MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  suppssof1 Structured version   Visualization version   GIF version

Theorem suppssof1 8203
Description: Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
Hypotheses
Ref Expression
suppssof1.s (𝜑 → (𝐴 supp 𝑌) ⊆ 𝐿)
suppssof1.o ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
suppssof1.a (𝜑𝐴:𝐷𝑉)
suppssof1.b (𝜑𝐵:𝐷𝑅)
suppssof1.d (𝜑𝐷𝑊)
suppssof1.y (𝜑𝑌𝑈)
Assertion
Ref Expression
suppssof1 (𝜑 → ((𝐴f 𝑂𝐵) supp 𝑍) ⊆ 𝐿)
Distinct variable groups:   𝜑,𝑣   𝑣,𝐵   𝑣,𝑂   𝑣,𝑅   𝑣,𝑌   𝑣,𝑍
Allowed substitution hints:   𝐴(𝑣)   𝐷(𝑣)   𝑈(𝑣)   𝐿(𝑣)   𝑉(𝑣)   𝑊(𝑣)

Proof of Theorem suppssof1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 suppssof1.a . . . . 5 (𝜑𝐴:𝐷𝑉)
21ffnd 6718 . . . 4 (𝜑𝐴 Fn 𝐷)
3 suppssof1.b . . . . 5 (𝜑𝐵:𝐷𝑅)
43ffnd 6718 . . . 4 (𝜑𝐵 Fn 𝐷)
5 suppssof1.d . . . 4 (𝜑𝐷𝑊)
6 inidm 4213 . . . 4 (𝐷𝐷) = 𝐷
7 eqidd 2726 . . . 4 ((𝜑𝑥𝐷) → (𝐴𝑥) = (𝐴𝑥))
8 eqidd 2726 . . . 4 ((𝜑𝑥𝐷) → (𝐵𝑥) = (𝐵𝑥))
92, 4, 5, 5, 6, 7, 8offval 7691 . . 3 (𝜑 → (𝐴f 𝑂𝐵) = (𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))))
109oveq1d 7431 . 2 (𝜑 → ((𝐴f 𝑂𝐵) supp 𝑍) = ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) supp 𝑍))
111feqmptd 6962 . . . . 5 (𝜑𝐴 = (𝑥𝐷 ↦ (𝐴𝑥)))
1211oveq1d 7431 . . . 4 (𝜑 → (𝐴 supp 𝑌) = ((𝑥𝐷 ↦ (𝐴𝑥)) supp 𝑌))
13 suppssof1.s . . . 4 (𝜑 → (𝐴 supp 𝑌) ⊆ 𝐿)
1412, 13eqsstrrd 4012 . . 3 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑥)) supp 𝑌) ⊆ 𝐿)
15 suppssof1.o . . 3 ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
16 fvexd 6907 . . 3 ((𝜑𝑥𝐷) → (𝐴𝑥) ∈ V)
173ffvelcdmda 7089 . . 3 ((𝜑𝑥𝐷) → (𝐵𝑥) ∈ 𝑅)
18 suppssof1.y . . 3 (𝜑𝑌𝑈)
1914, 15, 16, 17, 18suppssov1 8201 . 2 (𝜑 → ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) supp 𝑍) ⊆ 𝐿)
2010, 19eqsstrd 4011 1 (𝜑 → ((𝐴f 𝑂𝐵) supp 𝑍) ⊆ 𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3463  wss 3939  cmpt 5226  wf 6539  cfv 6543  (class class class)co 7416  f cof 7680   supp csupp 8163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-supp 8164
This theorem is referenced by:  frlmup1  21736  psrbagev1  22028  psrbagev1OLD  22029  jensen  26939  offinsupp1  32554
  Copyright terms: Public domain W3C validator