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Theorem suppssof1 7215
 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)
2 ffn 5958 . . . . 5 (𝐴:𝐷𝑉𝐴 Fn 𝐷)
31, 2syl 17 . . . 4 (𝜑𝐴 Fn 𝐷)
4 suppssof1.b . . . . 5 (𝜑𝐵:𝐷𝑅)
5 ffn 5958 . . . . 5 (𝐵:𝐷𝑅𝐵 Fn 𝐷)
64, 5syl 17 . . . 4 (𝜑𝐵 Fn 𝐷)
7 suppssof1.d . . . 4 (𝜑𝐷𝑊)
8 inidm 3784 . . . 4 (𝐷𝐷) = 𝐷
9 eqidd 2611 . . . 4 ((𝜑𝑥𝐷) → (𝐴𝑥) = (𝐴𝑥))
10 eqidd 2611 . . . 4 ((𝜑𝑥𝐷) → (𝐵𝑥) = (𝐵𝑥))
113, 6, 7, 7, 8, 9, 10offval 6802 . . 3 (𝜑 → (𝐴𝑓 𝑂𝐵) = (𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))))
1211oveq1d 6564 . 2 (𝜑 → ((𝐴𝑓 𝑂𝐵) supp 𝑍) = ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) supp 𝑍))
131feqmptd 6159 . . . . 5 (𝜑𝐴 = (𝑥𝐷 ↦ (𝐴𝑥)))
1413oveq1d 6564 . . . 4 (𝜑 → (𝐴 supp 𝑌) = ((𝑥𝐷 ↦ (𝐴𝑥)) supp 𝑌))
15 suppssof1.s . . . 4 (𝜑 → (𝐴 supp 𝑌) ⊆ 𝐿)
1614, 15eqsstr3d 3603 . . 3 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑥)) supp 𝑌) ⊆ 𝐿)
17 suppssof1.o . . 3 ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
18 fvex 6113 . . . 4 (𝐴𝑥) ∈ V
1918a1i 11 . . 3 ((𝜑𝑥𝐷) → (𝐴𝑥) ∈ V)
204ffvelrnda 6267 . . 3 ((𝜑𝑥𝐷) → (𝐵𝑥) ∈ 𝑅)
21 suppssof1.y . . 3 (𝜑𝑌𝑈)
2216, 17, 19, 20, 21suppssov1 7214 . 2 (𝜑 → ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) supp 𝑍) ⊆ 𝐿)
2312, 22eqsstrd 3602 1 (𝜑 → ((𝐴𝑓 𝑂𝐵) supp 𝑍) ⊆ 𝐿)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540   ↦ cmpt 4643   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793   supp csupp 7182 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-supp 7183 This theorem is referenced by:  psrbagev1  19331  frlmup1  19956  jensen  24515
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