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Theorem gsum2d2lem 18296
 Description: Lemma for gsum2d2 18297: 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 2621 . . . 4 (𝑗𝐴, 𝑘𝐶𝑋) = (𝑗𝐴, 𝑘𝐶𝑋)
21mpt2fun 6718 . . 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 2965 . . . . 5 (𝜑 → ∀𝑗𝐴𝑘𝐶 𝑋𝐵)
71fmpt2x 7184 . . . . 5 (∀𝑗𝐴𝑘𝐶 𝑋𝐵 ↔ (𝑗𝐴, 𝑘𝐶𝑋): 𝑗𝐴 ({𝑗} × 𝐶)⟶𝐵)
86, 7sylib 208 . . . 4 (𝜑 → (𝑗𝐴, 𝑘𝐶𝑋): 𝑗𝐴 ({𝑗} × 𝐶)⟶𝐵)
9 relxp 5190 . . . . . . . 8 Rel ({𝑗} × 𝐶)
109rgenw 2919 . . . . . . 7 𝑗𝐴 Rel ({𝑗} × 𝐶)
11 reliun 5202 . . . . . . 7 (Rel 𝑗𝐴 ({𝑗} × 𝐶) ↔ ∀𝑗𝐴 Rel ({𝑗} × 𝐶))
1210, 11mpbir 221 . . . . . 6 Rel 𝑗𝐴 ({𝑗} × 𝐶)
13 eldifi 3712 . . . . . . 7 (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) → 𝑧 𝑗𝐴 ({𝑗} × 𝐶))
1413adantl 482 . . . . . 6 ((𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → 𝑧 𝑗𝐴 ({𝑗} × 𝐶))
15 elrel 5185 . . . . . 6 ((Rel 𝑗𝐴 ({𝑗} × 𝐶) ∧ 𝑧 𝑗𝐴 ({𝑗} × 𝐶)) → ∃𝑗𝑘 𝑧 = ⟨𝑗, 𝑘⟩)
1612, 14, 15sylancr 694 . . . . 5 ((𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → ∃𝑗𝑘 𝑧 = ⟨𝑗, 𝑘⟩)
17 nfv 1840 . . . . . . 7 𝑗𝜑
18 nfiu1 4518 . . . . . . . . 9 𝑗 𝑗𝐴 ({𝑗} × 𝐶)
19 nfcv 2761 . . . . . . . . 9 𝑗𝑈
2018, 19nfdif 3711 . . . . . . . 8 𝑗( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)
2120nfcri 2755 . . . . . . 7 𝑗 𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)
2217, 21nfan 1825 . . . . . 6 𝑗(𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
23 nfmpt21 6678 . . . . . . . 8 𝑗(𝑗𝐴, 𝑘𝐶𝑋)
24 nfcv 2761 . . . . . . . 8 𝑗𝑧
2523, 24nffv 6157 . . . . . . 7 𝑗((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧)
2625nfeq1 2774 . . . . . 6 𝑗((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0
27 nfv 1840 . . . . . . 7 𝑘(𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
28 nfmpt22 6679 . . . . . . . . 9 𝑘(𝑗𝐴, 𝑘𝐶𝑋)
29 nfcv 2761 . . . . . . . . 9 𝑘𝑧
3028, 29nffv 6157 . . . . . . . 8 𝑘((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧)
3130nfeq1 2774 . . . . . . 7 𝑘((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0
32 simprr 795 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑧 = ⟨𝑗, 𝑘⟩)
3332fveq2d 6154 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = ((𝑗𝐴, 𝑘𝐶𝑋)‘⟨𝑗, 𝑘⟩))
34 df-ov 6610 . . . . . . . . . 10 (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘) = ((𝑗𝐴, 𝑘𝐶𝑋)‘⟨𝑗, 𝑘⟩)
35 simprl 793 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
3632, 35eqeltrrd 2699 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ⟨𝑗, 𝑘⟩ ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
3736eldifad 3568 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ⟨𝑗, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶))
38 opeliunxp 5133 . . . . . . . . . . . . 13 (⟨𝑗, 𝑘⟩ ∈ 𝑗𝐴 ({𝑗} × 𝐶) ↔ (𝑗𝐴𝑘𝐶))
3937, 38sylib 208 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑗𝐴𝑘𝐶))
4039simpld 475 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑗𝐴)
4139simprd 479 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑘𝐶)
4239, 5syldan 487 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑋𝐵)
431ovmpt4g 6739 . . . . . . . . . . 11 ((𝑗𝐴𝑘𝐶𝑋𝐵) → (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘) = 𝑋)
4440, 41, 42, 43syl3anc 1323 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑗(𝑗𝐴, 𝑘𝐶𝑋)𝑘) = 𝑋)
4534, 44syl5eqr 2669 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗𝐴, 𝑘𝐶𝑋)‘⟨𝑗, 𝑘⟩) = 𝑋)
46 eldifn 3713 . . . . . . . . . . . . 13 (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) → ¬ 𝑧𝑈)
4746ad2antrl 763 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ¬ 𝑧𝑈)
4832eleq1d 2683 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑧𝑈 ↔ ⟨𝑗, 𝑘⟩ ∈ 𝑈))
49 df-br 4616 . . . . . . . . . . . . 13 (𝑗𝑈𝑘 ↔ ⟨𝑗, 𝑘⟩ ∈ 𝑈)
5048, 49syl6bbr 278 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑧𝑈𝑗𝑈𝑘))
5147, 50mtbid 314 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ¬ 𝑗𝑈𝑘)
5239, 51jca 554 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘))
53 gsum2d2.n . . . . . . . . . 10 ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )
5452, 53syldan 487 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑋 = 0 )
5533, 45, 543eqtrd 2659 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0 )
5655expr 642 . . . . . . 7 ((𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → (𝑧 = ⟨𝑗, 𝑘⟩ → ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0 ))
5727, 31, 56exlimd 2085 . . . . . 6 ((𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → (∃𝑘 𝑧 = ⟨𝑗, 𝑘⟩ → ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0 ))
5822, 26, 57exlimd 2085 . . . . 5 ((𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → (∃𝑗𝑘 𝑧 = ⟨𝑗, 𝑘⟩ → ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0 ))
5916, 58mpd 15 . . . 4 ((𝜑𝑧 ∈ ( 𝑗𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → ((𝑗𝐴, 𝑘𝐶𝑋)‘𝑧) = 0 )
608, 59suppss 7273 . . 3 (𝜑 → ((𝑗𝐴, 𝑘𝐶𝑋) supp 0 ) ⊆ 𝑈)
61 ssfi 8127 . . 3 ((𝑈 ∈ Fin ∧ ((𝑗𝐴, 𝑘𝐶𝑋) supp 0 ) ⊆ 𝑈) → ((𝑗𝐴, 𝑘𝐶𝑋) supp 0 ) ∈ Fin)
624, 60, 61syl2anc 692 . 2 (𝜑 → ((𝑗𝐴, 𝑘𝐶𝑋) supp 0 ) ∈ Fin)
63 gsum2d2.a . . . 4 (𝜑𝐴𝑉)
64 gsum2d2.r . . . . 5 ((𝜑𝑗𝐴) → 𝐶𝑊)
6564ralrimiva 2960 . . . 4 (𝜑 → ∀𝑗𝐴 𝐶𝑊)
661mpt2exxg 7192 . . . 4 ((𝐴𝑉 ∧ ∀𝑗𝐴 𝐶𝑊) → (𝑗𝐴, 𝑘𝐶𝑋) ∈ V)
6763, 65, 66syl2anc 692 . . 3 (𝜑 → (𝑗𝐴, 𝑘𝐶𝑋) ∈ V)
68 gsum2d2.z . . . . 5 0 = (0g𝐺)
69 fvex 6160 . . . . 5 (0g𝐺) ∈ V
7068, 69eqeltri 2694 . . . 4 0 ∈ V
7170a1i 11 . . 3 (𝜑0 ∈ V)
72 isfsupp 8226 . . 3 (((𝑗𝐴, 𝑘𝐶𝑋) ∈ V ∧ 0 ∈ V) → ((𝑗𝐴, 𝑘𝐶𝑋) finSupp 0 ↔ (Fun (𝑗𝐴, 𝑘𝐶𝑋) ∧ ((𝑗𝐴, 𝑘𝐶𝑋) supp 0 ) ∈ Fin)))
7367, 71, 72syl2anc 692 . 2 (𝜑 → ((𝑗𝐴, 𝑘𝐶𝑋) finSupp 0 ↔ (Fun (𝑗𝐴, 𝑘𝐶𝑋) ∧ ((𝑗𝐴, 𝑘𝐶𝑋) supp 0 ) ∈ Fin)))
743, 62, 73mpbir2and 956 1 (𝜑 → (𝑗𝐴, 𝑘𝐶𝑋) finSupp 0 )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480  ∃wex 1701   ∈ wcel 1987  ∀wral 2907  Vcvv 3186   ∖ cdif 3553   ⊆ wss 3556  {csn 4150  ⟨cop 4156  ∪ ciun 4487   class class class wbr 4615   × cxp 5074  Rel wrel 5081  Fun wfun 5843  ⟶wf 5845  ‘cfv 5849  (class class class)co 6607   ↦ cmpt2 6609   supp csupp 7243  Fincfn 7902   finSupp cfsupp 8222  Basecbs 15784  0gc0g 16024  CMndccmn 18117 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-supp 7244  df-er 7690  df-en 7903  df-fin 7906  df-fsupp 8223 This theorem is referenced by:  gsum2d2  18297  gsumcom2  18298
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