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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 6103 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = (PwSer1‘𝑅)) | |
3 | ply1val.2 | . . . . . 6 ⊢ 𝑆 = (PwSer1‘𝑅) | |
4 | 2, 3 | syl6eqr 2662 | . . . . 5 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = 𝑆) |
5 | oveq2 6557 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (1𝑜 mPoly 𝑟) = (1𝑜 mPoly 𝑅)) | |
6 | 5 | fveq2d 6107 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘(1𝑜 mPoly 𝑟)) = (Base‘(1𝑜 mPoly 𝑅))) |
7 | 4, 6 | oveq12d 6567 | . . . 4 ⊢ (𝑟 = 𝑅 → ((PwSer1‘𝑟) ↾s (Base‘(1𝑜 mPoly 𝑟))) = (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅)))) |
8 | df-ply1 19373 | . . . 4 ⊢ Poly1 = (𝑟 ∈ V ↦ ((PwSer1‘𝑟) ↾s (Base‘(1𝑜 mPoly 𝑟)))) | |
9 | ovex 6577 | . . . 4 ⊢ (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅))) ∈ V | |
10 | 7, 8, 9 | fvmpt 6191 | . . 3 ⊢ (𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅)))) |
11 | fvprc 6097 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅) | |
12 | ress0 15761 | . . . . 5 ⊢ (∅ ↾s (Base‘(1𝑜 mPoly 𝑅))) = ∅ | |
13 | 11, 12 | syl6eqr 2662 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (∅ ↾s (Base‘(1𝑜 mPoly 𝑅)))) |
14 | fvprc 6097 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅) | |
15 | 3, 14 | syl5eq 2656 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑆 = ∅) |
16 | 15 | oveq1d 6564 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅))) = (∅ ↾s (Base‘(1𝑜 mPoly 𝑅)))) |
17 | 13, 16 | eqtr4d 2647 | . . 3 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅)))) |
18 | 10, 17 | pm2.61i 175 | . 2 ⊢ (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅))) |
19 | 1, 18 | eqtri 2632 | 1 ⊢ 𝑃 = (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 1𝑜c1o 7440 Basecbs 15695 ↾s cress 15696 mPoly cmpl 19174 PwSer1cps1 19366 Poly1cpl1 19368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-slot 15699 df-base 15700 df-ress 15702 df-ply1 19373 |
This theorem is referenced by: ply1bas 19386 ply1crng 19389 ply1assa 19390 ply1bascl 19394 ply1plusg 19416 ply1vsca 19417 ply1mulr 19418 ply1ring 19439 ply1lmod 19443 ply1sca 19444 |
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