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Theorem mptscmfsupp0 18909
Description: A mapping to a scalar product is finitely supported if the mapping to the scalar is finitely supported. (Contributed by AV, 5-Oct-2019.)
Hypotheses
Ref Expression
mptscmfsupp0.d (𝜑𝐷𝑉)
mptscmfsupp0.q (𝜑𝑄 ∈ LMod)
mptscmfsupp0.r (𝜑𝑅 = (Scalar‘𝑄))
mptscmfsupp0.k 𝐾 = (Base‘𝑄)
mptscmfsupp0.s ((𝜑𝑘𝐷) → 𝑆𝐵)
mptscmfsupp0.w ((𝜑𝑘𝐷) → 𝑊𝐾)
mptscmfsupp0.0 0 = (0g𝑄)
mptscmfsupp0.z 𝑍 = (0g𝑅)
mptscmfsupp0.m = ( ·𝑠𝑄)
mptscmfsupp0.f (𝜑 → (𝑘𝐷𝑆) finSupp 𝑍)
Assertion
Ref Expression
mptscmfsupp0 (𝜑 → (𝑘𝐷 ↦ (𝑆 𝑊)) finSupp 0 )
Distinct variable groups:   𝐵,𝑘   𝐷,𝑘   𝑘,𝐾   𝜑,𝑘   ,𝑘
Allowed substitution hints:   𝑄(𝑘)   𝑅(𝑘)   𝑆(𝑘)   𝑉(𝑘)   𝑊(𝑘)   0 (𝑘)   𝑍(𝑘)

Proof of Theorem mptscmfsupp0
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 mptscmfsupp0.d . . 3 (𝜑𝐷𝑉)
2 mptexg 6469 . . 3 (𝐷𝑉 → (𝑘𝐷 ↦ (𝑆 𝑊)) ∈ V)
31, 2syl 17 . 2 (𝜑 → (𝑘𝐷 ↦ (𝑆 𝑊)) ∈ V)
4 funmpt 5914 . . 3 Fun (𝑘𝐷 ↦ (𝑆 𝑊))
54a1i 11 . 2 (𝜑 → Fun (𝑘𝐷 ↦ (𝑆 𝑊)))
6 mptscmfsupp0.0 . . . 4 0 = (0g𝑄)
7 fvex 6188 . . . 4 (0g𝑄) ∈ V
86, 7eqeltri 2695 . . 3 0 ∈ V
98a1i 11 . 2 (𝜑0 ∈ V)
10 mptscmfsupp0.f . . 3 (𝜑 → (𝑘𝐷𝑆) finSupp 𝑍)
1110fsuppimpd 8267 . 2 (𝜑 → ((𝑘𝐷𝑆) supp 𝑍) ∈ Fin)
12 simpr 477 . . . . . . . 8 ((𝜑𝑑𝐷) → 𝑑𝐷)
13 mptscmfsupp0.s . . . . . . . . . . 11 ((𝜑𝑘𝐷) → 𝑆𝐵)
1413ralrimiva 2963 . . . . . . . . . 10 (𝜑 → ∀𝑘𝐷 𝑆𝐵)
1514adantr 481 . . . . . . . . 9 ((𝜑𝑑𝐷) → ∀𝑘𝐷 𝑆𝐵)
16 rspcsbela 3997 . . . . . . . . 9 ((𝑑𝐷 ∧ ∀𝑘𝐷 𝑆𝐵) → 𝑑 / 𝑘𝑆𝐵)
1712, 15, 16syl2anc 692 . . . . . . . 8 ((𝜑𝑑𝐷) → 𝑑 / 𝑘𝑆𝐵)
18 eqid 2620 . . . . . . . . 9 (𝑘𝐷𝑆) = (𝑘𝐷𝑆)
1918fvmpts 6272 . . . . . . . 8 ((𝑑𝐷𝑑 / 𝑘𝑆𝐵) → ((𝑘𝐷𝑆)‘𝑑) = 𝑑 / 𝑘𝑆)
2012, 17, 19syl2anc 692 . . . . . . 7 ((𝜑𝑑𝐷) → ((𝑘𝐷𝑆)‘𝑑) = 𝑑 / 𝑘𝑆)
2120eqeq1d 2622 . . . . . 6 ((𝜑𝑑𝐷) → (((𝑘𝐷𝑆)‘𝑑) = 𝑍𝑑 / 𝑘𝑆 = 𝑍))
22 oveq1 6642 . . . . . . . . 9 (𝑑 / 𝑘𝑆 = 𝑍 → (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊) = (𝑍 𝑑 / 𝑘𝑊))
23 mptscmfsupp0.z . . . . . . . . . . . 12 𝑍 = (0g𝑅)
24 mptscmfsupp0.r . . . . . . . . . . . . . 14 (𝜑𝑅 = (Scalar‘𝑄))
2524adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑑𝐷) → 𝑅 = (Scalar‘𝑄))
2625fveq2d 6182 . . . . . . . . . . . 12 ((𝜑𝑑𝐷) → (0g𝑅) = (0g‘(Scalar‘𝑄)))
2723, 26syl5eq 2666 . . . . . . . . . . 11 ((𝜑𝑑𝐷) → 𝑍 = (0g‘(Scalar‘𝑄)))
2827oveq1d 6650 . . . . . . . . . 10 ((𝜑𝑑𝐷) → (𝑍 𝑑 / 𝑘𝑊) = ((0g‘(Scalar‘𝑄)) 𝑑 / 𝑘𝑊))
29 mptscmfsupp0.q . . . . . . . . . . . 12 (𝜑𝑄 ∈ LMod)
3029adantr 481 . . . . . . . . . . 11 ((𝜑𝑑𝐷) → 𝑄 ∈ LMod)
31 mptscmfsupp0.w . . . . . . . . . . . . . 14 ((𝜑𝑘𝐷) → 𝑊𝐾)
3231ralrimiva 2963 . . . . . . . . . . . . 13 (𝜑 → ∀𝑘𝐷 𝑊𝐾)
3332adantr 481 . . . . . . . . . . . 12 ((𝜑𝑑𝐷) → ∀𝑘𝐷 𝑊𝐾)
34 rspcsbela 3997 . . . . . . . . . . . 12 ((𝑑𝐷 ∧ ∀𝑘𝐷 𝑊𝐾) → 𝑑 / 𝑘𝑊𝐾)
3512, 33, 34syl2anc 692 . . . . . . . . . . 11 ((𝜑𝑑𝐷) → 𝑑 / 𝑘𝑊𝐾)
36 mptscmfsupp0.k . . . . . . . . . . . 12 𝐾 = (Base‘𝑄)
37 eqid 2620 . . . . . . . . . . . 12 (Scalar‘𝑄) = (Scalar‘𝑄)
38 mptscmfsupp0.m . . . . . . . . . . . 12 = ( ·𝑠𝑄)
39 eqid 2620 . . . . . . . . . . . 12 (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
4036, 37, 38, 39, 6lmod0vs 18877 . . . . . . . . . . 11 ((𝑄 ∈ LMod ∧ 𝑑 / 𝑘𝑊𝐾) → ((0g‘(Scalar‘𝑄)) 𝑑 / 𝑘𝑊) = 0 )
4130, 35, 40syl2anc 692 . . . . . . . . . 10 ((𝜑𝑑𝐷) → ((0g‘(Scalar‘𝑄)) 𝑑 / 𝑘𝑊) = 0 )
4228, 41eqtrd 2654 . . . . . . . . 9 ((𝜑𝑑𝐷) → (𝑍 𝑑 / 𝑘𝑊) = 0 )
4322, 42sylan9eqr 2676 . . . . . . . 8 (((𝜑𝑑𝐷) ∧ 𝑑 / 𝑘𝑆 = 𝑍) → (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊) = 0 )
44 csbov12g 6674 . . . . . . . . . . . . . 14 (𝑑𝐷𝑑 / 𝑘(𝑆 𝑊) = (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊))
4544adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑑𝐷) → 𝑑 / 𝑘(𝑆 𝑊) = (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊))
46 ovex 6663 . . . . . . . . . . . . 13 (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊) ∈ V
4745, 46syl6eqel 2707 . . . . . . . . . . . 12 ((𝜑𝑑𝐷) → 𝑑 / 𝑘(𝑆 𝑊) ∈ V)
48 eqid 2620 . . . . . . . . . . . . 13 (𝑘𝐷 ↦ (𝑆 𝑊)) = (𝑘𝐷 ↦ (𝑆 𝑊))
4948fvmpts 6272 . . . . . . . . . . . 12 ((𝑑𝐷𝑑 / 𝑘(𝑆 𝑊) ∈ V) → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 𝑑 / 𝑘(𝑆 𝑊))
5012, 47, 49syl2anc 692 . . . . . . . . . . 11 ((𝜑𝑑𝐷) → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 𝑑 / 𝑘(𝑆 𝑊))
5150, 45eqtrd 2654 . . . . . . . . . 10 ((𝜑𝑑𝐷) → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊))
5251eqeq1d 2622 . . . . . . . . 9 ((𝜑𝑑𝐷) → (((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 0 ↔ (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊) = 0 ))
5352adantr 481 . . . . . . . 8 (((𝜑𝑑𝐷) ∧ 𝑑 / 𝑘𝑆 = 𝑍) → (((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 0 ↔ (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊) = 0 ))
5443, 53mpbird 247 . . . . . . 7 (((𝜑𝑑𝐷) ∧ 𝑑 / 𝑘𝑆 = 𝑍) → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 0 )
5554ex 450 . . . . . 6 ((𝜑𝑑𝐷) → (𝑑 / 𝑘𝑆 = 𝑍 → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 0 ))
5621, 55sylbid 230 . . . . 5 ((𝜑𝑑𝐷) → (((𝑘𝐷𝑆)‘𝑑) = 𝑍 → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 0 ))
5756necon3d 2812 . . . 4 ((𝜑𝑑𝐷) → (((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) ≠ 0 → ((𝑘𝐷𝑆)‘𝑑) ≠ 𝑍))
5857ss2rabdv 3675 . . 3 (𝜑 → {𝑑𝐷 ∣ ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) ≠ 0 } ⊆ {𝑑𝐷 ∣ ((𝑘𝐷𝑆)‘𝑑) ≠ 𝑍})
59 ovex 6663 . . . . . 6 (𝑆 𝑊) ∈ V
6059rgenw 2921 . . . . 5 𝑘𝐷 (𝑆 𝑊) ∈ V
6148fnmpt 6007 . . . . 5 (∀𝑘𝐷 (𝑆 𝑊) ∈ V → (𝑘𝐷 ↦ (𝑆 𝑊)) Fn 𝐷)
6260, 61mp1i 13 . . . 4 (𝜑 → (𝑘𝐷 ↦ (𝑆 𝑊)) Fn 𝐷)
63 suppvalfn 7287 . . . 4 (((𝑘𝐷 ↦ (𝑆 𝑊)) Fn 𝐷𝐷𝑉0 ∈ V) → ((𝑘𝐷 ↦ (𝑆 𝑊)) supp 0 ) = {𝑑𝐷 ∣ ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) ≠ 0 })
6462, 1, 9, 63syl3anc 1324 . . 3 (𝜑 → ((𝑘𝐷 ↦ (𝑆 𝑊)) supp 0 ) = {𝑑𝐷 ∣ ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) ≠ 0 })
6518fnmpt 6007 . . . . 5 (∀𝑘𝐷 𝑆𝐵 → (𝑘𝐷𝑆) Fn 𝐷)
6614, 65syl 17 . . . 4 (𝜑 → (𝑘𝐷𝑆) Fn 𝐷)
67 fvex 6188 . . . . . 6 (0g𝑅) ∈ V
6823, 67eqeltri 2695 . . . . 5 𝑍 ∈ V
6968a1i 11 . . . 4 (𝜑𝑍 ∈ V)
70 suppvalfn 7287 . . . 4 (((𝑘𝐷𝑆) Fn 𝐷𝐷𝑉𝑍 ∈ V) → ((𝑘𝐷𝑆) supp 𝑍) = {𝑑𝐷 ∣ ((𝑘𝐷𝑆)‘𝑑) ≠ 𝑍})
7166, 1, 69, 70syl3anc 1324 . . 3 (𝜑 → ((𝑘𝐷𝑆) supp 𝑍) = {𝑑𝐷 ∣ ((𝑘𝐷𝑆)‘𝑑) ≠ 𝑍})
7258, 64, 713sstr4d 3640 . 2 (𝜑 → ((𝑘𝐷 ↦ (𝑆 𝑊)) supp 0 ) ⊆ ((𝑘𝐷𝑆) supp 𝑍))
73 suppssfifsupp 8275 . 2 ((((𝑘𝐷 ↦ (𝑆 𝑊)) ∈ V ∧ Fun (𝑘𝐷 ↦ (𝑆 𝑊)) ∧ 0 ∈ V) ∧ (((𝑘𝐷𝑆) supp 𝑍) ∈ Fin ∧ ((𝑘𝐷 ↦ (𝑆 𝑊)) supp 0 ) ⊆ ((𝑘𝐷𝑆) supp 𝑍))) → (𝑘𝐷 ↦ (𝑆 𝑊)) finSupp 0 )
743, 5, 9, 11, 72, 73syl32anc 1332 1 (𝜑 → (𝑘𝐷 ↦ (𝑆 𝑊)) finSupp 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wne 2791  wral 2909  {crab 2913  Vcvv 3195  csb 3526  wss 3567   class class class wbr 4644  cmpt 4720  Fun wfun 5870   Fn wfn 5871  cfv 5876  (class class class)co 6635   supp csupp 7280  Fincfn 7940   finSupp cfsupp 8260  Basecbs 15838  Scalarcsca 15925   ·𝑠 cvsca 15926  0gc0g 16081  LModclmod 18844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-supp 7281  df-er 7727  df-en 7941  df-fin 7944  df-fsupp 8261  df-0g 16083  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-grp 17406  df-ring 18530  df-lmod 18846
This theorem is referenced by:  mptscmfsuppd  18910  gsumsmonply1  19654  pm2mpcl  20583  mply1topmatcllem  20589  mp2pm2mplem5  20596  pm2mpghmlem2  20598  chcoeffeqlem  20671
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