Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ply1ascl | Structured version Visualization version GIF version |
Description: The univariate polynomial ring inherits the multivariate ring's scalar function. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by Fan Zheng, 26-Jun-2016.) |
Ref | Expression |
---|---|
ply1ascl.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1ascl.a | ⊢ 𝐴 = (algSc‘𝑃) |
Ref | Expression |
---|---|
ply1ascl | ⊢ 𝐴 = (algSc‘(1𝑜 mPoly 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1ascl.a | . 2 ⊢ 𝐴 = (algSc‘𝑃) | |
2 | eqid 2610 | . . . 4 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
3 | eqid 2610 | . . . 4 ⊢ (Scalar‘(1𝑜 mPoly 𝑅)) = (Scalar‘(1𝑜 mPoly 𝑅)) | |
4 | ply1ascl.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | 4 | ply1sca 19444 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 = (Scalar‘𝑃)) |
6 | 5 | fveq2d 6107 | . . . 4 ⊢ (𝑅 ∈ V → (Base‘𝑅) = (Base‘(Scalar‘𝑃))) |
7 | eqid 2610 | . . . . . 6 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅) | |
8 | 1on 7454 | . . . . . . 7 ⊢ 1𝑜 ∈ On | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ V → 1𝑜 ∈ On) |
10 | id 22 | . . . . . 6 ⊢ (𝑅 ∈ V → 𝑅 ∈ V) | |
11 | 7, 9, 10 | mplsca 19266 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 = (Scalar‘(1𝑜 mPoly 𝑅))) |
12 | 11 | fveq2d 6107 | . . . 4 ⊢ (𝑅 ∈ V → (Base‘𝑅) = (Base‘(Scalar‘(1𝑜 mPoly 𝑅)))) |
13 | eqid 2610 | . . . . . . 7 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
14 | 4, 7, 13 | ply1vsca 19417 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘(1𝑜 mPoly 𝑅)) |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ V → ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘(1𝑜 mPoly 𝑅))) |
16 | 15 | oveqdr 6573 | . . . 4 ⊢ ((𝑅 ∈ V ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ V)) → (𝑥( ·𝑠 ‘𝑃)𝑦) = (𝑥( ·𝑠 ‘(1𝑜 mPoly 𝑅))𝑦)) |
17 | eqid 2610 | . . . . . 6 ⊢ (1r‘𝑃) = (1r‘𝑃) | |
18 | 7, 4, 17 | ply1mpl1 19448 | . . . . 5 ⊢ (1r‘𝑃) = (1r‘(1𝑜 mPoly 𝑅)) |
19 | 18 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ V → (1r‘𝑃) = (1r‘(1𝑜 mPoly 𝑅))) |
20 | fvex 6113 | . . . . 5 ⊢ (1r‘𝑃) ∈ V | |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ V → (1r‘𝑃) ∈ V) |
22 | 2, 3, 6, 12, 16, 19, 21 | asclpropd 19167 | . . 3 ⊢ (𝑅 ∈ V → (algSc‘𝑃) = (algSc‘(1𝑜 mPoly 𝑅))) |
23 | fvprc 6097 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅) | |
24 | 4, 23 | syl5eq 2656 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑃 = ∅) |
25 | reldmmpl 19248 | . . . . . 6 ⊢ Rel dom mPoly | |
26 | 25 | ovprc2 6583 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (1𝑜 mPoly 𝑅) = ∅) |
27 | 24, 26 | eqtr4d 2647 | . . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑃 = (1𝑜 mPoly 𝑅)) |
28 | 27 | fveq2d 6107 | . . 3 ⊢ (¬ 𝑅 ∈ V → (algSc‘𝑃) = (algSc‘(1𝑜 mPoly 𝑅))) |
29 | 22, 28 | pm2.61i 175 | . 2 ⊢ (algSc‘𝑃) = (algSc‘(1𝑜 mPoly 𝑅)) |
30 | 1, 29 | eqtri 2632 | 1 ⊢ 𝐴 = (algSc‘(1𝑜 mPoly 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 Oncon0 5640 ‘cfv 5804 (class class class)co 6549 1𝑜c1o 7440 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 1rcur 18324 algSccascl 19132 mPoly cmpl 19174 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-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-tset 15787 df-ple 15788 df-0g 15925 df-mgp 18313 df-ur 18325 df-ascl 19135 df-psr 19177 df-mpl 19179 df-opsr 19181 df-psr1 19371 df-ply1 19373 |
This theorem is referenced by: subrg1ascl 19450 subrg1asclcl 19451 evls1sca 19509 evl1sca 19519 pf1ind 19540 deg1le0 23675 |
Copyright terms: Public domain | W3C validator |