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Theorem mptscmfsupp0 20925
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)
21mptexd 7244 . 2 (𝜑 → (𝑘𝐷 ↦ (𝑆 𝑊)) ∈ V)
3 funmpt 6604 . . 3 Fun (𝑘𝐷 ↦ (𝑆 𝑊))
43a1i 11 . 2 (𝜑 → Fun (𝑘𝐷 ↦ (𝑆 𝑊)))
5 mptscmfsupp0.0 . . . 4 0 = (0g𝑄)
65fvexi 6920 . . 3 0 ∈ V
76a1i 11 . 2 (𝜑0 ∈ V)
8 mptscmfsupp0.f . . 3 (𝜑 → (𝑘𝐷𝑆) finSupp 𝑍)
98fsuppimpd 9409 . 2 (𝜑 → ((𝑘𝐷𝑆) supp 𝑍) ∈ Fin)
10 simpr 484 . . . . . . . 8 ((𝜑𝑑𝐷) → 𝑑𝐷)
11 mptscmfsupp0.s . . . . . . . . . . 11 ((𝜑𝑘𝐷) → 𝑆𝐵)
1211ralrimiva 3146 . . . . . . . . . 10 (𝜑 → ∀𝑘𝐷 𝑆𝐵)
1312adantr 480 . . . . . . . . 9 ((𝜑𝑑𝐷) → ∀𝑘𝐷 𝑆𝐵)
14 rspcsbela 4438 . . . . . . . . 9 ((𝑑𝐷 ∧ ∀𝑘𝐷 𝑆𝐵) → 𝑑 / 𝑘𝑆𝐵)
1510, 13, 14syl2anc 584 . . . . . . . 8 ((𝜑𝑑𝐷) → 𝑑 / 𝑘𝑆𝐵)
16 eqid 2737 . . . . . . . . 9 (𝑘𝐷𝑆) = (𝑘𝐷𝑆)
1716fvmpts 7019 . . . . . . . 8 ((𝑑𝐷𝑑 / 𝑘𝑆𝐵) → ((𝑘𝐷𝑆)‘𝑑) = 𝑑 / 𝑘𝑆)
1810, 15, 17syl2anc 584 . . . . . . 7 ((𝜑𝑑𝐷) → ((𝑘𝐷𝑆)‘𝑑) = 𝑑 / 𝑘𝑆)
1918eqeq1d 2739 . . . . . 6 ((𝜑𝑑𝐷) → (((𝑘𝐷𝑆)‘𝑑) = 𝑍𝑑 / 𝑘𝑆 = 𝑍))
20 oveq1 7438 . . . . . . . . 9 (𝑑 / 𝑘𝑆 = 𝑍 → (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊) = (𝑍 𝑑 / 𝑘𝑊))
21 mptscmfsupp0.z . . . . . . . . . . . 12 𝑍 = (0g𝑅)
22 mptscmfsupp0.r . . . . . . . . . . . . . 14 (𝜑𝑅 = (Scalar‘𝑄))
2322adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑑𝐷) → 𝑅 = (Scalar‘𝑄))
2423fveq2d 6910 . . . . . . . . . . . 12 ((𝜑𝑑𝐷) → (0g𝑅) = (0g‘(Scalar‘𝑄)))
2521, 24eqtrid 2789 . . . . . . . . . . 11 ((𝜑𝑑𝐷) → 𝑍 = (0g‘(Scalar‘𝑄)))
2625oveq1d 7446 . . . . . . . . . 10 ((𝜑𝑑𝐷) → (𝑍 𝑑 / 𝑘𝑊) = ((0g‘(Scalar‘𝑄)) 𝑑 / 𝑘𝑊))
27 mptscmfsupp0.q . . . . . . . . . . . 12 (𝜑𝑄 ∈ LMod)
2827adantr 480 . . . . . . . . . . 11 ((𝜑𝑑𝐷) → 𝑄 ∈ LMod)
29 mptscmfsupp0.w . . . . . . . . . . . . . 14 ((𝜑𝑘𝐷) → 𝑊𝐾)
3029ralrimiva 3146 . . . . . . . . . . . . 13 (𝜑 → ∀𝑘𝐷 𝑊𝐾)
3130adantr 480 . . . . . . . . . . . 12 ((𝜑𝑑𝐷) → ∀𝑘𝐷 𝑊𝐾)
32 rspcsbela 4438 . . . . . . . . . . . 12 ((𝑑𝐷 ∧ ∀𝑘𝐷 𝑊𝐾) → 𝑑 / 𝑘𝑊𝐾)
3310, 31, 32syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑑𝐷) → 𝑑 / 𝑘𝑊𝐾)
34 mptscmfsupp0.k . . . . . . . . . . . 12 𝐾 = (Base‘𝑄)
35 eqid 2737 . . . . . . . . . . . 12 (Scalar‘𝑄) = (Scalar‘𝑄)
36 mptscmfsupp0.m . . . . . . . . . . . 12 = ( ·𝑠𝑄)
37 eqid 2737 . . . . . . . . . . . 12 (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
3834, 35, 36, 37, 5lmod0vs 20893 . . . . . . . . . . 11 ((𝑄 ∈ LMod ∧ 𝑑 / 𝑘𝑊𝐾) → ((0g‘(Scalar‘𝑄)) 𝑑 / 𝑘𝑊) = 0 )
3928, 33, 38syl2anc 584 . . . . . . . . . 10 ((𝜑𝑑𝐷) → ((0g‘(Scalar‘𝑄)) 𝑑 / 𝑘𝑊) = 0 )
4026, 39eqtrd 2777 . . . . . . . . 9 ((𝜑𝑑𝐷) → (𝑍 𝑑 / 𝑘𝑊) = 0 )
4120, 40sylan9eqr 2799 . . . . . . . 8 (((𝜑𝑑𝐷) ∧ 𝑑 / 𝑘𝑆 = 𝑍) → (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊) = 0 )
42 csbov12g 7477 . . . . . . . . . . . . . 14 (𝑑𝐷𝑑 / 𝑘(𝑆 𝑊) = (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊))
4342adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑑𝐷) → 𝑑 / 𝑘(𝑆 𝑊) = (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊))
44 ovex 7464 . . . . . . . . . . . . 13 (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊) ∈ V
4543, 44eqeltrdi 2849 . . . . . . . . . . . 12 ((𝜑𝑑𝐷) → 𝑑 / 𝑘(𝑆 𝑊) ∈ V)
46 eqid 2737 . . . . . . . . . . . . 13 (𝑘𝐷 ↦ (𝑆 𝑊)) = (𝑘𝐷 ↦ (𝑆 𝑊))
4746fvmpts 7019 . . . . . . . . . . . 12 ((𝑑𝐷𝑑 / 𝑘(𝑆 𝑊) ∈ V) → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 𝑑 / 𝑘(𝑆 𝑊))
4810, 45, 47syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑑𝐷) → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 𝑑 / 𝑘(𝑆 𝑊))
4948, 43eqtrd 2777 . . . . . . . . . 10 ((𝜑𝑑𝐷) → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊))
5049eqeq1d 2739 . . . . . . . . 9 ((𝜑𝑑𝐷) → (((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 0 ↔ (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊) = 0 ))
5150adantr 480 . . . . . . . 8 (((𝜑𝑑𝐷) ∧ 𝑑 / 𝑘𝑆 = 𝑍) → (((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 0 ↔ (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊) = 0 ))
5241, 51mpbird 257 . . . . . . 7 (((𝜑𝑑𝐷) ∧ 𝑑 / 𝑘𝑆 = 𝑍) → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 0 )
5352ex 412 . . . . . 6 ((𝜑𝑑𝐷) → (𝑑 / 𝑘𝑆 = 𝑍 → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 0 ))
5419, 53sylbid 240 . . . . 5 ((𝜑𝑑𝐷) → (((𝑘𝐷𝑆)‘𝑑) = 𝑍 → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 0 ))
5554necon3d 2961 . . . 4 ((𝜑𝑑𝐷) → (((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) ≠ 0 → ((𝑘𝐷𝑆)‘𝑑) ≠ 𝑍))
5655ss2rabdv 4076 . . 3 (𝜑 → {𝑑𝐷 ∣ ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) ≠ 0 } ⊆ {𝑑𝐷 ∣ ((𝑘𝐷𝑆)‘𝑑) ≠ 𝑍})
57 ovex 7464 . . . . . 6 (𝑆 𝑊) ∈ V
5857rgenw 3065 . . . . 5 𝑘𝐷 (𝑆 𝑊) ∈ V
5946fnmpt 6708 . . . . 5 (∀𝑘𝐷 (𝑆 𝑊) ∈ V → (𝑘𝐷 ↦ (𝑆 𝑊)) Fn 𝐷)
6058, 59mp1i 13 . . . 4 (𝜑 → (𝑘𝐷 ↦ (𝑆 𝑊)) Fn 𝐷)
61 suppvalfn 8193 . . . 4 (((𝑘𝐷 ↦ (𝑆 𝑊)) Fn 𝐷𝐷𝑉0 ∈ V) → ((𝑘𝐷 ↦ (𝑆 𝑊)) supp 0 ) = {𝑑𝐷 ∣ ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) ≠ 0 })
6260, 1, 7, 61syl3anc 1373 . . 3 (𝜑 → ((𝑘𝐷 ↦ (𝑆 𝑊)) supp 0 ) = {𝑑𝐷 ∣ ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) ≠ 0 })
6316fnmpt 6708 . . . . 5 (∀𝑘𝐷 𝑆𝐵 → (𝑘𝐷𝑆) Fn 𝐷)
6412, 63syl 17 . . . 4 (𝜑 → (𝑘𝐷𝑆) Fn 𝐷)
6521fvexi 6920 . . . . 5 𝑍 ∈ V
6665a1i 11 . . . 4 (𝜑𝑍 ∈ V)
67 suppvalfn 8193 . . . 4 (((𝑘𝐷𝑆) Fn 𝐷𝐷𝑉𝑍 ∈ V) → ((𝑘𝐷𝑆) supp 𝑍) = {𝑑𝐷 ∣ ((𝑘𝐷𝑆)‘𝑑) ≠ 𝑍})
6864, 1, 66, 67syl3anc 1373 . . 3 (𝜑 → ((𝑘𝐷𝑆) supp 𝑍) = {𝑑𝐷 ∣ ((𝑘𝐷𝑆)‘𝑑) ≠ 𝑍})
6956, 62, 683sstr4d 4039 . 2 (𝜑 → ((𝑘𝐷 ↦ (𝑆 𝑊)) supp 0 ) ⊆ ((𝑘𝐷𝑆) supp 𝑍))
70 suppssfifsupp 9420 . 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 )
712, 4, 7, 9, 69, 70syl32anc 1380 1 (𝜑 → (𝑘𝐷 ↦ (𝑆 𝑊)) finSupp 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  {crab 3436  Vcvv 3480  csb 3899  wss 3951   class class class wbr 5143  cmpt 5225  Fun wfun 6555   Fn wfn 6556  cfv 6561  (class class class)co 7431   supp csupp 8185  Fincfn 8985   finSupp cfsupp 9401  Basecbs 17247  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17484  LModclmod 20858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-supp 8186  df-1o 8506  df-en 8986  df-fin 8989  df-fsupp 9402  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-ring 20232  df-lmod 20860
This theorem is referenced by:  mptscmfsuppd  20926  gsumsmonply1  22311  pm2mpcl  22803  mply1topmatcllem  22809  mp2pm2mplem5  22816  pm2mpghmlem2  22818  chcoeffeqlem  22891  lbsdiflsp0  33677  fedgmullem2  33681
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