Theorem prjcrvval 41952

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 6885	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝐸‘𝑓) = (𝐸‘𝐹))
2	1	imaeq1d 6052	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝐸‘𝑓) “ 𝑝) = ((𝐸‘𝐹) “ 𝑝))
3	2	eqeq1d 2728	. . 3 ⊢ (𝑓 = 𝐹 → (((𝐸‘𝑓) “ 𝑝) = { 0 } ↔ ((𝐸‘𝐹) “ 𝑝) = { 0 }))
4	3	rabbidv 3434	. 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 41951	. 2 ⊢ (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))
12		prjcrvval.f	. 2 ⊢ (𝜑 → 𝐹 ∈ ∪ ran 𝐻)
13	7	ovexi 7439	. . . 4 ⊢ 𝑃 ∈ V
14	13	rabex 5325	. . 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 41614	1 ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘𝐹) = {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }})

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  {crab 3426  Vcvv 3468  {csn 4623  ∪ cuni 4902  ran crn 5670   “ cima 5672  ‘cfv 6537  (class class class)co 7405  0cc0 11112  ℕ₀cn0 12476  ...cfz 13490  0_gc0g 17394  Fieldcfield 20588   eval cevl 21976   mHomP cmhp 22014  ℙ𝕣𝕠𝕛_ncprjspn 41934  ℙ𝕣𝕠𝕛Crvcprjcrv 41949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-prjcrv 41950

This theorem is referenced by:  prjcrv0  41953