Theorem suppssof1 8149

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	⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)
suppssof1.a	⊢ (𝜑 → 𝐴:𝐷⟶𝑉)
suppssof1.b	⊢ (𝜑 → 𝐵:𝐷⟶𝑅)
suppssof1.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)
suppssof1.y	⊢ (𝜑 → 𝑌 ∈ 𝑈)

Assertion

Ref	Expression
suppssof1	⊢ (𝜑 → ((𝐴 ∘f 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.

Step	Hyp	Ref	Expression
1		suppssof1.a	. . . . 5 ⊢ (𝜑 → 𝐴:𝐷⟶𝑉)
2	1	ffnd 6669	. . . 4 ⊢ (𝜑 → 𝐴 Fn 𝐷)
3		suppssof1.b	. . . . 5 ⊢ (𝜑 → 𝐵:𝐷⟶𝑅)
4	3	ffnd 6669	. . . 4 ⊢ (𝜑 → 𝐵 Fn 𝐷)
5		suppssof1.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
6		inidm 4167	. . . 4 ⊢ (𝐷 ∩ 𝐷) = 𝐷
7		eqidd 2737	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) = (𝐴‘𝑥))
8		eqidd 2737	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) = (𝐵‘𝑥))
9	2, 4, 5, 5, 6, 7, 8	offval 7640	. . 3 ⊢ (𝜑 → (𝐴 ∘f 𝑂𝐵) = (𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))))
10	9	oveq1d 7382	. 2 ⊢ (𝜑 → ((𝐴 ∘f 𝑂𝐵) supp 𝑍) = ((𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) supp 𝑍))
11	1	feqmptd 6908	. . . . 5 ⊢ (𝜑 → 𝐴 = (𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)))
12	11	oveq1d 7382	. . . 4 ⊢ (𝜑 → (𝐴 supp 𝑌) = ((𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) supp 𝑌))
13		suppssof1.s	. . . 4 ⊢ (𝜑 → (𝐴 supp 𝑌) ⊆ 𝐿)
14	12, 13	eqsstrrd 3957	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) supp 𝑌) ⊆ 𝐿)
15		suppssof1.o	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)
16		fvexd 6855	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ∈ V)
17	3	ffvelcdmda 7036	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) ∈ 𝑅)
18		suppssof1.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
19	14, 15, 16, 17, 18	suppssov1 8147	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) supp 𝑍) ⊆ 𝐿)
20	10, 19	eqsstrd 3956	1 ⊢ (𝜑 → ((𝐴 ∘f 𝑂𝐵) supp 𝑍) ⊆ 𝐿)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ⊆ wss 3889   ↦ cmpt 5166  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ∘_f cof 7629   supp csupp 8110

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-supp 8111

This theorem is referenced by:  frlmup1  21778  psrbagev1  22055  jensen  26952  offinsupp1  32799