Theorem fisuppov1 32664

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 5260	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
4	3	mptexd 7158	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) ∈ V)
5		fisuppov1.1	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
6		funmpt 6519	. . 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 6891	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)))
11	10	oveq1d 7361	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐷) supp 0 ) = ((𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) supp 0 ))
12	9, 1	fexd 7161	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
13		fisuppov1.2	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝑋)
14		ressuppss 8113	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 0 ∈ 𝑋) → ((𝐹 ↾ 𝐷) supp 0 ) ⊆ (𝐹 supp 0 ))
15	12, 13, 14	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐷) supp 0 ) ⊆ (𝐹 supp 0 ))
16	11, 15	eqsstrrd 3965	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) supp 0 ) ⊆ (𝐹 supp 0 ))
17		fisuppov1.8	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ( 0 𝑂𝑦) = 𝑍)
18		fvexd 6837	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ V)
19		fisuppov1.5	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑌)
20	16, 17, 18, 19, 13	suppssov1 8127	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) supp 𝑍) ⊆ (𝐹 supp 0 ))
21	4, 5, 7, 8, 20	fsuppsssuppgd 9266	1 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) finSupp 𝑍)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ⊆ wss 3897   class class class wbr 5089   ↦ cmpt 5170   ↾ cres 5616  Fun wfun 6475  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   supp csupp 8090   finSupp cfsupp 9245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-supp 8091  df-1o 8385  df-en 8870  df-fin 8873  df-fsupp 9246

This theorem is referenced by:  elrgspnlem1  33209  elrgspnlem2  33210  fldextrspunlsplem  33686