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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 16561 | . . . . . . . . . . 11 ⊢ (𝑆 ∈ CRing → (𝑆 ↾s 𝐵) = 𝑆) |
6 | 5 | eqcomd 2829 | . . . . . . . . . 10 ⊢ (𝑆 ∈ CRing → 𝑆 = (𝑆 ↾s 𝐵)) |
7 | 3, 6 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (𝑆 ↾s 𝐵)) |
8 | 7 | oveq2d 7174 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 mPoly 𝑆) = (𝐼 mPoly (𝑆 ↾s 𝐵))) |
9 | 2, 8 | syl5eq 2870 | . . . . . . 7 ⊢ (𝜑 → 𝑃 = (𝐼 mPoly (𝑆 ↾s 𝐵))) |
10 | 9 | fveq2d 6676 | . . . . . 6 ⊢ (𝜑 → (algSc‘𝑃) = (algSc‘(𝐼 mPoly (𝑆 ↾s 𝐵)))) |
11 | 1, 10 | syl5eq 2870 | . . . . 5 ⊢ (𝜑 → 𝐶 = (algSc‘(𝐼 mPoly (𝑆 ↾s 𝐵)))) |
12 | 11 | fveq1d 6674 | . . . 4 ⊢ (𝜑 → (𝐶‘𝑋) = ((algSc‘(𝐼 mPoly (𝑆 ↾s 𝐵)))‘𝑋)) |
13 | 12 | fveq2d 6676 | . . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘(𝐶‘𝑋)) = (((𝐼 evalSub 𝑆)‘𝐵)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝐵)))‘𝑋))) |
14 | eqid 2823 | . . . 4 ⊢ ((𝐼 evalSub 𝑆)‘𝐵) = ((𝐼 evalSub 𝑆)‘𝐵) | |
15 | eqid 2823 | . . . 4 ⊢ (𝐼 mPoly (𝑆 ↾s 𝐵)) = (𝐼 mPoly (𝑆 ↾s 𝐵)) | |
16 | eqid 2823 | . . . 4 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵) | |
17 | eqid 2823 | . . . 4 ⊢ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝐵))) = (algSc‘(𝐼 mPoly (𝑆 ↾s 𝐵))) | |
18 | evlsscasrng.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
19 | crngring 19310 | . . . . 5 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
20 | 4 | subrgid 19539 | . . . . 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 19538 | . . . . . 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 3970 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
27 | 14, 15, 16, 4, 17, 18, 3, 21, 26 | evlssca 20304 | . . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝐵)))‘𝑋)) = ((𝐵 ↑m 𝐼) × {𝑋})) |
28 | 13, 27 | eqtrd 2858 | . 2 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘(𝐶‘𝑋)) = ((𝐵 ↑m 𝐼) × {𝑋})) |
29 | evlsscasrng.o | . . . . 5 ⊢ 𝑂 = (𝐼 eval 𝑆) | |
30 | 29, 4 | evlval 20310 | . . . 4 ⊢ 𝑂 = ((𝐼 evalSub 𝑆)‘𝐵) |
31 | 30 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑂 = ((𝐼 evalSub 𝑆)‘𝐵)) |
32 | 31 | fveq1d 6674 | . 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 20304 | . 2 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐵 ↑m 𝐼) × {𝑋})) |
38 | 28, 32, 37 | 3eqtr4rd 2869 | 1 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝑂‘(𝐶‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 {csn 4569 × cxp 5555 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 Basecbs 16485 ↾s cress 16486 Ringcrg 19299 CRingccrg 19300 SubRingcsubrg 19533 algSccascl 20086 mPoly cmpl 20135 evalSub ces 20286 eval cevl 20287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-hom 16591 df-cco 16592 df-0g 16717 df-gsum 16718 df-prds 16723 df-pws 16725 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-srg 19258 df-ring 19301 df-cring 19302 df-rnghom 19469 df-subrg 19535 df-lmod 19638 df-lss 19706 df-lsp 19746 df-assa 20087 df-asp 20088 df-ascl 20089 df-psr 20138 df-mvr 20139 df-mpl 20140 df-evls 20288 df-evl 20289 |
This theorem is referenced by: evlsca 20313 |
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