MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mptscmfsupp0 Structured version   Visualization version   GIF version

Theorem mptscmfsupp0 19150
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 6649 . . 3 (𝐷𝑉 → (𝑘𝐷 ↦ (𝑆 𝑊)) ∈ V)
31, 2syl 17 . 2 (𝜑 → (𝑘𝐷 ↦ (𝑆 𝑊)) ∈ V)
4 funmpt 6087 . . 3 Fun (𝑘𝐷 ↦ (𝑆 𝑊))
54a1i 11 . 2 (𝜑 → Fun (𝑘𝐷 ↦ (𝑆 𝑊)))
6 mptscmfsupp0.0 . . . 4 0 = (0g𝑄)
7 fvex 6363 . . . 4 (0g𝑄) ∈ V
86, 7eqeltri 2835 . . 3 0 ∈ V
98a1i 11 . 2 (𝜑0 ∈ V)
10 mptscmfsupp0.f . . 3 (𝜑 → (𝑘𝐷𝑆) finSupp 𝑍)
1110fsuppimpd 8449 . 2 (𝜑 → ((𝑘𝐷𝑆) supp 𝑍) ∈ Fin)
12 simpr 479 . . . . . . . 8 ((𝜑𝑑𝐷) → 𝑑𝐷)
13 mptscmfsupp0.s . . . . . . . . . . 11 ((𝜑𝑘𝐷) → 𝑆𝐵)
1413ralrimiva 3104 . . . . . . . . . 10 (𝜑 → ∀𝑘𝐷 𝑆𝐵)
1514adantr 472 . . . . . . . . 9 ((𝜑𝑑𝐷) → ∀𝑘𝐷 𝑆𝐵)
16 rspcsbela 4149 . . . . . . . . 9 ((𝑑𝐷 ∧ ∀𝑘𝐷 𝑆𝐵) → 𝑑 / 𝑘𝑆𝐵)
1712, 15, 16syl2anc 696 . . . . . . . 8 ((𝜑𝑑𝐷) → 𝑑 / 𝑘𝑆𝐵)
18 eqid 2760 . . . . . . . . 9 (𝑘𝐷𝑆) = (𝑘𝐷𝑆)
1918fvmpts 6448 . . . . . . . 8 ((𝑑𝐷𝑑 / 𝑘𝑆𝐵) → ((𝑘𝐷𝑆)‘𝑑) = 𝑑 / 𝑘𝑆)
2012, 17, 19syl2anc 696 . . . . . . 7 ((𝜑𝑑𝐷) → ((𝑘𝐷𝑆)‘𝑑) = 𝑑 / 𝑘𝑆)
2120eqeq1d 2762 . . . . . 6 ((𝜑𝑑𝐷) → (((𝑘𝐷𝑆)‘𝑑) = 𝑍𝑑 / 𝑘𝑆 = 𝑍))
22 oveq1 6821 . . . . . . . . 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 472 . . . . . . . . . . . . 13 ((𝜑𝑑𝐷) → 𝑅 = (Scalar‘𝑄))
2625fveq2d 6357 . . . . . . . . . . . 12 ((𝜑𝑑𝐷) → (0g𝑅) = (0g‘(Scalar‘𝑄)))
2723, 26syl5eq 2806 . . . . . . . . . . 11 ((𝜑𝑑𝐷) → 𝑍 = (0g‘(Scalar‘𝑄)))
2827oveq1d 6829 . . . . . . . . . 10 ((𝜑𝑑𝐷) → (𝑍 𝑑 / 𝑘𝑊) = ((0g‘(Scalar‘𝑄)) 𝑑 / 𝑘𝑊))
29 mptscmfsupp0.q . . . . . . . . . . . 12 (𝜑𝑄 ∈ LMod)
3029adantr 472 . . . . . . . . . . 11 ((𝜑𝑑𝐷) → 𝑄 ∈ LMod)
31 mptscmfsupp0.w . . . . . . . . . . . . . 14 ((𝜑𝑘𝐷) → 𝑊𝐾)
3231ralrimiva 3104 . . . . . . . . . . . . 13 (𝜑 → ∀𝑘𝐷 𝑊𝐾)
3332adantr 472 . . . . . . . . . . . 12 ((𝜑𝑑𝐷) → ∀𝑘𝐷 𝑊𝐾)
34 rspcsbela 4149 . . . . . . . . . . . 12 ((𝑑𝐷 ∧ ∀𝑘𝐷 𝑊𝐾) → 𝑑 / 𝑘𝑊𝐾)
3512, 33, 34syl2anc 696 . . . . . . . . . . 11 ((𝜑𝑑𝐷) → 𝑑 / 𝑘𝑊𝐾)
36 mptscmfsupp0.k . . . . . . . . . . . 12 𝐾 = (Base‘𝑄)
37 eqid 2760 . . . . . . . . . . . 12 (Scalar‘𝑄) = (Scalar‘𝑄)
38 mptscmfsupp0.m . . . . . . . . . . . 12 = ( ·𝑠𝑄)
39 eqid 2760 . . . . . . . . . . . 12 (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
4036, 37, 38, 39, 6lmod0vs 19118 . . . . . . . . . . 11 ((𝑄 ∈ LMod ∧ 𝑑 / 𝑘𝑊𝐾) → ((0g‘(Scalar‘𝑄)) 𝑑 / 𝑘𝑊) = 0 )
4130, 35, 40syl2anc 696 . . . . . . . . . 10 ((𝜑𝑑𝐷) → ((0g‘(Scalar‘𝑄)) 𝑑 / 𝑘𝑊) = 0 )
4228, 41eqtrd 2794 . . . . . . . . 9 ((𝜑𝑑𝐷) → (𝑍 𝑑 / 𝑘𝑊) = 0 )
4322, 42sylan9eqr 2816 . . . . . . . 8 (((𝜑𝑑𝐷) ∧ 𝑑 / 𝑘𝑆 = 𝑍) → (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊) = 0 )
44 csbov12g 6853 . . . . . . . . . . . . . 14 (𝑑𝐷𝑑 / 𝑘(𝑆 𝑊) = (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊))
4544adantl 473 . . . . . . . . . . . . 13 ((𝜑𝑑𝐷) → 𝑑 / 𝑘(𝑆 𝑊) = (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊))
46 ovex 6842 . . . . . . . . . . . . 13 (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊) ∈ V
4745, 46syl6eqel 2847 . . . . . . . . . . . 12 ((𝜑𝑑𝐷) → 𝑑 / 𝑘(𝑆 𝑊) ∈ V)
48 eqid 2760 . . . . . . . . . . . . 13 (𝑘𝐷 ↦ (𝑆 𝑊)) = (𝑘𝐷 ↦ (𝑆 𝑊))
4948fvmpts 6448 . . . . . . . . . . . 12 ((𝑑𝐷𝑑 / 𝑘(𝑆 𝑊) ∈ V) → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 𝑑 / 𝑘(𝑆 𝑊))
5012, 47, 49syl2anc 696 . . . . . . . . . . 11 ((𝜑𝑑𝐷) → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 𝑑 / 𝑘(𝑆 𝑊))
5150, 45eqtrd 2794 . . . . . . . . . 10 ((𝜑𝑑𝐷) → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊))
5251eqeq1d 2762 . . . . . . . . 9 ((𝜑𝑑𝐷) → (((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 0 ↔ (𝑑 / 𝑘𝑆 𝑑 / 𝑘𝑊) = 0 ))
5352adantr 472 . . . . . . . 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 449 . . . . . 6 ((𝜑𝑑𝐷) → (𝑑 / 𝑘𝑆 = 𝑍 → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 0 ))
5621, 55sylbid 230 . . . . 5 ((𝜑𝑑𝐷) → (((𝑘𝐷𝑆)‘𝑑) = 𝑍 → ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) = 0 ))
5756necon3d 2953 . . . 4 ((𝜑𝑑𝐷) → (((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) ≠ 0 → ((𝑘𝐷𝑆)‘𝑑) ≠ 𝑍))
5857ss2rabdv 3824 . . 3 (𝜑 → {𝑑𝐷 ∣ ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) ≠ 0 } ⊆ {𝑑𝐷 ∣ ((𝑘𝐷𝑆)‘𝑑) ≠ 𝑍})
59 ovex 6842 . . . . . 6 (𝑆 𝑊) ∈ V
6059rgenw 3062 . . . . 5 𝑘𝐷 (𝑆 𝑊) ∈ V
6148fnmpt 6181 . . . . 5 (∀𝑘𝐷 (𝑆 𝑊) ∈ V → (𝑘𝐷 ↦ (𝑆 𝑊)) Fn 𝐷)
6260, 61mp1i 13 . . . 4 (𝜑 → (𝑘𝐷 ↦ (𝑆 𝑊)) Fn 𝐷)
63 suppvalfn 7471 . . . 4 (((𝑘𝐷 ↦ (𝑆 𝑊)) Fn 𝐷𝐷𝑉0 ∈ V) → ((𝑘𝐷 ↦ (𝑆 𝑊)) supp 0 ) = {𝑑𝐷 ∣ ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) ≠ 0 })
6462, 1, 9, 63syl3anc 1477 . . 3 (𝜑 → ((𝑘𝐷 ↦ (𝑆 𝑊)) supp 0 ) = {𝑑𝐷 ∣ ((𝑘𝐷 ↦ (𝑆 𝑊))‘𝑑) ≠ 0 })
6518fnmpt 6181 . . . . 5 (∀𝑘𝐷 𝑆𝐵 → (𝑘𝐷𝑆) Fn 𝐷)
6614, 65syl 17 . . . 4 (𝜑 → (𝑘𝐷𝑆) Fn 𝐷)
67 fvex 6363 . . . . . 6 (0g𝑅) ∈ V
6823, 67eqeltri 2835 . . . . 5 𝑍 ∈ V
6968a1i 11 . . . 4 (𝜑𝑍 ∈ V)
70 suppvalfn 7471 . . . 4 (((𝑘𝐷𝑆) Fn 𝐷𝐷𝑉𝑍 ∈ V) → ((𝑘𝐷𝑆) supp 𝑍) = {𝑑𝐷 ∣ ((𝑘𝐷𝑆)‘𝑑) ≠ 𝑍})
7166, 1, 69, 70syl3anc 1477 . . 3 (𝜑 → ((𝑘𝐷𝑆) supp 𝑍) = {𝑑𝐷 ∣ ((𝑘𝐷𝑆)‘𝑑) ≠ 𝑍})
7258, 64, 713sstr4d 3789 . 2 (𝜑 → ((𝑘𝐷 ↦ (𝑆 𝑊)) supp 0 ) ⊆ ((𝑘𝐷𝑆) supp 𝑍))
73 suppssfifsupp 8457 . 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 1485 1 (𝜑 → (𝑘𝐷 ↦ (𝑆 𝑊)) finSupp 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  {crab 3054  Vcvv 3340  csb 3674  wss 3715   class class class wbr 4804  cmpt 4881  Fun wfun 6043   Fn wfn 6044  cfv 6049  (class class class)co 6814   supp csupp 7464  Fincfn 8123   finSupp cfsupp 8442  Basecbs 16079  Scalarcsca 16166   ·𝑠 cvsca 16167  0gc0g 16322  LModclmod 19085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-supp 7465  df-er 7913  df-en 8124  df-fin 8127  df-fsupp 8443  df-0g 16324  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-grp 17646  df-ring 18769  df-lmod 19087
This theorem is referenced by:  mptscmfsuppd  19151  gsumsmonply1  19895  pm2mpcl  20824  mply1topmatcllem  20830  mp2pm2mplem5  20837  pm2mpghmlem2  20839  chcoeffeqlem  20912
  Copyright terms: Public domain W3C validator