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Theorem suppssof1 8185
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 6712 . . . 4 (𝜑𝐴 Fn 𝐷)
3 suppssof1.b . . . . 5 (𝜑𝐵:𝐷𝑅)
43ffnd 6712 . . . 4 (𝜑𝐵 Fn 𝐷)
5 suppssof1.d . . . 4 (𝜑𝐷𝑊)
6 inidm 4213 . . . 4 (𝐷𝐷) = 𝐷
7 eqidd 2727 . . . 4 ((𝜑𝑥𝐷) → (𝐴𝑥) = (𝐴𝑥))
8 eqidd 2727 . . . 4 ((𝜑𝑥𝐷) → (𝐵𝑥) = (𝐵𝑥))
92, 4, 5, 5, 6, 7, 8offval 7676 . . 3 (𝜑 → (𝐴f 𝑂𝐵) = (𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))))
109oveq1d 7420 . 2 (𝜑 → ((𝐴f 𝑂𝐵) supp 𝑍) = ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) supp 𝑍))
111feqmptd 6954 . . . . 5 (𝜑𝐴 = (𝑥𝐷 ↦ (𝐴𝑥)))
1211oveq1d 7420 . . . 4 (𝜑 → (𝐴 supp 𝑌) = ((𝑥𝐷 ↦ (𝐴𝑥)) supp 𝑌))
13 suppssof1.s . . . 4 (𝜑 → (𝐴 supp 𝑌) ⊆ 𝐿)
1412, 13eqsstrrd 4016 . . 3 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑥)) supp 𝑌) ⊆ 𝐿)
15 suppssof1.o . . 3 ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
16 fvexd 6900 . . 3 ((𝜑𝑥𝐷) → (𝐴𝑥) ∈ V)
173ffvelcdmda 7080 . . 3 ((𝜑𝑥𝐷) → (𝐵𝑥) ∈ 𝑅)
18 suppssof1.y . . 3 (𝜑𝑌𝑈)
1914, 15, 16, 17, 18suppssov1 8183 . 2 (𝜑 → ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) supp 𝑍) ⊆ 𝐿)
2010, 19eqsstrd 4015 1 (𝜑 → ((𝐴f 𝑂𝐵) supp 𝑍) ⊆ 𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  wss 3943  cmpt 5224  wf 6533  cfv 6537  (class class class)co 7405  f cof 7665   supp csupp 8146
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-supp 8147
This theorem is referenced by:  frlmup1  21693  psrbagev1  21980  psrbagev1OLD  21981  jensen  26876  offinsupp1  32459
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