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| Mirrors > Home > MPE Home > Th. List > Mathboxes > prjcrvval | Structured version Visualization version GIF version | ||
| Description: Value of the projective curve function. (Contributed by SN, 23-Nov-2024.) |
| 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) |
| Ref | Expression |
|---|---|
| prjcrvval | ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘𝐹) = {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveq2 6906 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝐸‘𝑓) = (𝐸‘𝐹)) | |
| 2 | 1 | imaeq1d 6077 | . . . 4 ⊢ (𝑓 = 𝐹 → ((𝐸‘𝑓) “ 𝑝) = ((𝐸‘𝐹) “ 𝑝)) |
| 3 | 2 | eqeq1d 2739 | . . 3 ⊢ (𝑓 = 𝐹 → (((𝐸‘𝑓) “ 𝑝) = { 0 } ↔ ((𝐸‘𝐹) “ 𝑝) = { 0 })) |
| 4 | 3 | rabbidv 3444 | . 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 42641 | . 2 ⊢ (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }})) |
| 12 | prjcrvval.f | . 2 ⊢ (𝜑 → 𝐹 ∈ ∪ ran 𝐻) | |
| 13 | 7 | ovexi 7465 | . . . 4 ⊢ 𝑃 ∈ V |
| 14 | 13 | rabex 5339 | . . 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 7040 | 1 ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘𝐹) = {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 {csn 4626 ∪ cuni 4907 ran crn 5686 “ cima 5688 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ℕ0cn0 12526 ...cfz 13547 0gc0g 17484 Fieldcfield 20730 eval cevl 22097 mHomP cmhp 22133 ℙ𝕣𝕠𝕛ncprjspn 42624 ℙ𝕣𝕠𝕛Crvcprjcrv 42639 |
| 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-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-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-id 5578 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-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-prjcrv 42640 |
| This theorem is referenced by: prjcrv0 42643 |
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