Theorem fisuppov1 32776

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 5253	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
4	3	mptexd 7169	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) ∈ V)
5		fisuppov1.1	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
6		funmpt 6524	. . 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 6897	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)))
11	10	oveq1d 7372	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐷) supp 0 ) = ((𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) supp 0 ))
12	9, 1	fexd 7172	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
13		fisuppov1.2	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝑋)
14		ressuppss 8124	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 0 ∈ 𝑋) → ((𝐹 ↾ 𝐷) supp 0 ) ⊆ (𝐹 supp 0 ))
15	12, 13, 14	syl2anc 590	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐷) supp 0 ) ⊆ (𝐹 supp 0 ))
16	11, 15	eqsstrrd 3950	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) supp 0 ) ⊆ (𝐹 supp 0 ))
17		fisuppov1.8	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ( 0 𝑂𝑦) = 𝑍)
18		fvexd 6843	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ V)
19		fisuppov1.5	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑌)
20	16, 17, 18, 19, 13	suppssov1 8138	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) supp 𝑍) ⊆ (𝐹 supp 0 ))
21	4, 5, 7, 8, 20	fsuppsssuppgd 9286	1 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) finSupp 𝑍)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ⊆ wss 3883   class class class wbr 5073   ↦ cmpt 5154   ↾ cres 5621  Fun wfun 6480  ⟶wf 6482  ‘cfv 6486  (class class class)co 7357   supp csupp 8101   finSupp cfsupp 9265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-supp 8102  df-1o 8396  df-en 8885  df-fin 8888  df-fsupp 9266

This theorem is referenced by:  elrgspnlem1  33324  elrgspnlem2  33325  selvply1rhmlemb  33712  evlextv  33735  fldextrspunlsplem  33866