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Theorem fisuppov1 32697
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 5329 . . 3 (𝜑𝐷 ∈ V)
43mptexd 7243 . 2 (𝜑 → (𝑥𝐷 ↦ ((𝐹𝑥)𝑂𝐵)) ∈ V)
5 fisuppov1.1 . 2 (𝜑𝑍𝑉)
6 funmpt 6605 . . 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 6977 . . . . 5 (𝜑 → (𝐹𝐷) = (𝑥𝐷 ↦ (𝐹𝑥)))
1110oveq1d 7445 . . . 4 (𝜑 → ((𝐹𝐷) supp 0 ) = ((𝑥𝐷 ↦ (𝐹𝑥)) supp 0 ))
129, 1fexd 7246 . . . . 5 (𝜑𝐹 ∈ V)
13 fisuppov1.2 . . . . 5 (𝜑0𝑋)
14 ressuppss 8206 . . . . 5 ((𝐹 ∈ V ∧ 0𝑋) → ((𝐹𝐷) supp 0 ) ⊆ (𝐹 supp 0 ))
1512, 13, 14syl2anc 584 . . . 4 (𝜑 → ((𝐹𝐷) supp 0 ) ⊆ (𝐹 supp 0 ))
1611, 15eqsstrrd 4034 . . 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 8220 . 2 (𝜑 → ((𝑥𝐷 ↦ ((𝐹𝑥)𝑂𝐵)) supp 𝑍) ⊆ (𝐹 supp 0 ))
214, 5, 7, 8, 20fsuppsssuppgd 9419 1 (𝜑 → (𝑥𝐷 ↦ ((𝐹𝑥)𝑂𝐵)) finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  Vcvv 3477  wss 3962   class class class wbr 5147  cmpt 5230  cres 5690  Fun wfun 6556  wf 6558  cfv 6562  (class class class)co 7430   supp csupp 8183   finSupp cfsupp 9398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-supp 8184  df-1o 8504  df-en 8984  df-fin 8987  df-fsupp 9399
This theorem is referenced by:  elrgspnlem1  33231  elrgspnlem2  33232
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