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Theorem mptscmfsupp0 20941
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 7243 . 2 (𝜑 → (𝑘𝐷 ↦ (𝑆 𝑊)) ∈ V)
3 funmpt 6605 . . 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 9406 . 2 (𝜑 → ((𝑘𝐷𝑆) supp 𝑍) ∈ Fin)
10 simpr 484 . . . . . . . 8 ((𝜑𝑑𝐷) → 𝑑𝐷)
11 mptscmfsupp0.s . . . . . . . . . . 11 ((𝜑𝑘𝐷) → 𝑆𝐵)
1211ralrimiva 3143 . . . . . . . . . 10 (𝜑 → ∀𝑘𝐷 𝑆𝐵)
1312adantr 480 . . . . . . . . 9 ((𝜑𝑑𝐷) → ∀𝑘𝐷 𝑆𝐵)
14 rspcsbela 4443 . . . . . . . . 9 ((𝑑𝐷 ∧ ∀𝑘𝐷 𝑆𝐵) → 𝑑 / 𝑘𝑆𝐵)
1510, 13, 14syl2anc 584 . . . . . . . 8 ((𝜑𝑑𝐷) → 𝑑 / 𝑘𝑆𝐵)
16 eqid 2734 . . . . . . . . 9 (𝑘𝐷𝑆) = (𝑘𝐷𝑆)
1716fvmpts 7018 . . . . . . . 8 ((𝑑𝐷𝑑 / 𝑘𝑆𝐵) → ((𝑘𝐷𝑆)‘𝑑) = 𝑑 / 𝑘𝑆)
1810, 15, 17syl2anc 584 . . . . . . 7 ((𝜑𝑑𝐷) → ((𝑘𝐷𝑆)‘𝑑) = 𝑑 / 𝑘𝑆)
1918eqeq1d 2736 . . . . . 6 ((𝜑𝑑𝐷) → (((𝑘𝐷𝑆)‘𝑑) = 𝑍𝑑 / 𝑘𝑆 = 𝑍))
20 oveq1 7437 . . . . . . . . 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 2786 . . . . . . . . . . 11 ((𝜑𝑑𝐷) → 𝑍 = (0g‘(Scalar‘𝑄)))
2625oveq1d 7445 . . . . . . . . . 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 3143 . . . . . . . . . . . . 13 (𝜑 → ∀𝑘𝐷 𝑊𝐾)
3130adantr 480 . . . . . . . . . . . 12 ((𝜑𝑑𝐷) → ∀𝑘𝐷 𝑊𝐾)
32 rspcsbela 4443 . . . . . . . . . . . 12 ((𝑑𝐷 ∧ ∀𝑘𝐷 𝑊𝐾) → 𝑑 / 𝑘𝑊𝐾)
3310, 31, 32syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑑𝐷) → 𝑑 / 𝑘𝑊𝐾)
34 mptscmfsupp0.k . . . . . . . . . . . 12 𝐾 = (Base‘𝑄)
35 eqid 2734 . . . . . . . . . . . 12 (Scalar‘𝑄) = (Scalar‘𝑄)
36 mptscmfsupp0.m . . . . . . . . . . . 12 = ( ·𝑠𝑄)
37 eqid 2734 . . . . . . . . . . . 12 (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
3834, 35, 36, 37, 5lmod0vs 20909 . . . . . . . . . . 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 2774 . . . . . . . . 9 ((𝜑𝑑𝐷) → (𝑍 𝑑 / 𝑘𝑊) = 0 )
4120, 40sylan9eqr 2796 . . . . . . . 8 (((𝜑𝑑𝐷) ∧ 𝑑 / 𝑘𝑆 = 𝑍) → (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊) = 0 )
42 csbov12g 7476 . . . . . . . . . . . . . 14 (𝑑𝐷𝑑 / 𝑘(𝑆 𝑊) = (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊))
4342adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑑𝐷) → 𝑑 / 𝑘(𝑆 𝑊) = (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊))
44 ovex 7463 . . . . . . . . . . . . 13 (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊) ∈ V
4543, 44eqeltrdi 2846 . . . . . . . . . . . 12 ((𝜑𝑑𝐷) → 𝑑 / 𝑘(𝑆 𝑊) ∈ V)
46 eqid 2734 . . . . . . . . . . . . 13 (𝑘𝐷 ↦ (𝑆 𝑊)) = (𝑘𝐷 ↦ (𝑆 𝑊))
4746fvmpts 7018 . . . . . . . . . . . 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 2774 . . . . . . . . . 10 ((𝜑𝑑𝐷) → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊))
5049eqeq1d 2736 . . . . . . . . 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 2958 . . . 4 ((𝜑𝑑𝐷) → (((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) ≠ 0 → ((𝑘𝐷𝑆)‘𝑑) ≠ 𝑍))
5655ss2rabdv 4085 . . 3 (𝜑 → {𝑑𝐷 ∣ ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) ≠ 0 } ⊆ {𝑑𝐷 ∣ ((𝑘𝐷𝑆)‘𝑑) ≠ 𝑍})
57 ovex 7463 . . . . . 6 (𝑆 𝑊) ∈ V
5857rgenw 3062 . . . . 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 8191 . . . 4 (((𝑘𝐷 ↦ (𝑆 𝑊)) Fn 𝐷𝐷𝑉0 ∈ V) → ((𝑘𝐷 ↦ (𝑆 𝑊)) supp 0 ) = {𝑑𝐷 ∣ ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) ≠ 0 })
6260, 1, 7, 61syl3anc 1370 . . 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 8191 . . . 4 (((𝑘𝐷𝑆) Fn 𝐷𝐷𝑉𝑍 ∈ V) → ((𝑘𝐷𝑆) supp 𝑍) = {𝑑𝐷 ∣ ((𝑘𝐷𝑆)‘𝑑) ≠ 𝑍})
6864, 1, 66, 67syl3anc 1370 . . 3 (𝜑 → ((𝑘𝐷𝑆) supp 𝑍) = {𝑑𝐷 ∣ ((𝑘𝐷𝑆)‘𝑑) ≠ 𝑍})
6956, 62, 683sstr4d 4042 . 2 (𝜑 → ((𝑘𝐷 ↦ (𝑆 𝑊)) supp 0 ) ⊆ ((𝑘𝐷𝑆) supp 𝑍))
70 suppssfifsupp 9417 . 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 1377 1 (𝜑 → (𝑘𝐷 ↦ (𝑆 𝑊)) finSupp 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wne 2937  wral 3058  {crab 3432  Vcvv 3477  csb 3907  wss 3962   class class class wbr 5147  cmpt 5230  Fun wfun 6556   Fn wfn 6557  cfv 6562  (class class class)co 7430   supp csupp 8183  Fincfn 8983   finSupp cfsupp 9398  Basecbs 17244  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17485  LModclmod 20874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-supp 8184  df-1o 8504  df-en 8984  df-fin 8987  df-fsupp 9399  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966  df-ring 20252  df-lmod 20876
This theorem is referenced by:  mptscmfsuppd  20942  gsumsmonply1  22326  pm2mpcl  22818  mply1topmatcllem  22824  mp2pm2mplem5  22831  pm2mpghmlem2  22833  chcoeffeqlem  22906  lbsdiflsp0  33653  fedgmullem2  33657
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