Theorem prjcrvval 42593

Description: Value of the projective curve function. (Contributed by SN, 23-Nov-2024.)

Hypotheses

Ref	Expression
prjcrvfval.h	⊢ 𝐻 = ((0...𝑁) mHomP 𝐾)
prjcrvfval.e	⊢ 𝐸 = ((0...𝑁) eval 𝐾)
prjcrvfval.p	⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)
prjcrvfval.0	⊢ 0 = (0g‘𝐾)
prjcrvfval.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
prjcrvfval.k	⊢ (𝜑 → 𝐾 ∈ Field)
prjcrvval.f	⊢ (𝜑 → 𝐹 ∈ ∪ ran 𝐻)

Assertion

Ref	Expression
prjcrvval	⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘𝐹) = {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }})

Distinct variable groups:   𝑁,𝑝   𝐾,𝑝   𝑃,𝑝   𝐹,𝑝

Allowed substitution hints:   𝜑(𝑝)   𝐸(𝑝)   𝐻(𝑝)   0 (𝑝)

Proof of Theorem prjcrvval

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6840	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝐸‘𝑓) = (𝐸‘𝐹))
2	1	imaeq1d 6019	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝐸‘𝑓) “ 𝑝) = ((𝐸‘𝐹) “ 𝑝))
3	2	eqeq1d 2731	. . 3 ⊢ (𝑓 = 𝐹 → (((𝐸‘𝑓) “ 𝑝) = { 0 } ↔ ((𝐸‘𝐹) “ 𝑝) = { 0 }))
4	3	rabbidv 3410	. 2 ⊢ (𝑓 = 𝐹 → {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }} = {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }})
5		prjcrvfval.h	. . 3 ⊢ 𝐻 = ((0...𝑁) mHomP 𝐾)
6		prjcrvfval.e	. . 3 ⊢ 𝐸 = ((0...𝑁) eval 𝐾)
7		prjcrvfval.p	. . 3 ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)
8		prjcrvfval.0	. . 3 ⊢ 0 = (0g‘𝐾)
9		prjcrvfval.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
10		prjcrvfval.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ Field)
11	5, 6, 7, 8, 9, 10	prjcrvfval 42592	. 2 ⊢ (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))
12		prjcrvval.f	. 2 ⊢ (𝜑 → 𝐹 ∈ ∪ ran 𝐻)
13	7	ovexi 7403	. . . 4 ⊢ 𝑃 ∈ V
14	13	rabex 5289	. . 3 ⊢ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }} ∈ V
15	14	a1i 11	. 2 ⊢ (𝜑 → {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }} ∈ V)
16	4, 11, 12, 15	fvmptd4 6974	1 ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘𝐹) = {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3402  Vcvv 3444  {csn 4585  ∪ cuni 4867  ran crn 5632   “ cima 5634  ‘cfv 6499  (class class class)co 7369  0cc0 11044  ℕ₀cn0 12418  ...cfz 13444  0_gc0g 17378  Fieldcfield 20615   eval cevl 21956   mHomP cmhp 21992  ℙ𝕣𝕠𝕛_ncprjspn 42575  ℙ𝕣𝕠𝕛Crvcprjcrv 42590

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  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-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-prjcrv 42591

This theorem is referenced by:  prjcrv0  42594