Theorem fisuppov1 32656

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 5274	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
4	3	mptexd 7180	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) ∈ V)
5		fisuppov1.1	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
6		funmpt 6538	. . 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 6912	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)))
11	10	oveq1d 7384	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐷) supp 0 ) = ((𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) supp 0 ))
12	9, 1	fexd 7183	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
13		fisuppov1.2	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝑋)
14		ressuppss 8139	. . . . 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 3979	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) supp 0 ) ⊆ (𝐹 supp 0 ))
17		fisuppov1.8	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ( 0 𝑂𝑦) = 𝑍)
18		fvexd 6855	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ V)
19		fisuppov1.5	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑌)
20	16, 17, 18, 19, 13	suppssov1 8153	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) supp 𝑍) ⊆ (𝐹 supp 0 ))
21	4, 5, 7, 8, 20	fsuppsssuppgd 9309	1 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) finSupp 𝑍)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ⊆ wss 3911   class class class wbr 5102   ↦ cmpt 5183   ↾ cres 5633  Fun wfun 6493  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   supp csupp 8116   finSupp cfsupp 9288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  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-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-supp 8117  df-1o 8411  df-en 8896  df-fin 8899  df-fsupp 9289

This theorem is referenced by:  elrgspnlem1  33209  elrgspnlem2  33210  fldextrspunlsplem  33661