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Mirrors > Home > MPE Home > Th. List > ply1bascl | Structured version Visualization version GIF version |
Description: A univariate polynomial is a univariate power series. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
Ref | Expression |
---|---|
ply1bascl.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1bascl.b | ⊢ 𝐵 = (Base‘𝑃) |
Ref | Expression |
---|---|
ply1bascl | ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(PwSer1‘𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1bascl.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
2 | ply1bascl.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | eqid 2733 | . . . . 5 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
4 | 2, 3 | ply1val 21687 | . . . 4 ⊢ 𝑃 = ((PwSer1‘𝑅) ↾s (Base‘(1o mPoly 𝑅))) |
5 | eqid 2733 | . . . 4 ⊢ (Base‘(PwSer1‘𝑅)) = (Base‘(PwSer1‘𝑅)) | |
6 | 4, 5 | ressbasss 17170 | . . 3 ⊢ (Base‘𝑃) ⊆ (Base‘(PwSer1‘𝑅)) |
7 | 1, 6 | eqsstri 4014 | . 2 ⊢ 𝐵 ⊆ (Base‘(PwSer1‘𝑅)) |
8 | 7 | sseli 3976 | 1 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(PwSer1‘𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6535 (class class class)co 7396 1oc1o 8446 Basecbs 17131 mPoly cmpl 21430 PwSer1cps1 21668 Poly1cpl1 21670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-1cn 11155 ax-addcl 11157 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-nn 12200 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-ply1 21675 |
This theorem is referenced by: coe1fval2 21703 coe1f 21704 ply1opprmul 21732 coe1mul 21763 |
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