Theorem fisuppov1 32775

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 5262	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
4	3	mptexd 7174	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) ∈ V)
5		fisuppov1.1	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
6		funmpt 6532	. . 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 6905	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)))
11	10	oveq1d 7377	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐷) supp 0 ) = ((𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) supp 0 ))
12	9, 1	fexd 7177	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
13		fisuppov1.2	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝑋)
14		ressuppss 8128	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 0 ∈ 𝑋) → ((𝐹 ↾ 𝐷) supp 0 ) ⊆ (𝐹 supp 0 ))
15	12, 13, 14	syl2anc 585	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐷) supp 0 ) ⊆ (𝐹 supp 0 ))
16	11, 15	eqsstrrd 3958	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) supp 0 ) ⊆ (𝐹 supp 0 ))
17		fisuppov1.8	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ( 0 𝑂𝑦) = 𝑍)
18		fvexd 6851	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ V)
19		fisuppov1.5	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑌)
20	16, 17, 18, 19, 13	suppssov1 8142	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) supp 𝑍) ⊆ (𝐹 supp 0 ))
21	4, 5, 7, 8, 20	fsuppsssuppgd 9290	1 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) finSupp 𝑍)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890   class class class wbr 5086   ↦ cmpt 5167   ↾ cres 5628  Fun wfun 6488  ⟶wf 6490  ‘cfv 6494  (class class class)co 7362   supp csupp 8105   finSupp cfsupp 9269

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-supp 8106  df-1o 8400  df-en 8889  df-fin 8892  df-fsupp 9270

This theorem is referenced by:  elrgspnlem1  33322  elrgspnlem2  33323  evlextv  33705  fldextrspunlsplem  33837