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Mirrors > Home > MPE Home > Th. List > ply1val | Structured version Visualization version GIF version |
Description: The value of the set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.) |
Ref | Expression |
---|---|
ply1val.1 | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1val.2 | ⊢ 𝑆 = (PwSer1‘𝑅) |
Ref | Expression |
---|---|
ply1val | ⊢ 𝑃 = (𝑆 ↾s (Base‘(1o mPoly 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1val.1 | . 2 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | fveq2 6665 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = (PwSer1‘𝑅)) | |
3 | ply1val.2 | . . . . . 6 ⊢ 𝑆 = (PwSer1‘𝑅) | |
4 | 2, 3 | syl6eqr 2874 | . . . . 5 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = 𝑆) |
5 | oveq2 7158 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (1o mPoly 𝑟) = (1o mPoly 𝑅)) | |
6 | 5 | fveq2d 6669 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘(1o mPoly 𝑟)) = (Base‘(1o mPoly 𝑅))) |
7 | 4, 6 | oveq12d 7168 | . . . 4 ⊢ (𝑟 = 𝑅 → ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟))) = (𝑆 ↾s (Base‘(1o mPoly 𝑅)))) |
8 | df-ply1 20344 | . . . 4 ⊢ Poly1 = (𝑟 ∈ V ↦ ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟)))) | |
9 | ovex 7183 | . . . 4 ⊢ (𝑆 ↾s (Base‘(1o mPoly 𝑅))) ∈ V | |
10 | 7, 8, 9 | fvmpt 6763 | . . 3 ⊢ (𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1o mPoly 𝑅)))) |
11 | fvprc 6658 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅) | |
12 | ress0 16552 | . . . . 5 ⊢ (∅ ↾s (Base‘(1o mPoly 𝑅))) = ∅ | |
13 | 11, 12 | syl6eqr 2874 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (∅ ↾s (Base‘(1o mPoly 𝑅)))) |
14 | fvprc 6658 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅) | |
15 | 3, 14 | syl5eq 2868 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑆 = ∅) |
16 | 15 | oveq1d 7165 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑆 ↾s (Base‘(1o mPoly 𝑅))) = (∅ ↾s (Base‘(1o mPoly 𝑅)))) |
17 | 13, 16 | eqtr4d 2859 | . . 3 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1o mPoly 𝑅)))) |
18 | 10, 17 | pm2.61i 184 | . 2 ⊢ (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1o mPoly 𝑅))) |
19 | 1, 18 | eqtri 2844 | 1 ⊢ 𝑃 = (𝑆 ↾s (Base‘(1o mPoly 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∅c0 4291 ‘cfv 6350 (class class class)co 7150 1oc1o 8089 Basecbs 16477 ↾s cress 16478 mPoly cmpl 20127 PwSer1cps1 20337 Poly1cpl1 20339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-slot 16481 df-base 16483 df-ress 16485 df-ply1 20344 |
This theorem is referenced by: ply1bas 20357 ply1crng 20360 ply1assa 20361 ply1bascl 20365 ply1plusg 20387 ply1vsca 20388 ply1mulr 20389 ply1ring 20410 ply1lmod 20414 ply1sca 20415 |
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