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Theorem suppssof1 7288
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 6012 . . . . 5 (𝐴:𝐷𝑉𝐴 Fn 𝐷)
31, 2syl 17 . . . 4 (𝜑𝐴 Fn 𝐷)
4 suppssof1.b . . . . 5 (𝜑𝐵:𝐷𝑅)
5 ffn 6012 . . . . 5 (𝐵:𝐷𝑅𝐵 Fn 𝐷)
64, 5syl 17 . . . 4 (𝜑𝐵 Fn 𝐷)
7 suppssof1.d . . . 4 (𝜑𝐷𝑊)
8 inidm 3806 . . . 4 (𝐷𝐷) = 𝐷
9 eqidd 2622 . . . 4 ((𝜑𝑥𝐷) → (𝐴𝑥) = (𝐴𝑥))
10 eqidd 2622 . . . 4 ((𝜑𝑥𝐷) → (𝐵𝑥) = (𝐵𝑥))
113, 6, 7, 7, 8, 9, 10offval 6869 . . 3 (𝜑 → (𝐴𝑓 𝑂𝐵) = (𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))))
1211oveq1d 6630 . 2 (𝜑 → ((𝐴𝑓 𝑂𝐵) supp 𝑍) = ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) supp 𝑍))
131feqmptd 6216 . . . . 5 (𝜑𝐴 = (𝑥𝐷 ↦ (𝐴𝑥)))
1413oveq1d 6630 . . . 4 (𝜑 → (𝐴 supp 𝑌) = ((𝑥𝐷 ↦ (𝐴𝑥)) supp 𝑌))
15 suppssof1.s . . . 4 (𝜑 → (𝐴 supp 𝑌) ⊆ 𝐿)
1614, 15eqsstr3d 3625 . . 3 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑥)) supp 𝑌) ⊆ 𝐿)
17 suppssof1.o . . 3 ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
18 fvexd 6170 . . 3 ((𝜑𝑥𝐷) → (𝐴𝑥) ∈ V)
194ffvelrnda 6325 . . 3 ((𝜑𝑥𝐷) → (𝐵𝑥) ∈ 𝑅)
20 suppssof1.y . . 3 (𝜑𝑌𝑈)
2116, 17, 18, 19, 20suppssov1 7287 . 2 (𝜑 → ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) supp 𝑍) ⊆ 𝐿)
2212, 21eqsstrd 3624 1 (𝜑 → ((𝐴𝑓 𝑂𝐵) supp 𝑍) ⊆ 𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  wss 3560  cmpt 4683   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  𝑓 cof 6860   supp csupp 7255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-supp 7256
This theorem is referenced by:  psrbagev1  19450  frlmup1  20077  jensen  24649
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