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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 6191 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = (PwSer1‘𝑅)) | |
3 | ply1val.2 | . . . . . 6 ⊢ 𝑆 = (PwSer1‘𝑅) | |
4 | 2, 3 | syl6eqr 2674 | . . . . 5 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = 𝑆) |
5 | oveq2 6658 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (1𝑜 mPoly 𝑟) = (1𝑜 mPoly 𝑅)) | |
6 | 5 | fveq2d 6195 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘(1𝑜 mPoly 𝑟)) = (Base‘(1𝑜 mPoly 𝑅))) |
7 | 4, 6 | oveq12d 6668 | . . . 4 ⊢ (𝑟 = 𝑅 → ((PwSer1‘𝑟) ↾s (Base‘(1𝑜 mPoly 𝑟))) = (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅)))) |
8 | df-ply1 19552 | . . . 4 ⊢ Poly1 = (𝑟 ∈ V ↦ ((PwSer1‘𝑟) ↾s (Base‘(1𝑜 mPoly 𝑟)))) | |
9 | ovex 6678 | . . . 4 ⊢ (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅))) ∈ V | |
10 | 7, 8, 9 | fvmpt 6282 | . . 3 ⊢ (𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅)))) |
11 | fvprc 6185 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅) | |
12 | ress0 15934 | . . . . 5 ⊢ (∅ ↾s (Base‘(1𝑜 mPoly 𝑅))) = ∅ | |
13 | 11, 12 | syl6eqr 2674 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (∅ ↾s (Base‘(1𝑜 mPoly 𝑅)))) |
14 | fvprc 6185 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅) | |
15 | 3, 14 | syl5eq 2668 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑆 = ∅) |
16 | 15 | oveq1d 6665 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅))) = (∅ ↾s (Base‘(1𝑜 mPoly 𝑅)))) |
17 | 13, 16 | eqtr4d 2659 | . . 3 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅)))) |
18 | 10, 17 | pm2.61i 176 | . 2 ⊢ (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅))) |
19 | 1, 18 | eqtri 2644 | 1 ⊢ 𝑃 = (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 ‘cfv 5888 (class class class)co 6650 1𝑜c1o 7553 Basecbs 15857 ↾s cress 15858 mPoly cmpl 19353 PwSer1cps1 19545 Poly1cpl1 19547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-slot 15861 df-base 15863 df-ress 15865 df-ply1 19552 |
This theorem is referenced by: ply1bas 19565 ply1crng 19568 ply1assa 19569 ply1bascl 19573 ply1plusg 19595 ply1vsca 19596 ply1mulr 19597 ply1ring 19618 ply1lmod 19622 ply1sca 19623 |
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