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Mirrors > Home > MPE Home > Th. List > evlsscasrng | Structured version Visualization version GIF version |
Description: The evaluation of a scalar of a subring yields the same result as evaluated as a scalar over the ring itself. (Contributed by AV, 12-Sep-2019.) |
Ref | Expression |
---|---|
evlsscasrng.q | ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) |
evlsscasrng.o | ⊢ 𝑂 = (𝐼 eval 𝑆) |
evlsscasrng.w | ⊢ 𝑊 = (𝐼 mPoly 𝑈) |
evlsscasrng.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evlsscasrng.p | ⊢ 𝑃 = (𝐼 mPoly 𝑆) |
evlsscasrng.b | ⊢ 𝐵 = (Base‘𝑆) |
evlsscasrng.a | ⊢ 𝐴 = (algSc‘𝑊) |
evlsscasrng.c | ⊢ 𝐶 = (algSc‘𝑃) |
evlsscasrng.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
evlsscasrng.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlsscasrng.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evlsscasrng.x | ⊢ (𝜑 → 𝑋 ∈ 𝑅) |
Ref | Expression |
---|---|
evlsscasrng | ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝑂‘(𝐶‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlsscasrng.c | . . . . . 6 ⊢ 𝐶 = (algSc‘𝑃) | |
2 | evlsscasrng.p | . . . . . . . 8 ⊢ 𝑃 = (𝐼 mPoly 𝑆) | |
3 | evlsscasrng.s | . . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
4 | evlsscasrng.b | . . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑆) | |
5 | 4 | ressid 15935 | . . . . . . . . . . 11 ⊢ (𝑆 ∈ CRing → (𝑆 ↾s 𝐵) = 𝑆) |
6 | 5 | eqcomd 2628 | . . . . . . . . . 10 ⊢ (𝑆 ∈ CRing → 𝑆 = (𝑆 ↾s 𝐵)) |
7 | 3, 6 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (𝑆 ↾s 𝐵)) |
8 | 7 | oveq2d 6666 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 mPoly 𝑆) = (𝐼 mPoly (𝑆 ↾s 𝐵))) |
9 | 2, 8 | syl5eq 2668 | . . . . . . 7 ⊢ (𝜑 → 𝑃 = (𝐼 mPoly (𝑆 ↾s 𝐵))) |
10 | 9 | fveq2d 6195 | . . . . . 6 ⊢ (𝜑 → (algSc‘𝑃) = (algSc‘(𝐼 mPoly (𝑆 ↾s 𝐵)))) |
11 | 1, 10 | syl5eq 2668 | . . . . 5 ⊢ (𝜑 → 𝐶 = (algSc‘(𝐼 mPoly (𝑆 ↾s 𝐵)))) |
12 | 11 | fveq1d 6193 | . . . 4 ⊢ (𝜑 → (𝐶‘𝑋) = ((algSc‘(𝐼 mPoly (𝑆 ↾s 𝐵)))‘𝑋)) |
13 | 12 | fveq2d 6195 | . . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘(𝐶‘𝑋)) = (((𝐼 evalSub 𝑆)‘𝐵)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝐵)))‘𝑋))) |
14 | eqid 2622 | . . . 4 ⊢ ((𝐼 evalSub 𝑆)‘𝐵) = ((𝐼 evalSub 𝑆)‘𝐵) | |
15 | eqid 2622 | . . . 4 ⊢ (𝐼 mPoly (𝑆 ↾s 𝐵)) = (𝐼 mPoly (𝑆 ↾s 𝐵)) | |
16 | eqid 2622 | . . . 4 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵) | |
17 | eqid 2622 | . . . 4 ⊢ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝐵))) = (algSc‘(𝐼 mPoly (𝑆 ↾s 𝐵))) | |
18 | evlsscasrng.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
19 | crngring 18558 | . . . . 5 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
20 | 4 | subrgid 18782 | . . . . 5 ⊢ (𝑆 ∈ Ring → 𝐵 ∈ (SubRing‘𝑆)) |
21 | 3, 19, 20 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑆)) |
22 | evlsscasrng.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
23 | 4 | subrgss 18781 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵) |
24 | 22, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ⊆ 𝐵) |
25 | evlsscasrng.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑅) | |
26 | 24, 25 | sseldd 3604 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
27 | 14, 15, 16, 4, 17, 18, 3, 21, 26 | evlssca 19522 | . . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝐵)))‘𝑋)) = ((𝐵 ↑𝑚 𝐼) × {𝑋})) |
28 | 13, 27 | eqtrd 2656 | . 2 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘(𝐶‘𝑋)) = ((𝐵 ↑𝑚 𝐼) × {𝑋})) |
29 | evlsscasrng.o | . . . . 5 ⊢ 𝑂 = (𝐼 eval 𝑆) | |
30 | 29, 4 | evlval 19524 | . . . 4 ⊢ 𝑂 = ((𝐼 evalSub 𝑆)‘𝐵) |
31 | 30 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑂 = ((𝐼 evalSub 𝑆)‘𝐵)) |
32 | 31 | fveq1d 6193 | . 2 ⊢ (𝜑 → (𝑂‘(𝐶‘𝑋)) = (((𝐼 evalSub 𝑆)‘𝐵)‘(𝐶‘𝑋))) |
33 | evlsscasrng.q | . . 3 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) | |
34 | evlsscasrng.w | . . 3 ⊢ 𝑊 = (𝐼 mPoly 𝑈) | |
35 | evlsscasrng.u | . . 3 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
36 | evlsscasrng.a | . . 3 ⊢ 𝐴 = (algSc‘𝑊) | |
37 | 33, 34, 35, 4, 36, 18, 3, 22, 25 | evlssca 19522 | . 2 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐵 ↑𝑚 𝐼) × {𝑋})) |
38 | 28, 32, 37 | 3eqtr4rd 2667 | 1 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝑂‘(𝐶‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 {csn 4177 × cxp 5112 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 Basecbs 15857 ↾s cress 15858 Ringcrg 18547 CRingccrg 18548 SubRingcsubrg 18776 algSccascl 19311 mPoly cmpl 19353 evalSub ces 19504 eval cevl 19505 |
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-inf2 8538 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-rmo 2920 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-int 4476 df-iun 4522 df-iin 4523 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-se 5074 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-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 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-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-srg 18506 df-ring 18549 df-cring 18550 df-rnghom 18715 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-assa 19312 df-asp 19313 df-ascl 19314 df-psr 19356 df-mvr 19357 df-mpl 19358 df-evls 19506 df-evl 19507 |
This theorem is referenced by: evlsca 19527 |
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