Theorem suppssof1 7328

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		ffn 6045	. . . . 5 ⊢ (𝐴:𝐷⟶𝑉 → 𝐴 Fn 𝐷)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 Fn 𝐷)
4		suppssof1.b	. . . . 5 ⊢ (𝜑 → 𝐵:𝐷⟶𝑅)
5		ffn 6045	. . . . 5 ⊢ (𝐵:𝐷⟶𝑅 → 𝐵 Fn 𝐷)
6	4, 5	syl 17	. . . 4 ⊢ (𝜑 → 𝐵 Fn 𝐷)
7		suppssof1.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
8		inidm 3822	. . . 4 ⊢ (𝐷 ∩ 𝐷) = 𝐷
9		eqidd 2623	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) = (𝐴‘𝑥))
10		eqidd 2623	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) = (𝐵‘𝑥))
11	3, 6, 7, 7, 8, 9, 10	offval 6904	. . 3 ⊢ (𝜑 → (𝐴 ∘𝑓 𝑂𝐵) = (𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))))
12	11	oveq1d 6665	. 2 ⊢ (𝜑 → ((𝐴 ∘𝑓 𝑂𝐵) supp 𝑍) = ((𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) supp 𝑍))
13	1	feqmptd 6249	. . . . 5 ⊢ (𝜑 → 𝐴 = (𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)))
14	13	oveq1d 6665	. . . 4 ⊢ (𝜑 → (𝐴 supp 𝑌) = ((𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) supp 𝑌))
15		suppssof1.s	. . . 4 ⊢ (𝜑 → (𝐴 supp 𝑌) ⊆ 𝐿)
16	14, 15	eqsstr3d 3640	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) supp 𝑌) ⊆ 𝐿)
17		suppssof1.o	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)
18		fvexd 6203	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ∈ V)
19	4	ffvelrnda 6359	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) ∈ 𝑅)
20		suppssof1.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
21	16, 17, 18, 19, 20	suppssov1 7327	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) supp 𝑍) ⊆ 𝐿)
22	12, 21	eqsstrd 3639	1 ⊢ (𝜑 → ((𝐴 ∘𝑓 𝑂𝐵) supp 𝑍) ⊆ 𝐿)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574   ↦ cmpt 4729   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895   supp csupp 7295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-supp 7296

This theorem is referenced by:  psrbagev1  19510  frlmup1  20137  jensen  24715