![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ply1mpl1 | Structured version Visualization version GIF version |
Description: The univariate polynomial ring has the same one as the corresponding multivariate polynomial ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.) |
Ref | Expression |
---|---|
ply1mpl1.m | ⊢ 𝑀 = (1𝑜 mPoly 𝑅) |
ply1mpl1.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1mpl1.o | ⊢ 1 = (1r‘𝑃) |
Ref | Expression |
---|---|
ply1mpl1 | ⊢ 1 = (1r‘𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1mpl1.o | . 2 ⊢ 1 = (1r‘𝑃) | |
2 | eqidd 2611 | . . . 4 ⊢ (⊤ → (Base‘𝑃) = (Base‘𝑃)) | |
3 | ply1mpl1.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | eqid 2610 | . . . . . . 7 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
5 | eqid 2610 | . . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
6 | 3, 4, 5 | ply1bas 19386 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘(1𝑜 mPoly 𝑅)) |
7 | ply1mpl1.m | . . . . . . 7 ⊢ 𝑀 = (1𝑜 mPoly 𝑅) | |
8 | 7 | fveq2i 6106 | . . . . . 6 ⊢ (Base‘𝑀) = (Base‘(1𝑜 mPoly 𝑅)) |
9 | 6, 8 | eqtr4i 2635 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑀) |
10 | 9 | a1i 11 | . . . 4 ⊢ (⊤ → (Base‘𝑃) = (Base‘𝑀)) |
11 | eqid 2610 | . . . . . . 7 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
12 | 3, 7, 11 | ply1mulr 19418 | . . . . . 6 ⊢ (.r‘𝑃) = (.r‘𝑀) |
13 | 12 | a1i 11 | . . . . 5 ⊢ (⊤ → (.r‘𝑃) = (.r‘𝑀)) |
14 | 13 | oveqdr 6573 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(.r‘𝑃)𝑦) = (𝑥(.r‘𝑀)𝑦)) |
15 | 2, 10, 14 | rngidpropd 18518 | . . 3 ⊢ (⊤ → (1r‘𝑃) = (1r‘𝑀)) |
16 | 15 | trud 1484 | . 2 ⊢ (1r‘𝑃) = (1r‘𝑀) |
17 | 1, 16 | eqtri 2632 | 1 ⊢ 1 = (1r‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 1𝑜c1o 7440 Basecbs 15695 .rcmulr 15769 1rcur 18324 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-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-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-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 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-dec 11370 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-ple 15788 df-0g 15925 df-mgp 18313 df-ur 18325 df-psr 19177 df-mpl 19179 df-opsr 19181 df-psr1 19371 df-ply1 19373 |
This theorem is referenced by: ply1ascl 19449 ply1nzb 23686 |
Copyright terms: Public domain | W3C validator |