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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 2623 | . . . 4 ⊢ (⊤ → (Base‘𝑃) = (Base‘𝑃)) | |
3 | ply1mpl1.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | eqid 2622 | . . . . . . 7 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
5 | eqid 2622 | . . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
6 | 3, 4, 5 | ply1bas 19565 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘(1𝑜 mPoly 𝑅)) |
7 | ply1mpl1.m | . . . . . . 7 ⊢ 𝑀 = (1𝑜 mPoly 𝑅) | |
8 | 7 | fveq2i 6194 | . . . . . 6 ⊢ (Base‘𝑀) = (Base‘(1𝑜 mPoly 𝑅)) |
9 | 6, 8 | eqtr4i 2647 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑀) |
10 | 9 | a1i 11 | . . . 4 ⊢ (⊤ → (Base‘𝑃) = (Base‘𝑀)) |
11 | eqid 2622 | . . . . . . 7 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
12 | 3, 7, 11 | ply1mulr 19597 | . . . . . 6 ⊢ (.r‘𝑃) = (.r‘𝑀) |
13 | 12 | a1i 11 | . . . . 5 ⊢ (⊤ → (.r‘𝑃) = (.r‘𝑀)) |
14 | 13 | oveqdr 6674 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(.r‘𝑃)𝑦) = (𝑥(.r‘𝑀)𝑦)) |
15 | 2, 10, 14 | rngidpropd 18695 | . . 3 ⊢ (⊤ → (1r‘𝑃) = (1r‘𝑀)) |
16 | 15 | trud 1493 | . 2 ⊢ (1r‘𝑃) = (1r‘𝑀) |
17 | 1, 16 | eqtri 2644 | 1 ⊢ 1 = (1r‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 1𝑜c1o 7553 Basecbs 15857 .rcmulr 15942 1rcur 18501 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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-dec 11494 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-ple 15961 df-0g 16102 df-mgp 18490 df-ur 18502 df-psr 19356 df-mpl 19358 df-opsr 19360 df-psr1 19550 df-ply1 19552 |
This theorem is referenced by: ply1ascl 19628 ply1nzb 23882 |
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