑗𝐴, 𝑘𝐶𝑋)‘⟨𝑗, 𝑘⟩))
34 df-ov 7136 . . . . . . . . 9 (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘) = ((𝑗𝐴, 𝑘𝐶𝑋)‘⟨𝑗, 𝑘⟩)
35 simprl 769 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
3632, 35eqeltrrd 2912 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ⟨𝑗, 𝑘⟩ ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
3736eldifad 3925 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ⟨𝑗, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶))
38 opeliunxp 5595 . . . . . . . . . . . 12 (⟨𝑗, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ (𝑗𝐴𝑘𝐶))
3937, 38sylib 220 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑗𝐴𝑘𝐶))
4039simpld 497 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑗𝐴)
4139simprd 498 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑘𝐶)
4239, 5syldan 593 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑋𝐵)
431ovmpt4g 7274 . . . . . . . . . 10 ((𝑗𝐴𝑘𝐶𝑋𝐵) → (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘) = 𝑋)
4440, 41, 42, 43syl3anc 1367 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘) = 𝑋)
4534, 44syl5eqr 2869 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗𝐴, 𝑘𝐶𝑋)‘⟨𝑗, 𝑘⟩) = 𝑋)
46 eldifn 4083 . . . . . . . . . . . 12 (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) → ¬ 𝑧𝑈)
4746ad2antrl 726 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ¬ 𝑧𝑈)
4832eleq1d 2895 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑧𝑈 ↔ ⟨𝑗, 𝑘⟩ ∈ 𝑈))
49 df-br 5043 . . . . . . . . . . . 12 (𝑗𝑈𝑘 ↔ ⟨𝑗, 𝑘⟩ ∈ 𝑈)
5048, 49syl6bbr 291 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑧𝑈𝑗𝑈𝑘))
5147, 50mtbid 326 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ¬ 𝑗𝑈𝑘)
5239, 51jca 514 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘))
53 gsum2d2.n . . . . . . . . 9 ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )
5452, 53syldan 593 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑋 = 0 )
5533, 45, 543eqtrd 2859 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0 )
5655expr 459 . . . . . 6 ((𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → (𝑧 = ⟨𝑗, 𝑘⟩ → ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0 ))
5727, 31, 56exlimd 2218 . . . . 5 ((𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → (∃𝑘 𝑧 = ⟨𝑗, 𝑘⟩ → ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0 ))
5814, 18, 26, 57exlimimdd 2219 . . . 4 ((𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0 )
598, 58suppss 7838 . . 3 (𝜑 → ((𝑗𝐴, 𝑘𝐶𝑋) supp 0 ) ⊆ 𝑈)
604, 59ssfid 8719 . 2 (𝜑 → ((𝑗𝐴, 𝑘𝐶𝑋) supp 0 ) ∈ Fin)
61 gsum2d2.a . . . 4 (𝜑𝐴𝑉)
62 gsum2d2.r . . . . 5 ((𝜑𝑗𝐴) → 𝐶𝑊)
6362ralrimiva 3169 . . . 4 (𝜑 → ∀𝑗𝐴 𝐶𝑊)
641mpoexxg 7751 . . . 4 ((𝐴𝑉 ∧ ∀𝑗𝐴 𝐶𝑊) → (𝑗𝐴, 𝑘𝐶𝑋) ∈ V)
6561, 63, 64syl2anc 586 . . 3 (𝜑 → (𝑗𝐴, 𝑘𝐶𝑋) ∈ V)
66 gsum2d2.z . . . . 5 0 = (0g𝐺)
6766fvexi 6660 . . . 4 0 ∈ V
6867a1i 11 . . 3 (𝜑0 ∈ V)
69 isfsupp 8815 . . 3 (((𝑗𝐴, 𝑘𝐶𝑋) ∈ V ∧ 0 ∈ V) → ((𝑗𝐴, 𝑘𝐶𝑋) finSupp 0 ↔ (Fun (𝑗𝐴, 𝑘𝐶𝑋) ∧ ((𝑗𝐴, 𝑘𝐶𝑋) supp 0 ) ∈ Fin)))
7065, 68, 69syl2anc 586 . 2 (𝜑 → ((𝑗𝐴, 𝑘𝐶𝑋) finSupp 0 ↔ (Fun (𝑗𝐴, 𝑘𝐶𝑋) ∧ ((𝑗𝐴, 𝑘𝐶𝑋) supp 0 ) ∈ Fin)))
713, 60, 70mpbir2and 711 1 (𝜑 → (𝑗𝐴, 𝑘𝐶𝑋) finSupp 0 )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1537  wex 1780  wcel 2114  wral 3125  Vcvv 3473  cdif 3910  {csn 4543  cop 4549   ciun 4895   class class class wbr 5042   × cxp 5529  Rel wrel 5536  Fun wfun 6325  wf 6327  cfv 6331  (class class class)co 7133  cmpo 7135   supp csupp 7808  Fincfn 8487   finSupp cfsupp 8811  Basecbs 16462  0gc0g 16692  CMndccmn 18885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-supp 7809  df-er 8267  df-en 8488  df-fin 8491  df-fsupp 8812
This theorem is referenced by:  gsum2d2  19073  gsumcom2  19074
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