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Mirrors > Home > MPE Home > Th. List > evlsvarsrng | Structured version Visualization version GIF version |
Description: The evaluation of the variable of polynomials over subring yields the same result as evaluated as variable of the polynomials over the ring itself. (Contributed by AV, 12-Sep-2019.) |
Ref | Expression |
---|---|
evlsvarsrng.q | ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) |
evlsvarsrng.o | ⊢ 𝑂 = (𝐼 eval 𝑆) |
evlsvarsrng.v | ⊢ 𝑉 = (𝐼 mVar 𝑈) |
evlsvarsrng.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evlsvarsrng.b | ⊢ 𝐵 = (Base‘𝑆) |
evlsvarsrng.i | ⊢ (𝜑 → 𝐼 ∈ 𝐴) |
evlsvarsrng.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlsvarsrng.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evlsvarsrng.x | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
Ref | Expression |
---|---|
evlsvarsrng | ⊢ (𝜑 → (𝑄‘(𝑉‘𝑋)) = (𝑂‘(𝑉‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlsvarsrng.q | . . 3 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) | |
2 | evlsvarsrng.v | . . 3 ⊢ 𝑉 = (𝐼 mVar 𝑈) | |
3 | evlsvarsrng.u | . . 3 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
4 | evlsvarsrng.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
5 | evlsvarsrng.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐴) | |
6 | evlsvarsrng.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
7 | evlsvarsrng.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
8 | evlsvarsrng.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | evlsvar 21022 | . 2 ⊢ (𝜑 → (𝑄‘(𝑉‘𝑋)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋))) |
10 | evlsvarsrng.o | . . . . . 6 ⊢ 𝑂 = (𝐼 eval 𝑆) | |
11 | 10, 4 | evlval 21027 | . . . . 5 ⊢ 𝑂 = ((𝐼 evalSub 𝑆)‘𝐵) |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑂 = ((𝐼 evalSub 𝑆)‘𝐵)) |
13 | 12 | fveq1d 6708 | . . 3 ⊢ (𝜑 → (𝑂‘(𝑉‘𝑋)) = (((𝐼 evalSub 𝑆)‘𝐵)‘(𝑉‘𝑋))) |
14 | 2 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑉 = (𝐼 mVar 𝑈)) |
15 | eqid 2734 | . . . . . . 7 ⊢ (𝐼 mVar 𝑆) = (𝐼 mVar 𝑆) | |
16 | 15, 5, 7, 3 | subrgmvr 20962 | . . . . . 6 ⊢ (𝜑 → (𝐼 mVar 𝑆) = (𝐼 mVar 𝑈)) |
17 | 4 | ressid 16761 | . . . . . . . . 9 ⊢ (𝑆 ∈ CRing → (𝑆 ↾s 𝐵) = 𝑆) |
18 | 6, 17 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑆 ↾s 𝐵) = 𝑆) |
19 | 18 | eqcomd 2740 | . . . . . . 7 ⊢ (𝜑 → 𝑆 = (𝑆 ↾s 𝐵)) |
20 | 19 | oveq2d 7218 | . . . . . 6 ⊢ (𝜑 → (𝐼 mVar 𝑆) = (𝐼 mVar (𝑆 ↾s 𝐵))) |
21 | 14, 16, 20 | 3eqtr2d 2780 | . . . . 5 ⊢ (𝜑 → 𝑉 = (𝐼 mVar (𝑆 ↾s 𝐵))) |
22 | 21 | fveq1d 6708 | . . . 4 ⊢ (𝜑 → (𝑉‘𝑋) = ((𝐼 mVar (𝑆 ↾s 𝐵))‘𝑋)) |
23 | 22 | fveq2d 6710 | . . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘(𝑉‘𝑋)) = (((𝐼 evalSub 𝑆)‘𝐵)‘((𝐼 mVar (𝑆 ↾s 𝐵))‘𝑋))) |
24 | eqid 2734 | . . . 4 ⊢ ((𝐼 evalSub 𝑆)‘𝐵) = ((𝐼 evalSub 𝑆)‘𝐵) | |
25 | eqid 2734 | . . . 4 ⊢ (𝐼 mVar (𝑆 ↾s 𝐵)) = (𝐼 mVar (𝑆 ↾s 𝐵)) | |
26 | eqid 2734 | . . . 4 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵) | |
27 | crngring 19546 | . . . . 5 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
28 | 4 | subrgid 19774 | . . . . 5 ⊢ (𝑆 ∈ Ring → 𝐵 ∈ (SubRing‘𝑆)) |
29 | 6, 27, 28 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑆)) |
30 | 24, 25, 26, 4, 5, 6, 29, 8 | evlsvar 21022 | . . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘((𝐼 mVar (𝑆 ↾s 𝐵))‘𝑋)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋))) |
31 | 13, 23, 30 | 3eqtrrd 2779 | . 2 ⊢ (𝜑 → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋)) = (𝑂‘(𝑉‘𝑋))) |
32 | 9, 31 | eqtrd 2774 | 1 ⊢ (𝜑 → (𝑄‘(𝑉‘𝑋)) = (𝑂‘(𝑉‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ↦ cmpt 5124 ‘cfv 6369 (class class class)co 7202 ↑m cmap 8497 Basecbs 16684 ↾s cress 16685 Ringcrg 19534 CRingccrg 19535 SubRingcsubrg 19768 mVar cmvr 20836 evalSub ces 21002 eval cevl 21003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-ofr 7459 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-sup 9047 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-fz 13079 df-fzo 13222 df-seq 13558 df-hash 13880 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-hom 16791 df-cco 16792 df-0g 16918 df-gsum 16919 df-prds 16924 df-pws 16926 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-mhm 18190 df-submnd 18191 df-grp 18340 df-minusg 18341 df-sbg 18342 df-mulg 18461 df-subg 18512 df-ghm 18592 df-cntz 18683 df-cmn 19144 df-abl 19145 df-mgp 19477 df-ur 19489 df-srg 19493 df-ring 19536 df-cring 19537 df-rnghom 19707 df-subrg 19770 df-lmod 19873 df-lss 19941 df-lsp 19981 df-assa 20787 df-asp 20788 df-ascl 20789 df-psr 20840 df-mvr 20841 df-mpl 20842 df-evls 21004 df-evl 21005 |
This theorem is referenced by: evlvar 21032 evls1var 21226 |
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