Theorem fisuppov1 33344

Description: Formula building theorem for finite support: operator with left annihilator. (Contributed by Thierry Arnoux, 5-Oct-2025.)

Hypotheses

Ref	Expression
fisuppov1.1	⊢ (𝜑 → 𝑍 ∈ 𝑉)
fisuppov1.2	⊢ (𝜑 → 0 ∈ 𝑋)
fisuppov1.3	⊢ (𝜑 → 𝐴 ∈ 𝑊)
fisuppov1.4	⊢ (𝜑 → 𝐷 ⊆ 𝐴)
fisuppov1.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑌)
fisuppov1.6	⊢ (𝜑 → 𝐹:𝐴⟶𝐸)
fisuppov1.7	⊢ (𝜑 → 𝐹 finSupp 0 )
fisuppov1.8	⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ( 0 𝑂𝑦) = 𝑍)

Assertion

Ref	Expression
fisuppov1	⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) finSupp 𝑍)

Distinct variable groups:   𝑥, 0   𝑦, 0   𝑥,𝐴   𝑦,𝐵   𝑥,𝐷   𝑥,𝐹   𝑦,𝑂   𝑦,𝑌   𝑥,𝑍   𝑦,𝑍   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥)   𝐷(𝑦)   𝐸(𝑥, 𝑦)   𝐹(𝑦)   𝑂(𝑥)   𝑉(𝑥, 𝑦)   𝑊(𝑥, 𝑦)   𝑋(𝑥, 𝑦)   𝑌(𝑥)

Proof of Theorem fisuppov1

Step	Hyp	Ref	Expression
1		fisuppov1.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
2		fisuppov1.4	. . . 4 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
3	1, 2	ssexd 5290	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
4	3	mptexd 7250	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) ∈ V)
5		fisuppov1.1	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
6		funmpt 6595	. . 3 ⊢ Fun (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵))
7	6	a1i 11	. 2 ⊢ (𝜑 → Fun (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)))
8		fisuppov1.7	. 2 ⊢ (𝜑 → 𝐹 finSupp 0 )
9		fisuppov1.6	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐸)
10	9, 2	feqresmpt 6974	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)))
11	10	oveq1d 7455	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐷) supp 0 ) = ((𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) supp 0 ))
12	9, 1	fexd 7253	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
13		fisuppov1.2	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝑋)
14		ressuppss 8225	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 0 ∈ 𝑋) → ((𝐹 ↾ 𝐷) supp 0 ) ⊆ (𝐹 supp 0 ))
15	12, 13, 14	syl2anc 596	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐷) supp 0 ) ⊆ (𝐹 supp 0 ))
16	11, 15	eqsstrrd 3948	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) supp 0 ) ⊆ (𝐹 supp 0 ))
17		fisuppov1.8	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ( 0 𝑂𝑦) = 𝑍)
18		fvexd 6920	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ V)
19		fisuppov1.5	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑌)
20	16, 17, 18, 19, 13	suppssov1 8239	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) supp 𝑍) ⊆ (𝐹 supp 0 ))
21	4, 5, 7, 8, 20	fsuppsssuppgd 9398	1 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) finSupp 𝑍)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   class class class wbr 5106   ↦ cmpt 5189   ↾ cres 5657  Fun wfun 6549  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   supp csupp 8201   finSupp cfsupp 9378

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-supp 8202  df-1o 8501  df-en 8999  df-fin 9002  df-fsupp 9379

This theorem is used by:  elrgspnlem1  33870  elrgspnlem2  33871  selvply1rhmlemb  34218  evlextv  34241  fldextrspunlsplem  34372