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Theorem fisuppov1 32692
Description: Formula building theorem for finite support: operator with left annihilator. (Contributed by Thierry Arnoux, 5-Oct-2025.)
Hypotheses
Ref Expression
fisuppov1.1 (𝜑𝑍𝑉)
fisuppov1.2 (𝜑0𝑋)
fisuppov1.3 (𝜑𝐴𝑊)
fisuppov1.4 (𝜑𝐷𝐴)
fisuppov1.5 ((𝜑𝑥𝐷) → 𝐵𝑌)
fisuppov1.6 (𝜑𝐹:𝐴𝐸)
fisuppov1.7 (𝜑𝐹 finSupp 0 )
fisuppov1.8 ((𝜑𝑦𝑌) → ( 0 𝑂𝑦) = 𝑍)
Assertion
Ref Expression
fisuppov1 (𝜑 → (𝑥𝐷 ↦ ((𝐹𝑥)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
StepHypRef Expression
1 fisuppov1.3 . . . 4 (𝜑𝐴𝑊)
2 fisuppov1.4 . . . 4 (𝜑𝐷𝐴)
31, 2ssexd 5324 . . 3 (𝜑𝐷 ∈ V)
43mptexd 7244 . 2 (𝜑 → (𝑥𝐷 ↦ ((𝐹𝑥)𝑂𝐵)) ∈ V)
5 fisuppov1.1 . 2 (𝜑𝑍𝑉)
6 funmpt 6604 . . 3 Fun (𝑥𝐷 ↦ ((𝐹𝑥)𝑂𝐵))
76a1i 11 . 2 (𝜑 → Fun (𝑥𝐷 ↦ ((𝐹𝑥)𝑂𝐵)))
8 fisuppov1.7 . 2 (𝜑𝐹 finSupp 0 )
9 fisuppov1.6 . . . . . 6 (𝜑𝐹:𝐴𝐸)
109, 2feqresmpt 6978 . . . . 5 (𝜑 → (𝐹𝐷) = (𝑥𝐷 ↦ (𝐹𝑥)))
1110oveq1d 7446 . . . 4 (𝜑 → ((𝐹𝐷) supp 0 ) = ((𝑥𝐷 ↦ (𝐹𝑥)) supp 0 ))
129, 1fexd 7247 . . . . 5 (𝜑𝐹 ∈ V)
13 fisuppov1.2 . . . . 5 (𝜑0𝑋)
14 ressuppss 8208 . . . . 5 ((𝐹 ∈ V ∧ 0𝑋) → ((𝐹𝐷) supp 0 ) ⊆ (𝐹 supp 0 ))
1512, 13, 14syl2anc 584 . . . 4 (𝜑 → ((𝐹𝐷) supp 0 ) ⊆ (𝐹 supp 0 ))
1611, 15eqsstrrd 4019 . . 3 (𝜑 → ((𝑥𝐷 ↦ (𝐹𝑥)) supp 0 ) ⊆ (𝐹 supp 0 ))
17 fisuppov1.8 . . 3 ((𝜑𝑦𝑌) → ( 0 𝑂𝑦) = 𝑍)
18 fvexd 6921 . . 3 ((𝜑𝑥𝐷) → (𝐹𝑥) ∈ V)
19 fisuppov1.5 . . 3 ((𝜑𝑥𝐷) → 𝐵𝑌)
2016, 17, 18, 19, 13suppssov1 8222 . 2 (𝜑 → ((𝑥𝐷 ↦ ((𝐹𝑥)𝑂𝐵)) supp 𝑍) ⊆ (𝐹 supp 0 ))
214, 5, 7, 8, 20fsuppsssuppgd 9422 1 (𝜑 → (𝑥𝐷 ↦ ((𝐹𝑥)𝑂𝐵)) finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951   class class class wbr 5143  cmpt 5225  cres 5687  Fun wfun 6555  wf 6557  cfv 6561  (class class class)co 7431   supp csupp 8185   finSupp cfsupp 9401
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-supp 8186  df-1o 8506  df-en 8986  df-fin 8989  df-fsupp 9402
This theorem is referenced by:  elrgspnlem1  33246  elrgspnlem2  33247  fldextrspunlsplem  33723
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