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Theorem suppssof1 7848
 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 6488 . . . 4 (𝜑𝐴 Fn 𝐷)
3 suppssof1.b . . . . 5 (𝜑𝐵:𝐷𝑅)
43ffnd 6488 . . . 4 (𝜑𝐵 Fn 𝐷)
5 suppssof1.d . . . 4 (𝜑𝐷𝑊)
6 inidm 4145 . . . 4 (𝐷𝐷) = 𝐷
7 eqidd 2799 . . . 4 ((𝜑𝑥𝐷) → (𝐴𝑥) = (𝐴𝑥))
8 eqidd 2799 . . . 4 ((𝜑𝑥𝐷) → (𝐵𝑥) = (𝐵𝑥))
92, 4, 5, 5, 6, 7, 8offval 7397 . . 3 (𝜑 → (𝐴f 𝑂𝐵) = (𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))))
109oveq1d 7150 . 2 (𝜑 → ((𝐴f 𝑂𝐵) supp 𝑍) = ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) supp 𝑍))
111feqmptd 6708 . . . . 5 (𝜑𝐴 = (𝑥𝐷 ↦ (𝐴𝑥)))
1211oveq1d 7150 . . . 4 (𝜑 → (𝐴 supp 𝑌) = ((𝑥𝐷 ↦ (𝐴𝑥)) supp 𝑌))
13 suppssof1.s . . . 4 (𝜑 → (𝐴 supp 𝑌) ⊆ 𝐿)
1412, 13eqsstrrd 3954 . . 3 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑥)) supp 𝑌) ⊆ 𝐿)
15 suppssof1.o . . 3 ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
16 fvexd 6660 . . 3 ((𝜑𝑥𝐷) → (𝐴𝑥) ∈ V)
173ffvelrnda 6828 . . 3 ((𝜑𝑥𝐷) → (𝐵𝑥) ∈ 𝑅)
18 suppssof1.y . . 3 (𝜑𝑌𝑈)
1914, 15, 16, 17, 18suppssov1 7847 . 2 (𝜑 → ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) supp 𝑍) ⊆ 𝐿)
2010, 19eqsstrd 3953 1 (𝜑 → ((𝐴f 𝑂𝐵) supp 𝑍) ⊆ 𝐿)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881   ↦ cmpt 5110  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∘f cof 7388   supp csupp 7815 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7390  df-supp 7816 This theorem is referenced by:  frlmup1  20491  psrbagev1  20753  jensen  25581  offinsupp1  30496
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