Theorem fsuppssov1 9297

Description: Formula building theorem for finite support: operator with left annihilator. Finite support version of suppssov1 8147. (Contributed by SN, 26-Apr-2025.)

Hypotheses

Ref	Expression
fsuppssov1.s	⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ 𝐴) finSupp 𝑌)
fsuppssov1.o	⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)
fsuppssov1.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉)
fsuppssov1.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅)
fsuppssov1.z	⊢ (𝜑 → 𝑍 ∈ 𝑊)

Assertion

Ref	Expression
fsuppssov1	⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) finSupp 𝑍)

Distinct variable groups:   𝜑,𝑣   𝜑,𝑥   𝑣,𝐵   𝑥,𝐷   𝑣,𝑂   𝑣,𝑅   𝑣,𝑌   𝑥,𝑌   𝑣,𝑍   𝑥,𝑍   𝑥,𝑉

Allowed substitution hints:   𝐴(𝑥,𝑣)   𝐵(𝑥)   𝐷(𝑣)   𝑅(𝑥)   𝑂(𝑥)   𝑉(𝑣)   𝑊(𝑥,𝑣)

Proof of Theorem fsuppssov1

Step	Hyp	Ref	Expression
1		fsuppssov1.s	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ 𝐴) finSupp 𝑌)
2		relfsupp 9276	. . . . . 6 ⊢ Rel finSupp
3	2	brrelex1i 5687	. . . . 5 ⊢ ((𝑥 ∈ 𝐷 ↦ 𝐴) finSupp 𝑌 → (𝑥 ∈ 𝐷 ↦ 𝐴) ∈ V)
4	1, 3	syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ 𝐴) ∈ V)
5		fsuppssov1.a	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉)
6	5	fmpttd 7067	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ 𝐴):𝐷⟶𝑉)
7		dmfex 7856	. . . 4 ⊢ (((𝑥 ∈ 𝐷 ↦ 𝐴) ∈ V ∧ (𝑥 ∈ 𝐷 ↦ 𝐴):𝐷⟶𝑉) → 𝐷 ∈ V)
8	4, 6, 7	syl2anc 585	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
9	8	mptexd 7179	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
10		fsuppssov1.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
11		funmpt 6536	. . 3 ⊢ Fun (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵))
12	11	a1i 11	. 2 ⊢ (𝜑 → Fun (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)))
13		ssidd 3945	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌) ⊆ ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌))
14		fsuppssov1.o	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)
15		fsuppssov1.b	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅)
16	2	brrelex2i 5688	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ↦ 𝐴) finSupp 𝑌 → 𝑌 ∈ V)
17	1, 16	syl 17	. . 3 ⊢ (𝜑 → 𝑌 ∈ V)
18	13, 14, 5, 15, 17	suppssov1 8147	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌))
19	9, 10, 12, 1, 18	fsuppsssuppgd 9295	1 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) finSupp 𝑍)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429   class class class wbr 5085   ↦ cmpt 5166  Fun wfun 6492  ⟶wf 6494  (class class class)co 7367   supp csupp 8110   finSupp cfsupp 9274

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  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 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-supp 8111  df-1o 8405  df-en 8894  df-fin 8897  df-fsupp 9275

This theorem is referenced by:  elrgspn  33307  elrgspnsubrunlem2  33309  evlextv  33686  mplvrpmrhm  33691  selvvvval  43018  evlselv  43020